fractal geometry and dynamical systems in pure and applied

601

Fractal Geometry and Dynamical Systemsin Pure and Applied Mathematics II:

Fractals in Applied MathematicsPISRS 2011 International Conference on Analysis, Fractal Geometry,

Dynamical Systems and EconomicsNovember 2011: Messina, Sicily, Italy

AMS Special Session on Fractal Geometry in Pure and Applied Mathematics:in Memory of Benoît Mandelbrot

January 2012: Boston, Massachusetts

AMS Special Session on Geometry and Analysis on Fractal SpacesMarch 2012: Honolulu, Hawaii

David CarfìMichel L. Lapidus

Erin P. J. PearseMachiel van Frankenhuijsen

Editors

American Mathematical Society









Editors

601









Editors

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Dennis DeTurck, Managing Editor

Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classification. Primary 28A80, 37D50, 37F10, 58J35, 58J50,58J65, 60J45, 60K35, 81Q35, 91G80.

Library of Congress Cataloging-in-Publication Data

Fractal geometry and dynamical systems in pure and applied mathematics / David Carfı,Michel L. Lapidus, Erin P. J. Pearse, Machiel van Frankenhuijsen, editors.

volumes cm. – (Contemporary mathematics; volumes 600, 601)PISRS 2011, First International Conference: Analysis, Fractal Geometry, Dynamical Systems andEconomics, November 8–12, 2011, Messina, Sicily, Italy.AMS Special Session, in memory of Benoıt Mandelbrot: Fractal Geometry in Pure and AppliedMathematics, January 4–7, 2012, Boston, MA.AMS Special Session: Geometry and Analysis on Fractal Spaces, March 3–4, 2012, Honolulu, HI.Includes bibliographical references.

ISBN 978-0-8218-9147-6 (alk. paper : v. I) – ISBN 978-0-8218-9148-3 (alk. paper : v. II)1. Fractals–Congresses. I. Carfı, David, 1971– II. Lapidus, Michel L. (Michel Laurent), 1956–III. Pearse, Erin P. J., 1975– IV. Frankenhuijsen, Machiel van, 1967– V. Mandelbrot, Benoıt B.

QC20.7.F73F715 2013 2013013894514′.742–dc23

Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/601

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Contents

Preface vii

Statistical Mechanics and Quantum Fields on FractalsEric Akkermans 1

Spectral Algebra of the Chernov and Bogoslovsky Finsler Metric TensorsVladimir Balan 23

Local Multifractal AnalysisJulien Barral, Arnaud Durand, Stephane Jaffard,

and Stephane Seuret 31

Extreme Risk and Fractal Regularity in FinanceLaurent E. Calvet and Adlai J. Fisher 65

An Algorithm for Dynamical Games with Fractal-Like TrajectoriesDavid Carfı and Angela Ricciardello 95

The Landscape of Anderson Localization in a Disordered MediumMarcel Filoche and Svitlana Mayboroda 113

Zeta Functions for Infinite Graphs and Functional EquationsDaniele Guido and Tommaso Isola 123

Vector Analysis on Fractals and ApplicationsMichael Hinz and Alexander Teplyaev 147

Non-Regularly Varying and Non-Periodic Oscillation of the On-DiagonalHeat Kernels on Self-Similar Fractals

Naotaka Kajino 165

Lattice Effects in the Scaling Limit of the Two-DimensionalSelf-Avoiding Walk

Tom Kennedy and Gregory F. Lawler 195

The Casimir Effect on Laakso SpacesRobert Kesler and Benjamin Steinhurst 211

The Decimation Method for Laplacians on Fractals: Spectra and ComplexDynamics

Nishu Lal and Michel L. Lapidus 227

The Current State of Fractal BilliardsMichel L. Lapidus and Robert G. Niemeyer 251

v

vi CONTENTS

Long-Range Dependence and the Rank of DecompositionsCeline Levy-Leduc and Murad S. Taqqu 289

Hitting Probabilities of the Random Covering SetsBing Li, Narn-Rueih Shieh, and Yimin Xiao 307

Fractal Oscillations Near the Domain Boundary of Radially SymmetricSolutions of p-Laplace Equations

Yuki Naito, Mervan Pasic, Satoshi Tanaka,

and Darko Zubrinic 325

Applications of the Contraction Mapping PrincipleJohn R. Quinn 345

Economics and Psychology. Perfect Rationality versus Bounded RationalityDaniele Schiliro 359

Preface

The Contemporary Mathematics volume

Fractal Geometry and Dynamical Systems in Pure and AppliedMathematics II: Fractals in Applied Mathematics

contains papers from talks given at three conferences held in 2011–2012, followingthe passing of Benoıt Mandelbrot (widely regarded as the father of fractal geometry)in October of 2010. These meetings are described in chronological order below.

On the occasion of the 2011 Anassilaos International Research Prize in Math-ematics, awarded to Michel L. Lapidus (University of California, Riverside), thePermanent International Session of Research Seminars (PISRS) held its first Inter-national Meeting

PISRS 2011: Analysis, Fractal Geometry, Dynamical Systemsand Economics.

The conference was held on November 8–12, 2011, at the University of Messina inSicily, Italy, and was attended by experts in the fields of Fractal Geometry, Dy-namical Systems, Number Theory, Noncommutative Geometry, Mathematical andTheoretical Physics, as well as Economics. In addition to approximately 40 expe-rienced researchers participating, the conference included more than 150 students,professors and experts following and attending the meeting. The Award Cere-mony for Michel Lapidus took place in Reggio Calabria on Saturday, November 12.The Scientific Committee of PISRS includes over 50 professors and scholars frommore than 25 outstanding universities around the world. It has several branches,including Applied Functional Analysis; Biomathematics; Decision and Game The-ory; Differential, Fractal and Noncommutative Geometry; Mathematical Methodsof Economics, Finance and Quantum Mechanics; Mathematical Physics and Dy-namical Systems. The Chairman of PISRS is David Carfı.

The 2012 AMS/MAA/SIAM Joint Mathematics National Meeting, held inBoston in January 2012, included an AMS Special Session on “Fractal Geometry inPure and Applied Mathematics” in memory of Benoıt Mandelbrot. Its organizerswere Michel Lapidus, Erin Pearse and Machiel van Frankenhuijsen. In five ses-sions (including sessions comprised of primarily applied topics), researchers fromaround the world presented their work in various areas of fractal mathematics.An entire session was devoted to the applications to Physics, Biology, Engineeringand Computer Science. During one of the breaks, an experiment was performedwhich demonstrated the capabilities of fractal antennas. Many speakers describedways in which their work was influenced by the work of Benoıt Mandelbrot, and a

vii

viii PREFACE

special dinner was organized in his honor. Several talks were attended by AlietteMandelbrot, Benoıt’s widow, who also gave a short but touching speech.

The Spring 2012 Meeting of the AMS Western Section, held in Honolulu,Hawaii, at the University of Hawaii at Manoa, included a Special Session on “Ge-ometry and Analysis on Fractal Spaces”. Its organizers were Michel Lapidus, Lu’Hung, John Rock and Machiel van Frankenhuijsen. In four sessions, researchersfrom around the world presented their work in various areas of fractal mathemat-ics.

This is a collection of papers on fractal geometry and dynamical systems inapplied mathematics and the applications to other sciences. It features articlesdiscussing a variety of connections between these subjects and other fields of sci-ence, including physics, engineering, computer science, technology, economics andfinance, as well as of mathematics (including probability theory in relation withstatistical physics and heat kernel estimates, geometric measure theory, partial dif-ferential equations in relation with condensed matter physics, global analysis onnonsmooth spaces, the theory of billiards, harmonic analysis and spectral theory).

These proceedings were conceived as a means of collecting some of the mostrecent developments in this active area of research, and also to bring together severalsurvey and research expository articles, as a means of introducing new researchersand graduate students to the forefront of the field. The present volume focuses onthe more applied aspects of the field, including the applications of fractal geometryand dynamical systems to other sciences. Its companion volume, entitled FractalGeometry and Dynamical Systems in Pure and Applied Mathematics I and subtitledFractals in Pure Mathematics, focuses on the more mathematical aspects of fractalgeometry and dynamical systems.

David Carfı,Michel L. Lapidus,Erin P. J. Pearse, andMachiel van Frankenhuijsen.

March 2013

Acknowledgements: The editors wish to acknowledge the support of the Na-tional Science Foundation (via M. L. Lapidus’ NSF grants DMS-0707524 and DMS-1107750) towards the preparation of these proceedings and especially towards thetravel and/or stay of several of the participants in the three conferences that gaverise to these proceedings.

Contemporary MathematicsVolume 601, 2013http://dx.doi.org/10.1090/conm/601/11962

Statistical Mechanics and Quantum Fields on Fractals

Eric Akkermans

This paper is dedicated to the memory of B. Mandelbrot.

Abstract. Fractals define a new and interesting realm for a discussion of ba-sic phenomena in quantum field theory and statistical mechanics. This interestresults from specific properties of fractals, e.g., their dilatation symmetry andthe corresponding absence of Fourier mode decomposition. Moreover, the ex-istence of a set of distinct dimensions characterizing the physical properties(spatial or spectral) of fractals make them a useful testing ground for dimen-sionality dependent physical problems. This paper addresses specific problemsincluding the behavior of the heat kernel and spectral zeta functions on frac-tals and their importance in the expression of spectral properties in quantumphysics. Finally, we apply these results to specific problems such as thermo-dynamics of quantum radiation by a fractal blackbody.

1. Introduction

The interest in the behavior of fractals (a word coined by B. Mandelbrot inthe 1970’s [1] but without well agreed definition) goes back to the study by math-ematicians of strange objects hardly defined by their topology such as the Kochcurve or the Sierpinski gasket. These objects are described by continuous but notdifferentiable functions. At about the same time, probabilists (Levy, Wiener, Doob,Ito, Kolmogorov) and physicists (Smoluchowski, Einstein, Perrin and Langevin toname a few) have been working to give a basis to the theory of brownian motion,yet another example of fractal object. More recently, physicists have recognizedthe ubiquitous character of fractals in almost all field of physics, including complexcondensed matter ([2]-[8]), phase transitions [9,10], turbulence [11], quantum fieldtheory ([12]-[16]) and aspects of stochastic processes [7,17,18]. An important ef-fort span over more than two decades led to new ideas and concepts to characterizefractals. Notions of self-similarity, iterative maps, fixed points and the identifica-tion of distinct fractal dimensions to characterize basic physical properties havebeen instrumental in the understanding of these objects.

Since the early 1980’s, mathematicians have opened new important directionsby being able to define properly brownian motion on some classes of fractals ([19]-[21]) and Laplacian operators on these structures [23, 24]. Progress along these

2010 Mathematics Subject Classification. Primary 81Q35, 28A80, 60G18, 82B10, 37F25,37F10.

Key words and phrases. Statistical mechanics, Quantum field theory, Fractals.The author was supported in part by the Israel Science Foundation Grant No.924/09.

c©2013 American Mathematical Society

1

2 ERIC AKKERMANS

two directions have led to a vast literature and it would be a hopeless task to listit exhaustively. Most of these results have been summarized in textbooks ([24]-[26]) and reviews [27]. These progresses prove to be instrumental in physics sincethey allow to go beyond phenomenological scaling relations towards a quantitativeanalysis of fractal structures.

A useful description in field theory is provided by path integrals and moregenerally functional integrals [28]. Eventually, their evaluation, within one-loopapproximation, boils down to the calculation of determinants of operators (e.g.the Laplacian) and their expression in terms of spectral functions among which themost useful and popular are the heat kernel and corresponding zeta functions ([30]-[32]). Their evaluation on Euclidean manifolds reveal instrumental in quantumfield theory [46], in statistical mechanics and the theory of phase transitions toname a few [29]. The new tools provided by mathematicians allow to extend theseapproaches to fractal structures ([33]-[35]).

It is the purpose of this paper to give an account of some of these new results.We shall present examples in the realm of wave and heat propagation on fractals,quantum mechanics, quantum field theory and statistical mechanics.

Why is it interesting to study physical phenomena on fractal structures? Frac-tals define a useful testing ground for dimensionality dependent physical phenom-ena. Indeed, many physical phenomena reveal being critically dependent uponspace dimensionality. Relevant examples include Anderson localization, Bose-Einstein condensation, onset of superfluidity (Mermin-Wagner-Coleman theorem).On Euclidean manifolds, there is a single space dimensionality so that it is usuallynot possible to identify the meaning of a dependence upon dimensionality. On afractal, as we shall see, there exist distinct dimensions which account for geometric,spectral or stochastic informations. A purpose of this paper is to discuss problemswhere this dependence plays a role. Fractals also provide a new playground for welldesigned new experimental setups [36]. An example is provided by spontaneousemission from atoms embedded in structures whose quantum vacuum has a fractalstructure [37].

2. Discrete scaling symmetry - Self similarity - Definitions

As opposed to Euclidean spaces characterized by translation invariance, self-similar (fractal) structures possess a dilatation symmetry of their physical proper-ties, each characterized by a specific fractal dimension. To illustrate our results, weshall consider throughout this paper simple examples of fractals such as the Siepin-ski gasket, families of diamond fractals represented on Fig.1 or Cantor sets, but thereader should keep in mind that our results apply to a broader class of self-similardeterministic fractals. Much less is known about other related but more complexsystems including random fractals (e.g. critical percolation clusters), multifractals,T-graphs, trees or more generally objects which do not exhibit an exact decimationsymmetry like in deterministic fractals considered here [38].

A fractal is an iterative structure. Let us give a simple example of dimensionalcharacterization. To that purpose, consider a triadic Cantor set obtained iterativelyby removing from a given initial interval of length L0 the middle third part. Definea uniform mass density on this interval so that the initial mass is M0 for a lengthL0. After the first iteration, the mass is 2M0 while the length is 3L0. After the n-thiteration, the mass is Mn = 2nM0 while the length becomes Ln = 3nL0, so that the

STATISTICAL MECHANICS AND QUANTUM FIELDS ON FRACTALS 3

� �

�

�

� �

�

�

�

� �

��

��

Figure 1. First 2 iterations of the diamond fractals D4,2, D6,2

and D6,3. Their respective branching factors (defined in the text)are B = 1, 2, 1.

fractal dimension describing the mass density, the geometric Hausdorff dimensiondh, is given by,

(2.1) dh = limn→∞lnMn

lnLn=

ln 2

ln 3

An alternative way to obtain this result, is to start from the scaling relationbetween masses M(L) at different lengths L, namely M(L) = M0(L) + 1

2M(3L)for the triadic Cantor set, where M0(L) is the initial mass. More generally, we areinterested in the solution of the equation

(2.2) f(x) = g(x) +1

bf(ax)

with determined scaling parameters a and b and a given initial function g(x). Thisform defines a discrete scaling symmetry as opposed to continuous scaling whichextends (2.2) to any rescaling a of the variable x. For the specific case g = 0, asolution of (2.2) can be sought under the form f(x) = xαF (x) for generally nonconstant F (x). Inserting into (2.2), and choosing α = ln b/ ln a, we obtain that Fmust fulfill F (ax) = F (x). Redefining F (x) ≡ G(lnx/ ln a) leads to G(lnx/ ln a +1) = G(lnx/ ln a), namely the general solution of (2.2) is of the form,

(2.3) f(x) = xln b/ ln a G

(lnx

ln a

)where G is a periodic function of its argument of period unity. The scaling form(2.2) can be iterated so that it rewrites

(2.4) f(x) =

∞∑n=0

b−n g(an x) .

This form makes explicit an important feature of discrete scaling symmetry namelythe scaling function f(x) can be written as a series with exponentially growing

4 ERIC AKKERMANS

coefficients b−n and an instead of polynomial growth. This is an essential andubiquitous feature of self-similarity that we shall encounter all along this paper.

2.1. Mellin transform. To study functions involving dilatation scaling sym-metry, it is desirable to find an appropriate transform equivalent to Laplace orFourier transforms for translation symmetry. The Mellin transform plays this role[39]. For a function f(t) defined on the positive real axis 0 < t < ∞, its Mellintransform

(2.5) Mf (s) ≡∫ ∞

0

dt ts−1 f(t)

is defined on the complex plane. The Mellin transform of f(t) is the two sidedLaplace transform of the function g(t) ≡ f (e−t), a property which precisely ac-counts for the dilatation symmetry in (2.2) or the exponential behavior in (2.4).An important property of the Mellin transform can be stated as follows. If f(t) isanalytic in 0 < t < ∞ and f(t) = O(t−α) for t → 0 and f(t) = O(t−β) for t → ∞,with α < β, then the Mellin transform Mf (s) is analytic in the strip α < s < β, and

f(t) = (1/2iπ)∫ σ+i∞σ−i∞ Mf (s) t−s ds where α < σ < β. Another suitable expression

of the Mellin transform is provided by the zeta function of the function f(x) definedby

(2.6) ζf (s) ≡ 1

Γ(s)

∫ ∞

0

dx xs−1 f(x)

which takes into account suitable properties of the Euler Γ function as we shall seelater. The inverse transform is thus

(2.7) f(x) =1

2iπ

∫ σ+i∞

σ−i∞ds x−sΓ(s) ζf(s)

A direct calculation of the zeta function of a function f(x) with a discrete scalingsymmetry (2.2) gives

(2.8) ζf (s) =b as ζg(s)

b as − 1

The behavior of f(x) is driven by the poles of ζf . Disregarding at this stage thepole structure of ζg, we focus on the poles in (2.8) which result from the scalingsymmetry. There are the solutions, sn, of b as = 1 namely,

(2.9) sn =ln (1/b)

ln a+

2iπ n

ln a

for integer n. The origin of these poles is a direct consequence of the exponentialbehavior of the coefficients in (2.4). These complex poles have been identifiedwith complex valued dimensions of self-similar fractal systems [26]. By an inversetransform, we have

(2.10) f(x) = x− ln(1/b)ln a

+∞∑n=−∞

Γ(sn) ζg(sn) e−2iπn ( ln xln a ) ≡ x− ln(1/b)

ln a G

(2π lnx

ln a

)namely, the general solution of (2.2) is the product of a power law behavior char-acterized by a fractal dimension ln (1/b) / ln a times a periodic function G(x+ 1) =G(x) of its argument 2π lnx/ ln a. This log-periodic behavior is tightly related to


the existence of a discrete scaling symmetry as expressed by (2.2) and it consti-tutes its fingerprint together with the power law prefactor determining the fractaldimension.

Log-periodic functions have a long history [40] including the well known Weier-strass function determined by the series,

(2.11) W (t) =

∞∑n=0

an cos (bn t) .

They appear in a broad range of problems such as the renormalization group andthe theory of phase transitions [9, 10], Markov processes in complex media [18],turbulence [11] and fractals [6,7]. Let us also mention that the scaling relation (2.2)with definite parameters (a, b) is a particular example of more general iterationprocesses described by the Poincare equation f(az) = P (f(z)) where P(x) is apolynomial [40]. This is actively studied in the mathematical literature [27].

Remark 2.1. The periodic function G(x) becomes constant when all residuesin the inverse Mellin transform vanish except for n = 0. This is the case forinstance for the mass density M(L) of the interval of length L or generally of anyEuclidean manifold of dimension d, where M(L) ∝ Ld is expected without log-periodic behavior. This can be checked either by a direct calculation or by noticingthat if the scaling symmetry M(aL) = bM(L) takes place for any value of a andnot only for a fixed one, then, averaging over a washes out the oscillations leavingonly the expected power law behavior.

2.2. A variational derivation. It is of interest to present another derivationof the form (2.10) not based on the inverse Mellin transform. This provides an-other point of view which is useful in cases where the pole structure cannot be easilyretrieved. To that purpose, we start from the iterated form (2.4) of f(x) and to esti-mate it, we use a saddle point approximation to find the value of n which dominatesf(x). By differentiating with respect to n, we obtain −(ln b)g + (ln a)an x g′ = 0.Defining u = an x, we have −(ln b)g(u) + (ln a)u g′(u) = 0. This equation in thevariable u admits a solution N such that u = aN x and then,

(2.12) N =lnu− ln x

ln a.

N is not necessarily an integer, namely N = n0 + t, where n0 is an integer and0 ≤ t ≤ 1. Then, we write f(x) in (2.4) under the form,(2.13)

f(x) =∞∑

m=−n0+1

b−(n0+m) g(an0+m x

)=

∞∑m=−n0+1

b−(N+m−t) g(aN+m−t x

).

Using (2.12) leads to b−N = (u/x)ln(1/b)/ ln a, and since, u and a are fixed quantities,then n0 → −∞ for x → 0, so that

(2.14) f(x) =(ux

) ln(1/b)ln a

∞∑m=−∞

b−(m−t) g(am−t u

)where the series is now a periodic function of t of period unity, namely using (2.12),a periodic function of lnx/ ln a of period unity. We thus recover (2.10).

6 ERIC AKKERMANS

3. Heat kernel and spectral functions - Generalities

The study of spectral functions on manifolds has a long and very successfulhistory which traces back to the early XXth century with considerations put for-ward by Lorentz on the blackbody radiation [41]. Given a Euclidean manifold, it ispossible to retrieve some of its geometric characteristics such as its volume, surface,curvature, Euler-Poincare characteristics by studying the spectral properties of theLaplace operator defined on the manifold [41,42,49]. In other words, the Lapla-cian or more precisely its eigenvalue spectrum can be viewed as a ”ruler” whichallows to span the manifold. Since stationary wave and heat equations are bothgoverned by the Laplacian, looking at the heat flow or at (scalar) electromagneticwave propagation are equivalent ways for extracting geometric characteristics ofEuclidean manifolds. This has led to seminal works in mathematics starting withH. Weyl [43] and the expansion which bears his name culminating in the celebratedquestion of M. Kac, ”Can you hear the shape of a drum ?” [44] and the negativeanswer provided by examples of isospectral domains of different shapes [45].

The direct relation between spectral functions of Laplace operators and pathintegral in quantum field theory [46] or equivalent functional forms of the partitionfunction in statistical mechanics are the underlying reasons for the enduring successof methods based on spectral functions of Laplace or corresponding Dirac operatorsin gauge theories [47].

Let us first illustrate these ideas using simple examples (see Chapter 5 in [50]).Consider first the diffusion of heat φ(x, t) along the infinite, unbounded real line(d = 1). The corresponding heat equation is

(3.1)∂φ

∂t= DΔφ

where the diffusion coefficient D sets units of length and time. The Green’s solutionof this equation is

(3.2) Pd=1(x, y, t) =1

(4πDt)1/2e−

(x−y)2

4Dt .

In a probabilistic interpretation, Pd=1(x, y, t) represents the probability density fora particle to diffuse from an initial position x to a final position y in a time t. Itsgeneralization Pd(x, y, t) to the d-dimensional free space is obtained from (3.2) byreplacing the exponent 1/2 in the denominator by d/2. A way to characterize thespace geometry which sustains the heat flow, is to consider the heat kernel definedby

(3.3) Zd(t) =

∫V

dxPd(x, x, t) =V

(4πDt)d/2

where the integral is over a d-dimensional volume V defined qualitatively withoutyet specifying boundary conditions. The heat kernel thus defined is an integralover all closed diffusing trajectories (starting and ending at a point x) within thevolume V . The Green’s function Pd is obtained for t > 0, from the normalizedeigenfunctions φn(x) and non negative eigenvalues λn (with degeneracy gn) of theheat equation (3.1) in d dimensions, as

(3.4) Pd(x, y, t) =∑n

gn φn(x)φ∗n(y) e−λn t


so that

(3.5) Zd(t) =∑n

gn e−λn t = Tr

(eD tΔ

).

This relation between the heat kernel Zd(t) and the Laplace operator −Δ, expressesthe distribution of closed diffusive trajectories within a manifold in terms of thespectrum of the Laplacian. This relation for the heat kernel is instrumental incalculating Euclidean path integrals [46], partition functions [29] and other spectraland transport quantities [50].

3.1. The Weyl expansion. Expression (3.5) is also very convenient to eval-uate Zd(t) for manifolds with boundaries. Consider the simple case of diffusionon an interval of length L. The corresponding eigenvalue spectrum of the Lapla-

cian (we set D = 1 for convenience) is given by λn = (nπ/L)2

where n is annon zero integer for Dirichlet (D) boundary conditions, φ(x = 0, t) = φ(L, t) = 0,whereas it includes the zero mode n = 0 for Neumann (N ) boundary conditions∂φ(x, t)|x=0 = ∂φ(x, t)|x=L = 0. The corresponding heat kernels ZN ,D(t) are thusrelated by

(3.6) ZN (t) =

∞∑n=0

e−(nπ/L)2 t = 1 + ZD(t)

The use of the Poisson formula allows to write the small time asymptotic expansion

(3.7) ZN ,D(t) =L√4π t

∓ 1

2+ · · ·

This is the simplest example of a Weyl expansion. For a two-dimensional domain ofarbitrary shape with surface S and boundary length L, we have the Weyl expansioncorresponding to Dirichlet boundary conditions [51],

(3.8) Z2(t) =S

4π t− L/4√

4π t+

1

6+ · · ·

where the constant term, 1/6, results from the integral of the local curvature of theboundary. More generally for a d-dimensional Euclidean manifold of hypervolumeV , hypersurface S, etc., the Weyl asymptotic expansion (restoring the diffusion

coefficient D), involves powers of 1/ (4πDt)(d−i)/2:

(3.9) Zd(t) V

(4πD t)d/2− αd

S

(4πD t)(d−1)/2+ · · ·

where αd is a constant which depends on boundary conditions [32, 41, 42]. TheWeyl asymptotic formula provides a small time expansion for Zd(t). Physically,it describes the behavior of a diffusive particle initially released at some point inthe manifold. At small time, it experiences a free space diffusion insensitive to theboundaries (volume term). At later times, the particle starts feeling the boundary(surface term), then its shape (local curvature term), etc..

To conclude this part, we discuss a point concerning the dispersion relation(i.e., the relation between time and length units) which will prove relevant whenconsidering diffusion on fractals. Zd is, as defined in (3.5), a dimensionless quan-tity. On the other hand, the Laplacian has dimensions of the inverse of a lengthsquared which allows to retrieve from the heat kernel geometric information aboutthe manifold. We thus need to insert a diffusion coefficient D in order for V 2/d/D

8 ERIC AKKERMANS

in (3.9) to have units of time. The diffusion coefficient D expresses the underlyingphysics of the diffusion flow, and it is related to the relevant sources of diffusionby an appropriate phenomenological relation (e.g. temperature T and viscosity σwhere D = RT/6π η aN = kBTσ) generally known as the Einstein relation [56].For instance, in the specific case of a covariant diffusion equation as studied in quan-tum mesoscopic physics [50], or in Euclidean time formulation of the Schrodingerequation of a particle of mass m, we have D = �/2m.

3.2. Spectral determinant - Density of states and spectral zeta func-tion. There is an important and useful relation between the heat kernel, its ge-ometrical content and the density of states ρ(λ) of the Laplacian defined on thecorresponding manifold. To find it, we define the spectral determinant associatedto the eigenvalue spectrum λn of −Δ,

(3.10) S(γ) = det (−DΔ + γ) =∏n

(λn + γ)

where γ is a real number. From the relation (3.5), it follows that

(3.11)

∫ ∞

0

Z(t)e−γtdt =∑n

gnγ + λn

=∂

∂γlnS(γ) .

The density of states ρ(λ) =∑

n δ(λ−λn) is thus directly related to the heat kerneland the spectral determinant through (see for instance Chapter 5 in [50])

(3.12) ρ(λ) = − 1

πlimη→0+Im

d

dγlnS(γ)

where γ is now complex valued, γ = −λ + iη. This last relation proves useful butuneasy to implement since the spectral determinant S(γ) is defined by the productof eigenvalues λn. This product is infinite, so that its definition is formal. To givean interpretation to S(γ), we resort to the spectral ζΔ function, associated to theLaplacian and defined by

(3.13) ζΔ(s) =∑n

1

λsn

.

This function is well-defined for all values of s for which the series converges. Usingthe identity

(3.14)1

λs=

1

Γ(s)

∫ ∞

0

dt ts−1e−tλ ,

we write

(3.15) ζΔ(s) =1

Γ(s)

∫ ∞

0

dt ts−1Tr(eDΔ t

).

ζΔ(s) thus defined, is the Mellin transform (2.5) of the heat kernel. It is convergentfor Res > d (d being the space dimension and the spectral dimension ds for afractal). Its analytic continuation in the complex plane defines a meromorphicfunction in s which is analytic at s = 0. We use this analyticity and the identity :

(3.16)d

dsλ−sn |s=0 = −lnλn

to express the spectral determinant as

(3.17) lnS(γ = 0) = − d

dsζΔ(s)|s=0


which is well defined.As an example, we consider the Laplacian on an interval of length L with

Dirichlet boundary conditions (and set D = 1). Inserting the expression λn =

(nπ/L)2 of the eigenvalues into (3.13), we obtain

(3.18) ζΔ(s) =∞∑

n=1

(L2

π2n2

)s

=

(L

π

)2s

ζR(2s) .

The Riemann zeta function ζR(2s) has a simple pole at 2s = 1. Thus, the inverseMellin transform (2.7) provides directly ZD(t) = (L/2π) Γ(1/2)t−1/2 + · · · namely(3.7). Using (3.12), we deduce the main contribution to the corresponding densityof states

(3.19) ρd=1(λ) L

2π√λ

and from (3.9), we obtain the generalization of these results to d-dimensional Eu-clidean manifolds,

ρ3(λ) =V

4π2

√λ− S

16π+ · · ·

ρ2(λ) =S

4π− L

8π

1√λ

+ · · ·(3.20)

To conclude this part, we note that the short time Weyl expansion of the heat kernelis related to the pole structure of the zeta function ζΔ associated to the Laplacian.

3.3. Counting function - Spectral zeta function - Wavelet transform.We consider the counting function N(λ) defined by

(3.21) N(λ) =∑λn<λ

1 .

The counting function N(λ) and the density of states ρ(λ) = dN(λ)dλ =

∑n δ(λ −

λn) are the sum of Heaviside (step) θ and Dirac δ functions characterizing thedistribution of eigenvalues λn of the Laplacian. The density of states ρ(λ) is relatedto the spectral determinant (3.12), to the heat kernel and to the spectral zetafunction. Unlike the counting function, the density of states is often ill-definedsince N(λ) is not differentiable (e.g. for a fractal spectrum).

Consider the number C(λ) of eigenvalues of the Laplace operator −Δ withinthe interval [0, λ] included into the support J of the whole spectrum. It is given interms of the counting function as

(3.22) C(λ) = N(λ) −N(0) =

∫J

dN(ε) θ (λ− ε)

where the integral extends over the whole spectrum J with the spectral measuredN(ε). By definition (3.5), the heat kernel Z(t) can be expressed as (with D = 1)

(3.23) Z(t) = Tr(etΔ)

=

∫J

dN(λ) e−t λ .

10 ERIC AKKERMANS

The spectral zeta function ζΔ(s) defined in (3.15) thus becomes,

ζΔ(s) =1

Γ(s)

∫ ∞

0

dt ts−1

∫J

dN(λ) e−t λ

=

∫J

dN(λ)λ−s(3.24)

as a result of the identity Γ(s) =∫∞0

dx xs−1 e−x. It is thus possible to relate theMellin transform MC(s) of C(λ) to ζΔ(s), namely,

MC(s) =

∫J

dC(λ)λ−s =

∫J

λ−s d

(∫J

dN(ε) θ (λ− ε)

)=

∫J

λ−s

∫J

dN(ε) δ(λ− ε)

= ζΔ(s)(3.25)

so that the spectral function ζΔ(s) is the Mellin transform of the counting function.This relation is a particular example of a more general property of the Mellintransform which can be viewed as a convolution theorem. To state it, consider anintegrable function g(x), suitably normalized to unity,

∫∞0

dt g(t) = 1, and whoseMellin transform Mg(s) is defined. The quantity

(3.26) Wg (λb, t) ≡∫J

dN(λ) g (t |λ− λb|) ,

defined for t > 0, is called the wavelet transform of the counting function N(λ). In-tuitively, the wavelet transform can be viewed as a mathematical microscope whichprobes the counting function at a point λb with an optics specified by the choice ofthe specific wavelet g(x). An important property of the wavelet transform is thatit preserves a discrete scaling symmetry (2.2) of the probed function. Performinga Mellin transform of Wg (λb, t) w.r.t. the variable t gives

(3.27) M [Wg (λb, t)] = ζΔ (s,−λb) Mg(s)

where ζΔ (s,−λb) ≡∫JdN(λ) |λ − λb|−s is a shifted zeta function. From the last

relation, it is immediate to obtain the relation between different wavelet transformsrespectively specified by the ”optics” f(t) and g(t) of Mellin transforms Mf andMg, namely

(3.28)M [Wg (λb, t)]

Mg(s)=

M [Wf (λb, t)]

Mf (s)= ζΔ (s,−λb) .

An interesting application of this convolution rule is obtained by taking λb = 0and the wavelet g(t) = 2 sin t

πt such that Mg(s) = (2/π) Γ(s − 1) sinπ(s − 1)/2 for0 < Re(s) < 2. We obtain from (3.26),

(3.29) M

[2

π

∫J

dN(λ)sinλ t

λ t

]=

2

πΓ(s− 1) sin

π

2(s− 1) ζΔ(s)

so that from the inverse Mellin transform,

(3.30)2

π

∫J

dN(λ)sinλ t

λ t=

1

2iπ

∫ σ+i∞

σ−i∞ds t−s sinc

[π(s− 1)

2

]Γ(s) ζΔ(s) .

Therefore different probes of the spectrum of the Laplacian can be related one toanother and expressed in terms of the spectral zeta function. This is a powerfulresult with interesting physical consequences since distinct physical phenomena


usually involve their own probing function (e.g. the Fermi golden rule describingthe response of the spectrum to an external perturbation (for instance spontaneousemission), involves the sinc probe g(t)) [37] and all are related to the heat kernelcharacterized by an exponential probe.

4. Laplacian on fractals - Heat kernel and spectral zeta function

This section is devoted to studying the heat kernel on fractals, namely onsystems whose geometry is characterized by a discrete scaling symmetry as definedin (2.2). In other words, we would like to extend the previous analysis and therelation between geometric and spectral properties to fractals. This is a vast subjectand we do not intend to be exhaustive but rather to study specific but genericenough examples in order to highlight some of the more salient results and openquestions.

We have seen that geometric characteristics of Euclidean manifolds can be re-trieved from the spectrum of the corresponding Laplace operator −Δ. The relevantspectral tools are the heat kernel (3.5) and the spectral zeta function ζΔ(s) definedby (3.15). How do they generalize to fractals ? Geometric information about afractal is characterized by its Hausdorff dimension dh. On the other hand, the verynotion of volume or surface of a fractal is rather ill-defined. Then, is it possible todefine a heat kernel and a Weyl expansion for fractals and if it is so, what kind ofinformation does it provide. Those questions have been and still are in the focusof intensive works from the mathematical community where an abundant numberof important results has been proved ([19]-[27]). It is not our purpose to reviewthem, but rather to show their relevance and usefulness in physics. An importantstep has been to prove, using either a probabilistic or an analytic approach, that aLaplace operator can be properly defined on fractal structures [24] as well as thecorresponding heat kernel and the spectral zeta-function [20].

As we have seen in Section 2, the characteristic feature of a discrete scalingsymmetry is the existence of a tower of complex poles in the zeta function associatedto the relevant quantity. These complex poles are a direct expression of the scalingform (2.2) and of the exponential behavior of the coefficients in the iteration series(2.4). Are there similar properties of the spectral zeta function of the Laplacian ona fractal ? The answer to this question is positive. There exist two characteristicparameters (a, b) ≡ (ldw , ldh), such that the heat kernel of the Laplacian on a self-similar fractal obeys the scaling relation (2.2). These scaling parameters involvethe geometric Hausdorff dimension dh, the walk dimension dw that we shall defineand discuss later on and an elementary step length l describing size scaling.

dh dw ds = 2dh/dw l = L1/nn

D4,2 2 2 2 2D6,2 ln 6/ ln 2 2 ln 6/ ln 2 2D6,3 ln 6/ ln 3 2 ln 6/ ln 3 3

Sierpinski ln 3/ ln 2 ln 5/ ln 2 2 ln 3/ ln 5 2

Figure 2. Fractal dimensions and size scaling factor for diamondfractals and for the Sierpinski gasket. For D6,2, the spectral di-mension is ds ≈ 2.58, and for D6,3, ds ≈ 1.63.

12 ERIC AKKERMANS

To proceed further, we consider the specific case of a diamond fractal (♦)(see Figs.1 and 2). It has been shown [33] that the corresponding heat kernelZ♦(t) has a closed expression of the form (2.4) where the initial function to beiterated is the heat kernel ZD(t) of the interval of unit length with Dirichlet (D)

boundary conditions, given by (3.6), namely, ZD(t) =∑∞

k=1 e−k2π2t (we set D = 1

for convenience), so that,

(4.1) Z♦ = ZD(t) + B∞∑

n=0

Ldhn ZD

(Ldwn t)

where we have defined the total length Ln ≡ ln of the diamond fractal at stepn of the iteration. The coefficient B ≡

(ldh−1 − 1

)is the branching factor of

the fractal (see Fig.1) and the integer ldh is the number of links into which agiven link is divided. As in (2.4), the series for Z♦(t) involves coefficients whichbehave exponentially with the iteration n. We therefore expect an expression ofthe form (2.10) with log-periodic oscillations with time. To show it, we calculatethe corresponding spectral zeta function,

(4.2) ζ♦ =1

Γ(s)

∫ ∞

0

dt ts−1 Z♦(t) .

An elementary calculation [33] leads to

ζ♦(s) =ζR(2s)

π2s

(1 + B

∞∑n=0

Ldh−dwsn

)

=ζR(2s)

π2sldh−1

(1 − l1−dws

1 − ldh−dws

),(4.3)

where ζR(2s) is the Riemann zeta function. Note that a similar structure exists forthe Sierpinski gasket [20,23], with the Riemann zeta function factor replaced byanother zeta function. ζ♦(s) has complex poles given by

(4.4) sm =dhdw

+2iπm

dw ln l≡ ds

2+

2iπm

dw ln l,

where m is an integer. These complex poles have been identified with complexdimensions for fractals [26]. The fractal dimension ds ≡ 2dh/dw is called spectraldimension. It has been obtained in earlier works in the physics community [4,5]and recognized as the relevant fractal dimension (unlike dh) underlying spectralproperties of self-similar fractals.

From the inverse Mellin transform (2.7), we obtain

Z♦(t) =Vs

tds/2

∞∑m=−∞

Γ(sm)ζR(2sm)/π2sme2iπ m ln t/(dw ln l)

=Vs

tds/2G♦

(2π ln t

dw ln l

)(4.5)

where G♦ is a periodic function of its argument of period unity. These log-periodicoscillations are represented on Fig.3 and we note that the higher order (with m)complex poles give much smaller contributions, a result related to the steep decreaseof the Euler Γ function along vertical lines. A similar behavior has been foundnumerically for the Sierpinksi gasket [23].


Figure 3. Heat kernel ZD(t) at small time, normalized by theleading non-oscillating term, for the fractal diamond D4,2. Thesolid [blue] curve is exact; the dashed [red] curve is the approxi-mate expression (4.5). At very small t, these curves are indistin-guishable, as shown in the inset plot. The relative amplitude ofthe oscillations remains constant as t → 0 [33].

For mathematical discussions of oscillations in heat kernel estimates see [52,53].In particular, there is a class of fractals where oscillations are related to large gapsin the spectrum. This topic is a subject of active current research [54, 55] andreferences therein.

4.1. Spectral volume - Anomalous walk dimension. We have seen thatfor a d-dimensions Euclidean manifold, the dominant contribution to the Weylexpansion takes the form (3.9), i.e., it is proportional to the volume V of themanifold. This dependence, as has been emphasized, is a consequence of the factthat the Laplacian sets units of length. In other words, the quantity td/2Tr

(e−tΔ

)has a well defined limit, V , for t → 0. This limit can be viewed as V = Ld where Lis the characteristic length set by the Laplacian. There is another consequence ofthe form Ld/td/2 of the dominant term of the Weyl expansion (3.9). Since Z(t) isdimensionless, this imposes the usual Euclidean dispersion L2 ∝ t characteristic ofa diffusion processes where units are matched by means of the diffusion coefficientD. This can be checked directly from the differential equation (3.1) or from itssolution (3.2). But for diffusion on a fractal there is no similar local equation norclosed expression of the local propagator P (r, r′, t). It is nevertheless possible toexpress an equivalent dispersion which is solely a property of the Laplace operator.

To that purpose, we note from (4.5), that the quantity tds/2 Z♦/G♦ has a finitelimit for t → 0 which we denoted Vs. Restoring length and time units (throughD) and comparing to (3.3) for the Euclidean case, leads necessarily to the formVs = Ldh

s where Ls is the characteristic spectral length associated to the Laplacian

14 ERIC AKKERMANS

on the diamond fractal and dh is the Hausdorff geometric dimension. Inserting thisexpression into Z♦ allows to rewrite the power law leading prefactor under the form(L2 dh/dss /t

)ds/2

. Since it is a dimensionless quantity, this implies that the disper-

sion for fractals is of the form L2 ∝ tds/dh . This new dispersion plays an essentialrole in the characterization of diffusion on fractals. It is a constitutive equationusually written under the form of a characteristic mean square displacement law,

(4.6) 〈r2(t)〉 ∝ t2/dw

where the exponent dw ≡ 2dh/ds is called ”anomalous walk dimension”. It is worthemphasizing that usually dw > 2 on a fractal which amounts to a slower diffusionas opposed to dw = 2 on a manifold. Thus, the expression,

(4.7) Z♦(t) =Ldhs

tds/2G♦

(2π ln t

dw ln l

)=

(Ldws

t

)ds/2

G♦

(2π ln t

dw ln l

)for the heat kernel on a fractal takes a form analogous to its Euclidean counterpart(3.1) but where the geometric volume V = Ld is now replaced by a spectral volumeVs = Ldh

s which results directly from the spectral properties of the Laplacian.

5. Thermodynamics on photons : The fractal blackbody [34]

5.1. Thermodynamics of the quantum radiation - Generalities. A di-rect application of the considerations developed in the previous sections is to thestudy of statistical mechanics of quantum radiation (photons) in fractal structures.Indeed, a basic aspect of it, namely the blackbody radiation precisely addressesthe relation between the electromagnetic modes inside a cavity (a manifold) andits geometric characteristics. Historically, the seminal work of H. Weyl which led,among other results, to the expansion (3.9) has been motivated by considerationsraised by H. Lorentz about the dependence of the Jean’s radiation law upon thevolume of the cavity at the exclusion of other geometric characteristics.

Although the derivation of the thermodynamic partition function of quantumradiation constitutes well known textbook materials, we wish to re-examine it to-wards its application to fractals in order to emphasize some key points and basicassumptions. The purpose of this derivation is to show that the partition functionis directly related to the spectral zeta function so that it can be calculated for Eu-clidean manifolds and fractal as well. Moreover the geometric information about amanifold retrieved from thermodynamics relates directly to the spectral geometryof the Laplace operator. This point is not always emphasized in textbooks whichprefer to consider combinatorics of mode counting in simple cubic geometries, anapproach which relies heavily on the existence of Fourier transform and phase spacequantization cells, a tool which is not available for fractals.

For convenience, we consider thermodynamics of a scalar massless field (scalarphotons) which simplifies the calculation due to the vanishing chemical potentialresulting from zero mass. Generalization to massive fields is possible though slightlymore cumbersome [59, 60]. The spectral partition function (including the zeromode Casimir contribution) of a quantum oscillator of frequency ω at temperature

T = 1/kBβ is lnZ(T, ω) = −β�ω2 − ln

(1 − e−β�ω

). Using the identity

(5.1) − ln(1 − e−β�ω

)=

�β

4π

∫ ∞

0

dτ

τ3/2e−ω2τ

∞∑n=1

e−n2 (�β)2/4τ


together with the Poisson formula

(5.2)+∞∑

n=−∞e−n2 τ =

√π

τ

+∞∑n=−∞

e−π2 n2/τ

leads to

(5.3) lnZ(T, ω) =1

2

∫ ∞

0

dτ

τe−ω2 τ

+∞∑n=−∞

e−( 2πn�β )2 τ .

The operator ∂20 ≡ ∂2

t defined with periodic boundary conditions φ(t + �β) = φ(t)admits a discrete spectrum M known as Matsubara frequencies 2πn/�β, so that

(5.4)+∞∑

n=−∞e−( 2πn

�β )2 τ = TrM

(e−τ∂2

0

).

Identifiying ω2 = c2k2 with the eigenvalues of c2Δ, and tracing over all modes,we obtain for the partition function of the quantum radiation at temperature T =1/kBβ,

(5.5) lnZ(T, V ) =1

2

∫ ∞

0

dτ

τTrM

(e−τc2Δ

)TrM

(e−τ∂2

0

)where TrM denotes the trace over the modes of the Laplacian defined on the mani-fold M which contains the anticipated dependence on the volume V . Finally usingthe identity lnO = −

∫∞0

dττ e−O τ , and ln DetO = Tr lnO, we obtain the elegant

and compact dimensionless form for the partition function,

lnZ(T, V ) = −1

2ln DetM×M

(∂20 + L2

β Δ)

(5.6)

=1

2

∫ ∞

0

dτ

τf(τ )TrM

(e−τL2

βΔ)

(5.7)

where we have defined the photon thermal wavelength Lβ ≡ �βc and the dimen-

sionless operator ∂20 = (�β)

2∂20 , so that f(τ ) = TrM

(e−τ∂2

0

)=∑+∞

n=−∞ e−(2πn)2τ

is a dimensionless Jacobi θ3-function.From this last expression of the partition function, we see that it is indeed

determined by the heat kernel of the Laplace operator on the manifold M

(5.8) KM(τ ) ≡ TrMe−τ L2β Δ .

This form is useful to describe in a closed way the thermodynamics of quantumradiation in a large hypercube of volume Ld, in d space dimensions. There the

modes �k are quantized in units of 2π�n/V 1/d where �n is a vector with integerscomponents defining elementary cells in the reciprocal phase space. The previousexpression of the partition function becomes

(5.9) lnZ(β, V = Ld) = lnZ(�βcV −1/d) = −1

2ln DetM×M

(∂20 + L2

βV−2/dΔ

),

where −Δ is a dimensionless Laplacian. Therefore Z is a function of the singlevariable LβV

−1/d. Standard thermodynamic quantities follow immediately fromthis specific scaling behavior. For example, the equation of state, PV = U/d,

16 ERIC AKKERMANS

relating the internal energy U of the radiation to its pressure P and the volume V ,follows immediately from

(5.10) U = − ∂

∂βlnZ(T, V ) = −

(d lnZ(x)

dx

)� c V −1/d

and

(5.11) P =1

β

(∂ lnZ∂V

)T

= −(d lnZ(x)

dx

)�cV −1/d

V d.

The Stefan-Boltzmann law for the internal energy U is a consequence of the equationof state and the thermodynamic relation

(∂U∂V

)T

= T(∂P∂T

)V−P , while noticing from

(5.11) that P depends on T only, in the thermodynamic limit. We then obtainU = aV T d+1 where a is a constant to be determined from (5.9). It is alreadyapparent from this simple case that the volume dependence in the thermodynamicequation of state, PV = U/d, comes from spectrum of the Laplacian.

For black-body radiation associated to Euclidean manifolds of complicatedshape, it is difficult to make an explicit mode decomposition and find an ex-plicit expression like (5.9) for the heat kernel. However, we can learn about thethermodynamic [large volume] limit from the Weyl expansion (3.9) of the heatkernel. Note that the large volume limit corresponds to V � Ld

β, which is a

”high temperature” limit kBT � �c/V 1/d. Keeping only the dominant volumeterm in (3.9), expression (5.7) leads immediately to the familiar thermodynamicexpressions [56] previously derived: lnZ = (V/Ld

β)ζR(d + 1)Γ(d+12

)/π(d+1)/2,

P = (kBT/Ldβ)ζR(d + 1)Γ

(d+12

)/π(d+1)/2. Away from the thermodynamic limit,

subdominant terms in (3.9) lead to corrections that depend on the exact geometryof the volume enclosing the radiation, but the equation of state PV = U/d is alwaysvalid [42]. This formulation in terms of heat kernel makes it clear that the Weylexpansion is directly related to the thermodynamic limit of a black-body radiationsystem, so we can use the leading term as a definition of the volume probed by thephotons as they attain thermal equilibrium.

5.2. Thermodynamics of the quantum radiation on fractals. Based onprevious results and especially expression (5.7) of the partition function, we are nowin position to study thermodynamics of quantum radiation on fractals [34]. Theheat kernel KF (τ ) equivalent of (5.8) but on a fractal F is obtained using (4.7), sothat

(5.12) KF (τ ) = TrF(e−L2

βτ Δ)

=(Ls/Lβ)

dh

τds/2GF

(2π ln τ

dw ln l

).

The thermodynamic partition function lnZ(�βcV

−1/dhs

)is a function of the single

variable LβV−1/dhs , where V

1/dhs = Ls is the spectral length associated to the

Laplacian on the corresponding fractal F . Thermodynamic quantities and relationsfollow immediately from this specific scaling behavior and from (5.10) and (5.11) togive the thermodynamic equation of state on a fractal as PVs = U/dh. The secondimportant conclusion is that the actual expressions for pressure P , internal energyU , etc... will be modified on a fractal, not only by the appearance of the spectraldimension ds and spectral volume Vs, but also by the appearance of oscillatoryterms arising from the behavior of the log-periodic function GF in (5.12).


5.3. Vacuum Casimir energy. Another straightforward consequence of(5.12) is the expression of the zero temperature free energy namely the vacuumCasimir energy calculated in various complex geometries and recently on quantumgraphs [57]. A general expression for the Casimir energy is obtained from theinverse Mellin transform of the spectral zeta function which gives

lnZ(T, V ) = −1

2

(Lβ

L

)ζM

(−1

2

)+

1

πi

∫C

(L

Lβ

)2s

Γ(2s) ζR(2s + 1) ζM(s) ds(5.13)

The first term gives the standard zero temperature “vacuum Casimir energy” con-tribution [58], proportional to ζM

(− 1

2

). On a fractal F , we have L = Ls and the

previous expression leads for the vacuum Casimir energy to:

(5.14) E0 =1

2

�c

LsζF

(−1

2

).

6. Conclusion and some open questions

We have presented general features of deterministic self similar fractals whichpossess an exact decimation symmetry. To describe quantitatively diffusion pro-cesses and wave propagation on these structures, we have defined and developspectral tools related to the Laplacian, such as the heat kernel and the spectralzeta function. After reviewing some of their main features for Euclidean manifolds,we have calculated them on fractals. This has enabled us to single out a number ofspecific features of fractals such as the spectral dimension ds, the spectral volumeVs, the existence of log-periodic oscillations of spectral quantities, an unusual dis-persion characterized by the walk dimension dw = 2dh/ds, dh being the geometricHausdorff dimension. These features show up when extending results of statisticalmechanics and quantum field theory on fractal structures. To see this at work, wehave considered the problem of thermodynamics of quantum radiation on fractalsin the simplified scalar version as well as the vacuum Casimir energy. The resultswe have obtained generalize straightforwardly to related problems such as massivebosons. In that case, it is a direct consequence of the form of the heat kernel thatBose-Einstein condensation occurs for ds > 2 i.e. independently of the geometricdimension dh [59, 60], a fact already recognized long ago [5]. More involved isthe problem of the occurrence of superfluidity (a phenomenon distinct from Bose-Einstein condensation) which depends on terms of higher order than the volumecontribution in the Weyl expansion [60]. Beyond equilibrium situations, the prob-lem of quantum emission of radiation (either spontaneous or stimulated) is highlyinteresting either from a fundamental point of view or for applications. It can beshown [37] that the probability of emission, also known as vacuum persistence inthe context of quantum field theory, is driven by another spectral kernel, the sinckernel (3.30) related to the heat kernel and which exhibits an analogous qualitativebehavior.

We have mostly discussed spectral quantities and not the behavior of non-diagonal terms in the propagator P (x, y, t) associated to diffusion. Very little isknown about it except for useful bounds [19] and a conjectured form supported bynumerical results [61]. This part of the characterization of fractals is nonethelessessential to better understand stochastic processes, out of equilibrium statistical

18 ERIC AKKERMANS

mechanics and large deviation physics. Preliminary results based on specific distri-butions of traps [18] or on an additivity principle [62] seem promising.

Let us mention finally that the absence of translational invariance in fractalsprevents using Fourier transforms. It is preferable instead to use Mellin trans-form as explained in section 2. This has important consequences in formulatingan uncertainty principle relating direct to reciprocal spaces. We have emphasizedthat while the direct space is driven by the Hausdorff dimension dh, candidatesfor reciprocal spaces obtained from the Laplacian are instead driven by ds. Thisissue has far reaching consequences if one wishes to properly formulate canonicalfield quantization on fractal structures including spin. This leads to yet anotherchallenge on fractals. A proper understanding of spin on Euclidean manifolds hasrequired tools such as heat kernel and Weyl expansion of Dirac operators. Topolog-ical properties related to spin thus appear under the form of powerful results knownas Index theorems [47]. Their extension to fractals requires a proper understandingof cohomology, connexions and definition of a Laplacian using Hodge theorem [63].

Acknowledgments

It is a pleasure to thank G. Dunne, A. Teplyaev, E. Gurevich, D. Gittelmanand O. Spielberg for collaboration and discussions.

References

[1] Benoit B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Co., San Francisco,Calif., 1982. Schriftenreihe fur den Referenten. [Series for the Referee]. MR665254 (84h:00021)

[2] T. Nakayama, K. Yakubo and R. L. Orbach, Dynamical properties of fractal networks: Scal-ing, numerical simulations, and physical realizations, Rev. Mod. Phys. 66, (1994), 381-443,S. Havlin and D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 36, (1987), 695-798 and Diffusion and Reactions in Fractals and Disordered Systems, (Cambridge UniversityPress, 2005).

[3] Eytan Domany, Shlomo Alexander, David Bensimon, and Leo P. Kadanoff, Solutions to theSchrodinger equation on some fractal lattices, Phys. Rev. B (3) 28 (1983), no. 6, 3110–3123.MR717348 (85h:82033)

[4] S. Alexander and R. Orbach, Density of States on fractals: fractons, J. Phys. Lett. 43,(1982), L625-L631. R. Rammal and G. Toulouse, Random walks on fractal structures andpercolation clusters, J. Phys. Lett. 44, (1983), L13-L22. R. Rammal, Harmonic Analysis InFractal Spaces: Random Walk Statistics And Spectrum Of The Schrodinger Equation, Phys.Rept. 103, (1984), 151-161.

[5] R. Rammal, Spectrum of harmonic excitations on fractals, J. Physique 45 (1984), no. 2,191–206. MR737523 (85d:82101)

[6] D. Bessis, J. S. Geronimo, and P. Moussa, Mellin transforms associated with Julia sets andphysical applications, J. Statist. Phys. 34 (1984), no. 1-2, 75–110, DOI 10.1007/BF01770350.MR739123 (85i:58067)

[7] Didier Sornette, Discrete-scale invariance and complex dimensions, Phys. Rep. 297 (1998),no. 5, 239–270, DOI 10.1016/S0370-1573(97)00076-8. MR1710267 (2000h:82036)

[8] C. W. Groth, J. Tworzydlo and C. W. J. Beenakker, Electronic shot noise in fractal conduc-tors, Phys. Rev. Lett. 100, (2008), 176804.

[9] Th. Niemeijer and J. van Leeuwen, in Phase Transitions and Critical Phenomena Vol. 6,C. Domb and M. Green (Eds.), (Academic Press, 1976), B. Hughes, M. Schlesinger andE. Montroll, Random walks with self-similar clusters, Proc. Nat. Acad. Sci. 78, (1981), 3287-3291, M. Nauenberg, Scaling representation for critical phenomena, J. Phys. A 8, (1975),925-928 and R. Jullien, K. Uzelac and P. Pfeuty, The Yang-Lee edge singularity studied bya four-level quantum renormalization-group blocking method, J. Physique 42, (1981), 1075-1080, B. Derrida, C. Itzykson and J. M. Luck, Oscillatory Critical Amplitudes in HierarchicalModels, Commun. Math. Phys. 94, (1984), 115-132 and R. B. Griffiths and M. Kaufman,


Spin systems on hierarchical lattices. Introduction and thermodynamic limit, Phys. Rev. B26, (1982), 5022-5027.

[10] Y. Meurice, Nonlinear aspects of the renormalization group flows of Dyson’s hierarchi-cal model, J. Phys. A 40 (2007), no. 23, R39–R102, DOI 10.1088/1751-8113/40/23/R01.MR2343505 (2008k:82044)

[11] E. A. Novikov, The effects of intermittency on statistical characteristics of turbulence andscale similarity of breakdown coefficients, Phys. Fluids A 2 (1990), no. 5, 814–820, DOI

10.1063/1.857629. MR1050014 (91d:76048)[12] V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov, Fractal structure of 2d-quantum

gravity, Mod. Phys. Lett. A 3, (1988), 819-827 and M. G. Harris and J. F. Wheater, TheHausdorff dimension in polymerized quantum gravity, Phys. Lett. B 448, (1999), 185

[13] J. Ambjorn, J. Jurkiewicz and Y. Watabiki, On the fractal structure of two-dimensionalquantum gravity, Nucl. Phys. B 454, 313 (1995); J. Ambjorn, J. Jurkiewicz and R. Loll,Spectral dimension of the universe, Phys. Rev. Lett. 95, (2005), 171-301.

[14] Oliver Lauscher and Martin Reuter, Fractal spacetime structure in asymptotically safegravity, J. High Energy Phys. 10 (2005), 050, 16 pp. (electronic), DOI 10.1088/1126-6708/2005/10/050. MR2180532 (2006e:83064)

[15] T. Jonsson and J. F. Wheater, The spectral dimension of the branched polymer phase of two-dimensional quantum gravity, Nucl. Phys. B 515, (1998), 549 and D. F. Litim, Fixed pointsof quantum gravity, Phys. Rev. Lett. 92, (2004), 201301 and G. Calcagni, Fractal universeand quantum gravity, Phys. Rev. Lett. 104, (2010), 251-301 [arXiv:0912.3142], Gravity on amultifractal, Phys. Lett. B 697, (2011), 251.

[16] Christopher T. Hill, Fractal theory space: spacetime of noninteger dimensionality, Phys.Rev. D (3) 67 (2003), no. 8, 085004, 13, DOI 10.1103/PhysRevD.67.085004. MR1995175(2004f:81257)

[17] S. Condamin, O. Benichou, V. Tejedor, R. Voituriez and J. Klafter, First passage times incomplex scale invariant media, Nature 450, (2007), 77, O. Benichou, B. Meyer, V. Tejedorand R. Voituriez, Zero constant formula for first-passage observables in bounded domains,Phys. Rev. Lett. 101, (2008), 130601 and V. Tejedor, O. Benichou and R. Voituriez, Globalmean first-passage time of random walks on complex networks, Phys. Rev. E 80, (2009),

065104.[18] J. Bernasconi and W. R. Schneider, Diffusion on a one-dimensional lattice with random

asymmetric transition rates, J. Phys. A 15 (1982), no. 12, L729–L734. MR682326 (84a:82037)[19] S. Goldstein, Random walks and diffusions on fractals. In Percolation theory and ergodic

theory of infinite particle systems, (Minneapolis, Minn., 1984-1985), volume 8 of IMA Vol.Math. Appl., pp. 121-129, Springer, New York, 1987, S. Kusuoka, A diffusion process on afractal, in Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), pp 251-274.Academic Press, Boston, MA, 1987, M.T. Barlow and E.A. Perkins, Brownian motion onthe Sierpinski gasket, Prob. Theor. Rel. Fields 79, (1988), 543, M. T. Barlow, Diffusions onfractals, Lecture Notes in Math 1690, (1988), 1-121, M. T. Barlow, R. F. Bass, The con-struction of Brownian motion on the Sierpinski carpet, Ann. Inst. Poincare Probab. Statist.25, (1989), 225 and M. T. Barlow, R. F. Bass, T. Kumagai, A. Teplyaev, Uniqueness ofBrownian Motion on Sierpinski Carpets, J. Eur. Math. Soc. 2010.

[20] Alexander Teplyaev, Spectral zeta functions of fractals and the complex dynamics ofpolynomials, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4339–4358 (electronic), DOI10.1090/S0002-9947-07-04150-5. MR2309188 (2008j:11119)

[21] Gregory Derfel, Peter J. Grabner, and Fritz Vogl, The zeta function of the Laplacian oncertain fractals, Trans. Amer. Math. Soc. 360 (2008), no. 2, 881–897 (electronic), DOI10.1090/S0002-9947-07-04240-7. MR2346475 (2008h:58062)

[22] J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math., 6, (1989),259-290, J. Kigami, Harmonic calculus on limits of networks and its application to dendrites,J. Functional Analysis 128 (1995), 48-86, J. Kigami, D. R. Sheldon and R. S. Strichartz,

Green’s functions on fractals, Fractals 8 (2000), 385-402 and J. Kigami, Harmonic analysisfor resistance forms, J. Functional Analysis 204 (2003), 399-444.

[23] Adam Allan, Michael Barany, and Robert S. Strichartz, Spectral operators on theSierpinski gasket. I, Complex Var. Elliptic Equ. 54 (2009), no. 6, 521–543, DOI10.1080/17476930802272978. MR2537254 (2010f:28011)

20 ERIC AKKERMANS

[24] Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, CambridgeUniversity Press, Cambridge, 2001. MR1840042 (2002c:28015)

[25] Robert S. Strichartz, Differential equations on fractals, Princeton University Press, Princeton,NJ, 2006. A tutorial. MR2246975 (2007f:35003)

[26] M. L. Lapidus and M. van Frankenhuysen, Fractal geometry, complex dimensions and zetafunctions, Geometry and spectra of fractal strings, Springer Monographs in Mathematics,(Springer, New York, 2006) and B. M. Hambly and M. L. Lapidus, Random fractal strings:

their zeta functions, complex dimensions and spectral asymptotics, Trans. Amer. Math. Soc.358, 285 (2006); B. M. Hambly and T. Kumagai, Diffusion on the scaling limit of the criticalpercolation cluster in the diamond hierarchical lattice, Comm. Math. Phys. 295, (2010) 29.

[27] G. Derfel, P. J. Grabner and F. Vogl, Laplace Operators on Fractals and Related FunctionalEquations, To be published in J. Phys. A, 2012.

[28] N. Birrell and P. Davies, Quantum Fields In Curved Space , (Cambridge Univ. Press, 1982)and J. S. Dowker and R. Critchley, Effective Lagrangian And Energy Momentum Tensor InDe Sitter Space, Phys. Rev. D 13, (1976), 3224.

[29] C. Itzykson and J. M. Drouffe, Statistical Field Theory, (Cambridge Univ. Press, 1989).[30] S. W. Hawking, Zeta Function Regularization Of Path Integrals In Curved Space-Time,

Comm. Math. Phys. 55, (1977) 133-148, E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Byt-senko and S. Zerbini, Zeta Regularization Techniques With Applications, (World Scientific,Singapore, 1994).

[31] K. Kirsten, Spectral functions in mathematics and physics, (Chapman and Hall/CRC, BocaRaton, FL, 2001).

[32] D. V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rep. 388 (2003), no. 5-6,279–360, DOI 10.1016/j.physrep.2003.09.002. MR2017528 (2004m:58044)

[33] E. Akkermans, G. V. Dunne and A. Teplyaev, Physical Consequences of Complex Dimensionsof Fractals, Europhys. Lett. 88, (2009), 40007.

[34] E. Akkermans, G. V. Dunne and A. Teplyaev, Thermodynamics of photons on fractals, Phys.Rev. Lett. 105, (2010), 230407.

[35] Gerald V. Dunne, Heat kernels and zeta functions on fractals, J. Phys. A 45 (2012), no. 37,374016, 22, DOI 10.1088/1751-8113/45/37/374016. MR2970533

[36] L. Dal Negro and S. V. Boriskina, Deterministic Aperiodic Nanostructures for Photonics andPlasmonics Applications, Laser and Photonics Reviews 5, (2011), 1-41.

[37] E. Akkermans and E. Gurevich, Spontaneous Emission from a Fractal Vacuum, Europhys.Lett. 103 (2013) 30009, 1-5. arXiv:1210.6895.

[38] Y. Gefen, B. B. Mandelbrot, and A. Aharony, Critical Phenomena on Fractal Lattices, Phys.Rev. Lett. 45, (1980), 855-858 and Y. Gefen, A. Aharony, B. B. Mandelbrot and S. Kirk-patrick, Solvable Fractal Family, and Its Possible Relation to the Backbone at Percolation,Phys. Rev. Lett. 47, (1981), 1771-1774.

[39] Bertrand, J., Bertrand, P., Ovarlez, J. The Mellin Transform, The Transforms and Applica-tions Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC,2000.

[40] G. Valiron, Lectures on the General Theory of Integral Functions, Private, Toulouse, 1923.[41] W. Arendt, R. Nittka, W. Peter, and F. Steiner, Weyl’s Law: Spectral Properties of the

Laplacian in Mathematics and Physics, in Mathematical Analysis of Evolution, Information,and Complexity, W. Arendt and W. P. Schleich (Eds), (Wiley, NY, 2009).

[42] Heinrich P. Baltes and Eberhard R. Hilf, Spectra of finite systems, Bibliographisches Insti-tut, Mannheim, 1976. A review of Weyl’s problem: the eigenvalue distribution of the waveequation for finite domains and its applications on the physics of small systems. MR0435624(55 #8582)

[43] Hermann Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Dif-ferentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math.Ann. 71 (1912), no. 4, 441–479, DOI 10.1007/BF01456804 (German). MR1511670

[44] Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23.MR0201237 (34 #1121)

[45] C. Gordon, D. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannianorbifolds, Inventiones Mathematicae, 110(1), (1992), 1-22. S, Rosenberg, The Laplacian on aRiemannian manifold, volume 31 of London Mathematical Society Student Texts. CambridgeUniversity Press, Cambridge, 1997.


[46] Gerald V. Dunne, Functional determinants in quantum field theory, J. Phys. A 41 (2008),no. 30, 304006, 15, DOI 10.1088/1751-8113/41/30/304006. MR2425830 (2009f:81126)

[47] Tohru Eguchi, Peter B. Gilkey, and Andrew J. Hanson, Gravitation, gauge theories and dif-ferential geometry, Phys. Rep. 66 (1980), no. 6, 213–393, DOI 10.1016/0370-1573(80)90130-1.MR598586 (81m:83002)

[48] S. Minakshisundaram, Zeta functions on the sphere, J. Ind. Math. Soc. 13, (1949), 41 andS. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace-

operator on Riemannian manifolds, Canadian J. Math. 1, (1949), 242-256.[49] S. A. Molcanov,Diffusion processes, and Riemannian geometry, Uspehi Mat. Nauk 30 (1975),

no. 1(181), 3–59 (Russian). MR0413289 (54 #1404)[50] E. Akkermans and G. Montambaux, Mesoscopic physics of electrons and photons (Cambridge

Univ. Press, 2007).[51] K. Stewartson and R.T. Waechter, On hearing the shape of a drum: further results, Proc.

Camb. Phil. Soc. 69, (1971), 353.[52] B. M. Hambly, Asymptotics for functions associated with heat flow on the Sierpinski carpet,

Canad. J. Math. 63 (2011), no. 1, 153–180, DOI 10.4153/CJM-2010-079-7. MR2779136(2012f:35548)

[53] Naotaka Kajino, Spectral asymptotics for Laplacians on self-similar sets, J. Funct. Anal. 258(2010), no. 4, 1310–1360, DOI 10.1016/j.jfa.2009.11.001. MR2565841 (2011j:31010)

[54] Robert S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series,Math. Res. Lett. 12 (2005), no. 2-3, 269–274. MR2150883 (2006e:28013)

[55] K. Hare, B. Steinhurst, A. Teplyaev and D. Zhou, Disconnected Julia sets and gaps inthe spectrum of Laplacians on symmetric finitely ramified fractals. to appear in the MRL,arXiv:1105.1747.

[56] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics I, (Pergamon, Oxford, 1980), R. Kubo,Statistical-Mechanical Theory of Irreversible Processes, Jour. Phys. Soc. Japan 12, (1957),570 and P. Martin and J. Schwinger, Theory of Many-Particle Systems. I. Phys. Rev. 115,(1959), 1342.

[57] J. M. Harrison and K. Kirsten, Zeta functions of quantum graphs, J. Phys. A 44 (2011),no. 23, 235301, 29, DOI 10.1088/1751-8113/44/23/235301. MR2800848 (2012i:81106)

[58] J. S. Dowker and Gerard Kennedy, Finite temperature and boundary effects in static space-times, J. Phys. A 11 (1978), no. 5, 895–920. MR0479266 (57 #18709)

[59] J. P. Chen, Statistical mechanics of Bose gas in Sierpinski carpets, arXiv:1202.1274.[60] E. Akkermans and D. Gittelman, Onset of Bose-Einstein condensation and superfluidity in

self-similar fractals, in preparation 2012.[61] E. Akkermans, O. Benichou, G V. Dunne, A. Teplyaev and R. Voituriez, Spatial Log Periodic

Oscillations of First-Passage Observables in Fractals, Phys. Rev. E 86, 061125 (2012).[62] Eric Akkermans, Thierry Bodineau, Bernard Derrida and Ohad Shpielberg, Universal current

fluctuations in the symmetric exclusion process and other diffusive systems, Europhys. Lett.103 (2013), 20001, 1–6. arXiv:1306.3145.

[63] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, volume 298of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1992 and E.Akkermans and K. Mallick, Geometrical description of vortices in Ginzburg-Landau billiards,in “Topological Aspects of low dimensional systems.” Les Houches Summer School (sessionLXIX), Springer (1999).

Department of Physics, Technion Israel Institute of Technology, Haifa 32000, Is-

rael

E-mail address: [email protected]


Spectral Algebra of the Chernov and Bogoslovsky FinslerMetric Tensors

Vladimir Balan

Abstract. The paper studies the Chernov and Bogoslovsky notable geometricstructures, which have been recently used as alternative models for extendedSpecial Relativity. The spectral theory of the associated metric tensor field isconsidered, and the Parafac tensor decomposition is obtained, in the case oflow dimensions.

Preliminaries

The recent interest to m−the root Finsler metrics originates in their quality ofproviding valid model candidates for Ecology (e.g., [1]), Diffusion Processes ([2])and Relativity ([20–22, 27]). The blending between the geometric properties ofthese metrics and the spectral algebraic data of the related subjacent tensors, wasrevealed both by algebraists ([23,24]), and geometers ([3,4]).

In the present work, we study the spectral data for the Finsler metric tensorfields - which generally extend the (pseudo-)Riemannian metric, by exhibiting thesame 0-homogeneity in the vector argument. Namely, we focus on the notable theChernov ([13]) and the Bogoslovsky ([7–9]) m−th root Finsler structures, whichhave recently been used as alternative candidates for relativistic models.

We start by providing a brief account of both the geometric Finsler and thealgebraic spectral backgrounds.

0.1. The Chernov and Bogoslovsky metric tensor fields. Generally, aFinsler structure is a pair (M,F ), where M is an n−dimensional real differentiablemanifold, and F : M → R is a mapping, called the fundamental Finsler functionor the Finsler norm). We shall denote by (x, y) ≡ (xi, ya) the local coordinates ofa point of the tangent space TM . The mapping F is subjected to the followingaxioms ([19]):

• F (x, y) > 0, ∀y ∈ TxM\{0x};• F (x, λy) = |λ|F (x, y), ∀λ ∈ R, ∀(x, y) ∈ TM ;

• F is continuous on TM and smooth on TM = TM\{0};

2010 Mathematics Subject Classification. Primary 65F30, 53B40; Secondary 15A18, 15A69.Key words and phrases. Finsler structure, Chernov metric, Bogoslovsky metric, spectra,

eigenvalues, eigenvalues, Parafac decomposition, best rank-one approximation.


23

24 VLADIMIR BALAN

• The coefficients of the y−Hessian

(0.1) gij =1

2

∂2F 2

∂yi∂yj

of the energy scalar field 12F

2 form a symmetric positive definite matrix

for all x ∈ M . These coefficients determine the Finsler metric tensor field1

g = gij(x, y)dyi ⊗ dyj .

Surprisingly, many Finsler-type models have been recently investigated, which onlypartially obey these restrictive requirements. The axioms are reconsidered in a re-laxed form, leaving way to pseudo-Finslerian extensions of the pseudo-Riemannianframework, as follows:

• F may be not necessarily non-negative;

• the absolute homogeneity may be replaced with the (weaker) positive-homogeneity property:

F (x, λy) = λF (x, y), ∀λ ∈ (0,∞) ⊂ R, ∀(x, y) ∈ TM ;

• the smoothness domain of the Finsler fundamental function F may be a

strict subset D ⊂ TM ;

• the matrix (gij(x, y))i,j∈1,n may be non-degenerate, but of constant sig-nature.

Among the existent intensively studied Finsler structures which comply to suchrelaxed requirements, we mention:

• the m−th root Finsler structures, for which the fundamental function hasthe form:2

F (x, y) = m√ai1 . . . aim(x) · yi1 · . . . · yim ,

where (ai1 . . . aim(x))i1,...,im∈1,n are the coefficients of an (0,m)−tensorfield on the manifold M and m ≥ 2. We note that for m = 2 and(aij)i,j∈1,n non-degenerate and of constant signature, one gets, in partic-

ular, the (pseudo-)Riemannian norm.

• the extended Randers structure, where

F (x, y) =√aij(x)yiyj + bi(x)yi,

where (aij(x))i,j∈1,n are the coefficients of a (pseudo-)Riemannian metric

tensor field, and (bk(x))k∈1,n are the coefficients of an 1-form on M .3

• the Kropina structure, an (α, β)−type Finsler metric ([10]), for which

F (x, y) =aij(x)yiyj

bi(x)yi,

with the ingredient tensors a and b considered as above.

1Throughout the paper, for brevity, we shall use the Einstein index summation conventionover repeated indices

2For consistency, for m even, it is customarily assumed that y ∈ TxM is always chosen suchthat either the transvected result under the root sign be positive, or the its absolute value isimplicitly considered.

3The non-degeneracy condition imposes here, as unique requirement, ||b||a ≡√

aijbibj �=0,∀x ∈ M , where aij is the dual tensor field given by aisasj = δij ,∀i, j ∈ 1, n.

SPECTRAL ALGEBRA OF FINSLER METRIC TENSORS 25

Among the m−th root Finsler structures, one can easily identify three remarkablemetric spaces considered in recent alternative models of Relativity, of which thefirst two are the subject of concern for our forecoming PCA spectral analysis:

• the Chernov structure:4

(0.2) F (y) =

(n∑

i=1

y1 · . . . yi . . . · yn

)1/m

, m = n− 1;

• the Bogoslovsky structure:

(0.3) F (y) =

(y1 · . . . · yn ·

n∑i=1

yi

)1/m

, m = n + 1;

• the Berwald-Moor structure:

F (y) = (y1 · . . . · yn)1/m , m = n.

We note that each m−the root structure uniquely defines its symmetric structuraltensor A ≡ (ai1 . . . aim(x))i1,...,im∈1,n, by means of the relation5

Fm = A(y, . . . , y︸︷︷︸m times

).

1. Spectral theory prerequisites

The applications of the spectral data is subject of intensive study ([11,12,14,15,25]), which includes as a proficient tool the recently developed E−determinantstheory ([16]). The spectra of the Chernov and Bogoslovsky structural tensors wereinvestigated in ([3–5]). In the following, we shall develop the spectral theory of theFinsler metric tensor associated to these structures.

Generally, for a given symmetric tensor field A ∈ T 0m(M), one can define ([24,

26]) its Z−eigenvalues λ ∈ R and the corresponding Z−eigenvectors y ∈ TxM bythe relation:6

Aym−1 = λy, ||y||2 ≡(

n∑i=1

y2i

)1/2

= 1,

and its H−eigenvalues λ ∈ R and the corresponding H−eigenvectors y ∈ TxM bythe relations:

(Aym−1)i = λyi, ∀i ∈ 1, n.

We note first that since the metric tensor (gij(y))i,j=1,n is of order two and is sym-metric, the HO-SVD analysis basically reduces to the classic SVD-PCA framework.As well, in this case, the Z−eigenvalues and the H−eigenvalues coincide with theclassical eigenvalues of the matrix associated to the metric tensor. As well, usingthe formula (0.1), one easily gets the following result:

4The hat denotes exclusion of the corresponding hatted term.5Taking into account that the two investigated models are of locally Minkowski type (i.e.,

x−independent) and that the reasonings are basically performed within an arbitrary given fiberTxM of the tangent bundle, we shall write all the tensor indices in subscript, reserving the super-script for powers.

6Here Aym−1 is a for A( · , y, . . . , y︸︷︷︸m−1 times

).

26 VLADIMIR BALAN

Lemma 1.1. a) The metric tensor of the Chernov Finsler structure (0.2) isexplicitly given by:

(1.1) gij =1

n− 1S

2(2−n)n−1

[S · Sij(1 − δij) +

3 − n

n− 1· SiSj

], i, j = 1, n,

where ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

S = y1 · . . . yi . . . · yn

Si =∂S

∂yi=

∑k∈1,n\{i}

y1 · . . . yi . . . yk . . . · yn

Sij =∂2S

∂yi∂yj=

∑k∈1,n\{i,j}

y1 · . . . yi . . . yj . . . yk . . . · yn.

b) The metric tensor of the Bogoslovsky structure (0.3) has the components:

(1.2) gij =1

n + 1S

−2nn+1

[S · Sij +

1 − n

n + 1· SiSj

], i, j = 1, n,

where we denoted⎧⎪⎪⎪⎨⎪⎪⎪⎩S = y1 · . . . · yn · (y1 + . . . + yn)

Si = y1 · . . . · yn + (y1 + . . . + yn)y1 · . . . yi . . . · ynSij = y1 · . . . yi . . . · yn + y1 · . . . yj . . . · yn

+(1 − δij)(y1 + . . . + yn)y1 · . . . yi . . . yj . . . · yn.We use the spectral data associated to the metric tensor, in order to determine

its best rank-one approximation.

2. Spectral results for low dimensions

We note that, while determining the spectral data of the metrics, the degreeof complexity increases as the dimension n goes high. In the following, we considerthe two Finsler structures for lower, more tractable dimensions, and discuss thesignature of the spectrum.

2.1. The Chernov metric tensor. The most tractable Chernov case is n = 3(m = 2); the Finsler metric is, in this case, just the Minkovski norm, subject to achange of basis. However, in this case one may straightforward find the spectraldata, and use this to determine the Candecomp decomposition of the metric tensor.More precisely, we have

Theorem 2.1. Within the low-dimensional case n = 3 (m = 2), the followinghold true:

a) The Chernov metric (0.2) has the coefficients [g] = 12

(0 1 11 0 11 1 0

)and the asso-

ciated spectral data is σZ(g) = σH(g) = {1,− 12 ,−

12}, with7

Sλ=1 = {±v1}, Sλ=− 12

= { u / ||u|| | u ∈ Span(v2, v3}},and

v1 =1√3(1, 1, 1)t, v2 =

1√2(1, 0,−1)t, v3 =

1√6(1,−2, 1)t.

7We shall indicate only the generating unit vectors within the corresponding eigenspaces.


b) The Parafac/Candecomp decomposition of g has the form

g = 1 · ω1 ⊗ ω1 − 12ω2 ⊗ ω2 − 1

2ω3 ⊗ ω3

⇔ [g] ≡ 12

(0 1 11 0 11 1 0

)= 1 · v1 · vt1 − 1

2 · v2 · vt2 − 12 · v3 · vt3,

where ωi = vti , i = 1, 3.

c) The best rank-one approximation of the tensor g is:8

g =1

2

∑i �=j

dxi ⊗ dxj ∼ λ∗ · v∗ ⊗ v∗ =1

3

∑i,j=1,3

dxi ⊗ dxj ,

with λ∗ = λ1 = maxσ(g) = 1, v∗ = v1 = 1√3(1, 1, 1).

However, for the case n = 4 (m = 3), we consider a scaled version the Chernovtensor g, whose associated symmetric matrix [h] = 9F 4[g] has the 10 essentialentries:9 ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

h11 = −(cb + bd + cd)2

h22 = −(ca + ad + cd)2

h33 = −(ab + ad + bd)2

h44 = −(ab + ca + cb)2

h12 = 4abcd + 2ab(c2 + d2) + 2cd(c + d)(a + b) − c2d2

h13 = 4abcd + 2ac(b2 + d2) + 2bd(b + d)(a + c) − b2d2

h14 = 4abcd + 2ad(b2 + d2) + 2bc(b + c)(a + d) − b2c2

h23 = 4abcd + 2bc(a2 + d2) + 2ad(a + d)(b + c) − a2d2

h24 = 4abcd + 2bd(a2 + c2) + 2ac(a + c)(b + d) − a2c2

h34 = 4abcd + 2cd(a2 + b2) + 2ab(a + b)(c + d) − a2b2

which exhibit a symmetry pattern produced by the total symmetry in the compo-nents of y = (a, b, c, d) of the fundamental function. One may easily check that thesign of the eigenvalues depends on the flag-change, e.g.

sign(σ(g))y=(1,1,1,1) = (+,−,−,−);

sign(σ(g))y=(2,−1,−1,2) = (+,−,−,+).

hence, the Chernov metric g does not have constant signature (is flag-dependent),and therefore within any fiber of the bundle of Finsler vector fields

(TM ×M TM, pr1, TM) the structure provided by g is Riemannian or pseudo-Riemannian, depending on direction only.

2.2. The Bogoslovsky metric tensor. The Bogoslovsky metric tensor (1.2)has, for n = 2 (m = 3), the form:

g =1

9F 4[b2(2a2 + 2ab− b2)dx1 ⊗ dx1 + a2(2b2 + 2ab− a2)dx2 ⊗ dx2+

+ab(a2 + ab + b2)(dx1 ⊗ dx2 + dx2 ⊗ dx1),

8The distance is provided here by the Frobenius norm of the difference of the tensors, whichis computed by means of the Euclidean flat scalar product.

9For the sake of simplicity, within the flag (x, y) ∈ TM , we denoted the vector componentsas y = (a, b, c, d).

28 VLADIMIR BALAN

with F given in (0.3), where we denoted for brevity y = (y1, y2) = (a, b). ThoughTrace(g) has no constant sign, the second Jacobi minor of the metric tensor,

Δ2 = det[g] =1

81F 8[−18a2b2(a2 + ab + b2)(a + b)2]

is always non-positive, and the Finsler structure admits opposite eigenvalues. Thesmoothness domain of F needs the removal of the three intersecting straight linesΓ : ab(a + b) = 0. The tensor is of Minkowski type over TM with Γ removed fromeach tangent fiber. Regarding the spectral data related to the metric tensor (1.2),we can state the following

Theorem 2.2. Within the low-dimensional case n = 2 (m = 3), the followinghold true:

a) The Bogoslovsky metric (0.3) has the associated spectral data

σZ(g) = σH(g) =

{λ± =

1

2[ab(a2 + b2) − (a2 − b2)2 ±

√Δ]

},

where we denoted

Δ = a8 + b8 − 4ab(a6 + b6) + 68a2b2(a4 + b4) + 228a3b3(a2 + b2) + 314a4b4 ≥ 0,

and

Sλ+= {±v+}, Sλ− = {±v−},

where

v+ =1

2

(A +

√Δ, A−

√Δ)t

, v− =

(2B

C −√

Δ,

2B

C +√

Δ

)t

,

with the abbreviations⎧⎪⎨⎪⎩A = 4a2b2 + 2ab(a2 + b2 − (a4 + b4)

B = ab(4a2 + 7ab + 4b2)

C = a4 − b4 + 2ab(b2 − a2).

b) The Bogoslovsky metric tensor admits the Parafac/Candecomp decomposi-tion

g = λ+ · ω+ ⊗ ω+ + λ−ω− ⊗ ω− ⇔ [g] = λ+ · v+ · vt+ + λ− · v− · vt−,

where ωi = vti , i = 1, 2.

c) The best rank-one approximation of the tensor g is g ∼ λ+ · ω+ ⊗ ω+.

For the case n = 3 (m = 4), the computational load increases; the essentialcoefficients develop symmetry patterns due to the total symmetry in F (0.3), and,e.g., denoting y = (a, b, c), one has10

g11 = −k(2bc + c2 + b2)b2c2, g12 = k{[3(ac + ab + bc) + 2(a2 + b2) + c2]abc2},where k = 1

16F 2 . The characteristic polynomial has the form

P (λ) = −λ3 + I · λ2 − J · λ + det([g]),

10The remaining components of g have corresponding similar shapes, obtained by appropri-ately reconsidering the roles of the variables a, b, c vs. the indexes of the metric tensor.


where⎧⎪⎨⎪⎩I = k(−a4b2 − 2a3b3 − a2b4 − a4c2 − 2a3c3 − a2c4 − 2b3c3 − b2c4 − b4c2)

J = k2[−4c2b2a2(ca2 + a2b + b2a + c2a + c2b + b2c− bca)(a + b + c)3]

det([g]) = k3[16a4c4b4(a2 + ab + ac + bc + b2 + c2)(a + b + c)4],

whence one easily concludes that at least one eigenvalue of the Bogoslovsky tensoris non-negative.

3. Conclusions

The explicit form of the Chernov and Bogoslovsky Finsler metric tensor wasobtained. The spectral data was obtained for the in low dimensions, and conse-quently, the Candecomp/Parafac decomposition was derived. The best rank-oneapproximation was pointed out, in these particular cases.

References

[1] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The theory of sprays and Finsler spaceswith applications in physics and biology, Fundamental Theories of Physics, vol. 58, KluwerAcademic Publishers Group, Dordrecht, 1993. MR1273129 (95e:53094)

[2] L. Astola and L. Florack, Finsler Geometry on higher order tensor fields and applications tohigh angular resolution diffusion imaging. LNCS 5567 (2009), 224–234.

[3] V. Balan, Numerical multilinear algebra of symmetric m-root structures. Spectral proper-ties and applications. In Symmetry: Culture and Science, Part 2; Geometric Approaches toSymmetry 2010; Budapest, Hungary, 21, 1–3 (2010), 119–131.

[4] V. Balan, Spectra of multilinear forms associated to notable m-root relativistic models. Lin-ear Algebra and Appl. (LAA), online http://dx.doi.org/10.1016/j.laa.2011.06.033;436, 1, 1(2012), 152–162.

[5] Vladimir Balan and Nikolay Perminov, Applications of resultants in the spectral m-rootframework, Appl. Sci. 12 (2010), 20–29. MR2609373 (2011e:53015)

[6] Aurel Bejancu, Coisotropic submanifolds of pseudo-Finsler manifolds, Facta Univ. Ser. Math.Inform. 15 (2000), 57–68. Dedicated to Professor Radosav Z. Jorjevic for his 65th birthday(Nis, 1998). MR2024619 (2004j:53096)

[7] George Yu. Bogoslovsky, Dynamic rearrangement of vacuum and the phase transitions inthe geometric structure of space-time, Int. J. Geom. Methods Mod. Phys. 9 (2012), no. 1,1250007, 20, DOI 10.1142/S0219887812500077. MR2891521

[8] George Yu. Bogoslovsky, Rapidities and observable 3-velocities in the flat Finslerian eventspace with entirely broken 3D isotropy, SIGMA Symmetry Integrability Geom. Methods Appl.4 (2008), Paper 045, 21, DOI 10.3842/SIGMA.2008.045. MR2425647 (2009f:53120)

[9] G.Yu. Bogoslovsky, H.F. Goenner, On the possibility of phase transitions in the geometric

structure of space-time, Phys. Lett. A 244, (1998), 222–228; arXiv:gr-qc/9804082v1.[10] Ioan Bucataru, Nonholonomic frames in Finsler geometry, Balkan J. Geom. Appl. 7 (2002),

no. 1, 13–27. MR1940562 (2003k:53087)[11] K. C. Chang, Kelly Pearson, and Tan Zhang, On eigenvalue problems of real symmetric

tensors, J. Math. Anal. Appl. 350 (2009), no. 1, 416–422, DOI 10.1016/j.jmaa.2008.09.067.MR2476927 (2009j:15034)

[12] Zhe Chang and Xin Li, Modified Newton’s gravity in Finsler space as a possible al-ternative to dark matter hypothesis, Phys. Lett. B 668 (2008), no. 5, 453–456, DOI10.1016/j.physletb.2008.09.010. MR2463269 (2009k:83075)

[13] V.M. Chernov, On defining equations for the elements of associative and commutative alge-bras and on associated metric forms, In: Space-Time Structure. Algebra and Geometry, D.G.Pavlov, Gh. Atanasiu, V. Balan (eds), Lilia Print, Moscow 2007, 189–209.

[14] P. Comon, Block methods for channel identification and source separation. IEEE Symposiumon Adaptive Systems for Signal Process, Commun. Control (Lake Louise, Alberta, Canada,Oct 1-4, 2000. Invited Plenary), 87–92.

30 VLADIMIR BALAN

[15] R. Coppi and S. Bolasco (eds.), Multiway data analysis, North-Holland Publishing Co., Am-sterdam, 1989. Papers from the International Meeting on the Analysis of Multiway DataMatrices held in Rome, March 28–30, 1988. MR1088948 (91j:62003)

[16] S. Hu, Z.-H. Huang, C. Ling, L. Qi, E-determinants of tensors, arXiv:1109.0348v3 [math.NA]2 Sep 2011.

[17] Eleftherios Kofidis and Phillip A. Regalia, Tensor approximation and signal processing ap-plications, I (Boulder, CO, 1999), Contemp. Math., vol. 280, Amer. Math. Soc., Providence,

RI, 2001, pp. 103–133, DOI 10.1090/conm/280/04625. MR1850404 (2002f:93039)[18] Makoto Matsumoto and Katsumi Okubo, Theory of Finsler spaces with mth root metric: con-

nections and main scalars, Tensor (N.S.) 56 (1995), no. 1, 93–104. 3rd International Confer-ence on Differential Geometry and its Applications (Athens, 1994). MR1376173 (97e:53033)

[19] Radu Miron and Mihai Anastasiei, The geometry of Lagrange spaces: theory and applications,Fundamental Theories of Physics, vol. 59, Kluwer Academic Publishers Group, Dordrecht,1994. MR1281613 (95f:53120)

[20] D.G. Pavlov, Four-dimensional time. Hypercomplex Numbers in Geom. Phys. 1 (1) (2004),31–39.

[21] D.G. Pavlov, Generalization of scalar product axioms. Hypercomplex Numbers in Geom.Phys. 1 (1) (2004), 5–18.

[22] D.G. Pavlov, S.S. Kokarev, Conformal gauges of the Berwald-Moor Geometry and theirinduced non-linear symmetries (in Russian), Hypercomplex Numbers in Geom. Phys. 2 (10)(2008), 5, 3–14.

[23] Liqun Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005),no. 6, 1302–1324, DOI 10.1016/j.jsc.2005.05.007. MR2178089 (2006j:15031)

[24] Liqun Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneouspolynomial and the algebraic hypersurface it defines, J. Symbolic Comput. 41 (2006), no. 12,1309–1327, DOI 10.1016/j.jsc.2006.02.011. MR2271327 (2007i:14064)

[25] L. Qi, The Spectral Theory of Tensors, (Rough Version), arxiv.org/pdf/1201.3424, 17 Jan2012.

[26] Liqun Qi, Wenyu Sun, and Yiju Wang, Numerical multilinear algebra and its applications,Front. Math. China 2 (2007), no. 4, 501–526, DOI 10.1007/s11464-007-0031-4. MR2346433

(2008g:15051)[27] Ian W. Roxburgh, Post-Newtonian tests of quartic metric theories of gravity, Rep.

Math. Phys. 31 (1992), no. 2, 171–178, DOI 10.1016/0034-4877(92)90011-O. MR1227039(94d:83061)

[28] H. Shimada, On Finsler spaces with the metric L = m√

ai1i2...imyi1 . . . ym, Tensor 3 (1979),366-372.

University Politehnica of Bucharest, Faculty of Applied Sciences, Department

Mathematics-Informatics, Splaiul Independentei 313, RO-060042, Bucharest, Romania.



Local Multifractal Analysis

Julien Barral, Arnaud Durand, Stephane Jaffard, and Stephane Seuret

Abstract. We introduce a local multifractal formalism adapted to functions,measures or distributions which display multifractal characteristics that canchange with time, or location. We develop this formalism in a general frame-work and we work out several examples of measures and functions where thissetting is relevant.

1. Introduction

Let f denote a function, a positive Radon measure, or, more generally, a distri-bution defined on a nonempty open set Ω. One often associates with f a pointwiseexponent, denoted by hf (x), which allows to quantify the local smoothness of f atx. On the mathematical side, the purpose of multifractal analysis is to determinethe fractal dimensions of the level sets of the function x �→ hf (x). Let

EH = {x : hf (x) = H}.The multifractal spectrum of f (associated with the regularity exponent hf ) is

df (H) = dim EH

(where dim denotes the Hausdorff dimension, see Definition 3). Multifractal spectrayield a description of the local singularities of the function, or measure, underconsideration.

Regularity exponents (and therefore the multifractal spectrum) of many func-tions, stochastic processes, or measures used in modeling can be theoretically de-termined directly from the definition. However, usually, one cannot recover theseresults numerically on simulations, because the exponents thus obtained turn outto be extremely erratic, everywhere discontinuous functions. It is for instance thecase of Levy processes [33], or of multiplicative cascades (see the book [10], and,in particular the review paper by J. Barral, A. Fan, and J. Peyriere) so that adirect determination of hf (x) leads to totally unstable computations. A fortiori,the estimation of the multifractal spectrum from its definition is unfeasible. The

2010 Mathematics Subject Classification. Primary 28A80, 60GXX; Secondary 37C40, 42C40,60J75.

Key words and phrases. Multifractal analysis, Hausdorff dimensions, stochastic processes,

geometric measure theory, wavelets, ergodic theory.The third author was supported in part by ANR AMATIS ANR-BS01-011-01.The first, second and fourth authors were supported in part by ANR MUTADIS ANR-11-

JS01-0009.


31

32 J. BARRAL, A. DURAND, S. JAFFARD, AND S. SEURET

multifractal formalism is a tentative way to bypass the intermediate step of thedetermination of the pointwise exponent, by relating the multifractal spectrumdirectly with averaged quantities that are effectively computable on experimentaldata. Such quantities can usually be interpreted as global regularity indices. Forinstance, the first one historically used in the function setting (ζf (p), referred to asKolmogorov scaling function) can be defined as follows; for the sake of simplicity,we only consider in this introduction the function setting and we assume here thatthe functions considered are defined on the whole Rd.

Recall that Lipschitz spaces are defined, for s ∈ (0, 1), and p ∈ [1,∞], byf ∈ Lip(s, Lp(Rd)) if f ∈ Lp and ∃C > 0 such that ∀h > 0,

(1.1) ‖ f(x + h) − f(x) ‖Lp≤ Chs.

(the definition for larger s requires the use of higher order differences, and theextension to p < 1 requires to replace Lebesgue spaces by Hardy spaces, see [35]).Then

(1.2) ζf (p) = p · sup{s : f ∈ Lip(s, Lp(Rd))}.Initially introduced by U. Frisch and G. Parisi in the mid 80s, the purpose of

multifractal analysis is to investigate the relationships between the pointwise regu-larity information supplied by df (H) and the global regularity information suppliedby ζf (p). Note that these quantities can be computed on the whole domain of defi-nition Ω of f , or can be restricted to an open subdomain ω ⊂ Ω. A natural questionis to understand how they depend on the region ω where they are computed. It isremarkable that, in many situations, there is no dependency at all on ω; we willthen say that the corresponding quantity is homogeneous. It is the case for severalclasses of stochastic processes. For instance, sample paths of Levy processes (andfields) [24,25,33], the Levy processes in multifractal time studied in [15], and frac-tional Brownian motions (FBM) almost surely have homogeneous Holder spectra,and, in the case of FBM, the Legendre spectrum also is homogeneous, see [34,36].In the random setting, it is also the case for many examples of multiplicative cas-cades, see [12]. Many deterministic functions or measures also are homogeneous(homogeneity is usually not explicitly stated as such in the corresponding papers,but is implicit in the determination of the spectra). This is for instance the case forself-similar or self-conformal measures when one assumes the so-called open set con-dition, or for Gibbs measures on conformal repellers (see for instance [50,51,53]).It is also the case for many applications, for instance the Legendre spectra raisingfrom natural experiments (such as turbulence, see [2,4] and references therein) arefound to be homogeneous.

On the opposite, many natural objects, either theoretical or coming from realdata, have been shown to be non-homogeneous : Their multifractal characteristicsdepend on the domain Ω over which they are observed:

• It is the case of some classes of Markov processes, see [9] and Section 5.2,and also of some Markov cascades studied in [8].

• Some self-similar measures when the open set condition is relaxed intothe weak-separation condition may satisfy the multifractal formalism onlywhen restricted to some intervals (see [30,31,59,60]).

• In applications, many types of signals, which have a human origin, canhave multifractal characteristics that change with time: A typical exampleis supplied by finance data, see [2], where changes can be attributed to

LOCAL MULTIFRACTAL ANALYSIS 33

outside phenomena such as political events, but also to the increasingsophistication of financial tools, which may lead to instabilities (financialcrises) and implies that some characteristic features of the data, possiblycaptured by multifractal analysis, evolve with time. This situation is alsonatural in image analysis because of the occlusion phenomenon; indeed, anatural image is a patchwork of textures with different characteristics, sothat its global spectrum of singularities reflects the multifractal nature ofeach component, and also of the boundaries (which may also be fractal)where discontinuities appear. Note that the notion of local Hausdorffdimension which plays a central role in this section, has been introducedin [39] precisely with the motivation of image analysis.

• Functions spaces with varying smoothness have been introduced moti-vated by the study of the relationship between general pseudo-differentialoperators and later by questions arising in PDEs, see [55] for a reviewon the subject; scaling functions with characteristics depending on thelocation are then the natural tool to measure optimal regularity in thiscontext. We will investigate this relationship in Section 7.

This paper will provide new examples of multifractal characteristics which de-pend on the domain of observation. In such situations, the determination of a localspectrum of singularities for each “component” ω ⊂ Ω will carry more informationthan the knowledge of the “global” one only. A natural question is to understandhow the different quantities which we have introduced depend on the region ω wherethey are computed.

Some of the notions studied in this paper have been already introduced in[9]; let us also mention that a local Lq-spectrum was independently introducedin [40,41], where the authors studied studied this notion for measures in doublingmetric spaces (as well as the notion of local homogeneity) and obtained, for instance,upper bounds for the dimensions of the sets of points with given lower and upperlocal dimensions using these local concepts. The goal of their approach in [40]was to investigate conical density and porosity questions. In our paper, on topof measures, we also deal with functions, get comparable upper bounds for thecorresponding multifractal spectra, and the examples we develop are very different.

Let us now make precise the notion we started with, namely pointwise regu-larity. The two most widely used exponents are the pointwise Holder exponent of(locally bounded) functions and the local dimension of measures. In the following,B(x0, r) denotes the open ball of center x0 and radius r.

Definition 1. Let μ be a positive Radon measure defined on an open subsetΩ ⊂ Rd. Let x0 ∈ Ω and let α ≥ 0. The measure μ belongs to hα(x0) if

(1.3) ∃C,R > 0, ∀r ≤ R, μ(B(x0, r)) ≤ Crα.

Let x0 belong to the support of μ. The lower local dimension of μ at x0 is

(1.4) hμ(x0) = sup{α : μ ∈ hα(x0)} = lim infr→0+

log μ(B(x0, r))

log r.

We now turn to the case of locally bounded functions. In this setting, the notioncorresponding to the lower local dimension is the pointwise Holder regularity.

Definition 2. Let x0 ∈ Rd and let α ≥ 0. Let f : Ω → R be a locally boundedfunction; f belongs to Cα(x0) if there exist C,R > 0 and a polynomial P of degree


at most α such that

(1.5) if |x− x0| ≤ R, then |f(x) − P (x− x0)| ≤ C|x− x0|α.The Holder exponent of f at x0 is

(1.6) hf (x0) = sup{α : f ∈ Cα(x0)}.

This paper is organized as follows:In Section 2, we recall the notions of dimensions that we will use (both in the

global and local case), we prove some basic results concerning the notion of localHausdorff dimension, and we recall the wavelet characterization of pointwise Holderregularity.

In Section 3 we recall the multifractal formalism on a domain in a generalabstract form which is adapted both to the function and the measure setting; thenthe corresponding version of local multifractal formalism is obtained, and we drawits relationship with the notion of germ space.

In Section 4, we investigate more precisely the local multifractal analysis ofmeasures, providing natural and new examples where this notion indeed containsmore information than the single multifractal spectrum. In particular, we introducenew cascade models the local characteristics of which change smoothly with thelocation; here again, we show that the local tools introduced in Section 2 yield theexact multifractal characteristics of these cascades.

In Section 5, we review the results concerning some Markov processes whichdo not have stationary increments; then we show that the notion of local spectrumallows to recover the exact pointwise behavior of the Multifractional BrownianMotion (in contradistinction with the usual “global” multifractal formalism).

In Section 6 we consider other regularity exponents characterized by dyadicfamilies, and show how they can be characterized in a similar way as the previousones, by log-log plot regressions of quantities defined on the dyadic cubes.

Finally, in Section 7 the relationship between the local scaling function andfunction spaces with varying smoothness is developed.

2. Properties of the local Hausdorff dimension and the localmultifractal spectrum

2.1. Some notations and recalls. In order to make precise the differentnotions of multifractal spectra, we need to recall the notion of dimension which willbe used.

Definition 3. Let A ⊂ Rd. If ε > 0 and δ ∈ [0, d], we denote

Mδε = inf

R

(∑i

|Ai|δ),

where R is an ε-covering of A, i.e. a covering of A by bounded sets {Ai}i∈N ofdiameters |Ai| ≤ ε. The infimum is therefore taken on all ε-coverings R.

For any δ ∈ [0, d], the δ-dimensional Hausdorff measure of A is

mesδ(A) = limε→0

Mδε .

There exists δ0 ∈ [0, d] such that

∀δ < δ0, mesδ(A) = +∞ and ∀δ > δ0, mesδ(A) = 0;


this critical value δ0 is called the Hausdorff dimension of A, and is denoted bydim(A). By convention, we set dim(∅) = −∞.

In practice, obtaining lower bounds for the Hausdorff dimension directly fromthe definition involve considering all possible coverings of the set, and is thereforenot practical. One rather uses the mass distribution principle which involves insteadthe construction of a well-adapted measure.

Proposition 1. Let A ⊂ Rd and let μ be a Radon measure such that μ(A) > 0;

if ∀x ∈ A, lim supr→0

μ(B(x, r))

rs≤ C then Hs(A) ≥ μ(A)

c.

We will see in Section 3.2 a local version of this result.Apart from the Hausdorff dimension, we will also need another notion of di-

mension: The packing dimension which was introduced by C. Tricot, see [61]:

Definition 4. Let A be a bounded subset of Rd; if ε > 0, we denote byNε(A) the smallest number of sets of radius ε required to cover A. The lower boxdimension of A is

dimB(A) = lim infε→0

logNε(A)

− log ε.

The packing dimension of a set A ⊂ Rd is

(2.1) dimp(A) = inf

{supi∈N

(dimBAi : A ⊂

∞⋃i=1

Ai

)}(the infimum is taken over all possible partitions of A into a countable collectionAi).

2.2. Local Hausdorff dimension. In situations where the spectra are nothomogeneous, the purpose of multifractal analysis is to understand how they changewith the location where they are considered. In the case of the multifractal spec-trum, this amounts to determine how the Hausdorff dimension of the set Ef (H)changes locally. This can be performed using the notion of local Hausdorff dimen-sion, which can be traced back to [39] (see also [8] where this notion is shown tobe fitted to the study of deranged Cantor sets).

Definition 5. Let A ⊂ Rd, and x ∈ Rd. The local Hausdorff dimension of Aat x is the function defined by

(2.2) ∀x ∈ A, dim(A, x) = limr→0

dim(A ∩B(x, r)).

Remarks:

• The limit exists because, if Ω1 ⊂ Ω2, then dim(Ω1) ≤ dim(Ω2); thereforethe right-hand side of (2.2), being a non-negative increasing function of r,has a limit when r → 0.

• We can also consider this quantity as defined on the whole Rd, in whichcase, it takes the value −∞ outside of A.

• The same definition allows to define a local dimension, associated withany other definition of fractional dimension; one gets for instance a notionof local packing dimension.


The following result shows that the local Hausdorff dimension encapsulates allthe information concerning the Hausdorff dimensions of the sets of the form A∩ω,for any open set ω.

Proposition 2. Let A ⊂ Rd; then for any open set ω which intersects A,

(2.3) dim(A ∩ ω) = supx∈ω

dim(A, x).

Proof. For r small enough, Br ⊂ ω; it follows that

∀x ∈ ω, dim(A, x) ≤ dim(A ∩ ω),

and therefore supx∈ω

dim(A, x) ≤ dim(A ∩ ω).

Let us now prove the converse inequality. Let (Kn)n∈N be an increasing se-quence of compact sets such that ∪Kn = ω; then

dim(A ∩ ω) = limn→∞

dim(A ∩Kn).

Let δ > 0 be given; then

∀x ∈ Kn, ∃r(x) > 0, dim(A ∩B(x, r)) − dim(A, x) ≤ δ.

We extract a finite covering of Kn from the collection {B(x, r(x))}x∈Knwhich yields

a finite number of points x1, · · ·xN ∈ ω such that Kn ⊂⋃B(xi, r(xi)); thus

dim(A ∩Kn) ≤ supi=1,··· ,N

dim(A ∩B(xi, r(xi)))

≤ supi=1,··· ,N

dim(A, xi) + δ ≤ supx∈ω

dim(A, x) + δ.

Taking δ → 0 and N → ∞ yields the required estimate. �Proposition 2 implies the following regularity for the local Hausdorff dimension.

Corollary 1. Let A be a given subset of Rd; then the function x �→ dim(A, x)is upper semi-continuous.

Proof. We have

dim(A, x) = limr→0

dim(A ∩B(x, r)) = limr→0

supy∈B(x,r)

dim(A, y) = lim supy→x

dim(A, y).

�2.3. Wavelets and wavelet leaders. In Section 3 we will describe a general

framework for deriving a multifractal formalism adapted to pointwise regularityexponents. The key property of these exponents that we will need is that they arederived from log-log plot regressions of quantities defined on the dyadic cubes. Letus first check that it is the case for the pointwise exponent of measures.

Recall that a dyadic cube of scale j ∈ Z is of the form

(2.4) λ =

[k12j

,k1 + 1

2j

)× · · · ×

[kd2j

,kd + 1

2j

),

where k = (k1, . . . kd) ∈ Zd. Each point x0 ∈ Rd is contained in a unique dyadiccube of scale j, denoted by λj(x0).

Let 3λj(x0) denote the cube with the same center as λj(x0) and three timeswider; it is easy to check that (1.3) and (1.4) can be rewritten as

hμ(x0) = lim infj→+∞

log μ(3λj(x0))

log 2−j.


We now show that the Holder exponent of a function can be recovered in asimilar way, from quantities derived from wavelet coefficients. Recall that orthonor-mal wavelet bases on Rd are of the following form: There exist a function ϕ and2d − 1 functions ψ(i) with the following properties: The ϕ(x− k) (k ∈ Zd) and the2dj/2ψ(i)(2jx− k) (k ∈ Zd, j ∈ Z) form an orthonormal basis of L2(Rd). This basisis r-smooth if ϕ and the ψ(i) are Cr and if the ∂αϕ, and the ∂αϕψ(i), for |α| ≤ r,have fast decay. Therefore, ∀f ∈ L2,

(2.5) f(x) =∑k∈Zd

ckϕ(x− k) +∞∑j=0

∑k∈Zd

∑i

cij,kψ(i)(2jx− k);

the ck and cij,k are the wavelet coefficients of f :

(2.6) cij,k = 2dj∫Rd

f(x)ψ(i)(2jx− k)dx, and ck =

∫Rd

f(x)ϕ(x− k)dx.

Note that (2.5) and (2.6) make sense even if f does not belong to L2; indeed,when using smooth enough wavelets, (2.6)can be interpreted as a duality productbetween smooth functions (the wavelets) and distributions, and (2.5) converges inthe corresponding function space (either for the strong or, sometimes, only theweak-∗ topology).

Instead of the three indices (i, j, k), wavelets will be indexed by dyadic cubesas follows: Since the wavelet index i takes 2d − 1 values, we can assume that ittakes values in {0, 1}d − (0, . . . , 0); we will use the notations

λ (= λ(i, j, k)) =k

2j+

i

2j+1+

[0,

1

2j+1

)d

, cλ = cij,k, ψλ(x) = ψ(i)(2jx− k).

Note that the cube λ which indexes the wavelet gives information about its loca-tion and scale; if one uses compactly supported wavelets, then ∃C > 0 such thatsupp (ψλ) ⊂ C · λ.

Finally, Λj will denote the set of dyadic cubes λ which index a wavelet of scale

j, i.e. wavelets of the form ψλ(x) = ψ(i)(2jx − k) (note that Λj is a subset of thedyadic cubes of side 2j+1). We take for norm on Rd

if x = (x1, . . . , xd), |x| = supi=1,...,d

|xi|;

so that the diameter of a dyadic cube of side 2−j is exactly 2−j .In the following, when dealing with Holder regularity of functions, we will

always assume that, if a function f is defined on an unbounded set Ω, then it hasslow increase, i.e. it satisfies

∃C,N > 0 |f(x)| ≤ C(1 + |x|)N ;

and, if Ω �= Rd, then the wavelet basis used is compactly supported, so that, ifx0 ∈ Ω, then the wavelet coefficients “close” to x0 are well defined for j largeenough.

Let f be a locally bounded function, with slow increase. The pointwise Holderregularity of f is characterized in terms of the wavelet leaders of f :

(2.7) dλ = supλ′⊂3λ

|cλ′ |.

The assumptions we made on f imply that wavelet leaders are well defined andfinite.


We note dj(x0) = dλj(x0). The following result allows to characterize the Holderexponent by the decay rate of the dλj(x0) when j → +∞, see [35].

Proposition 3. Let α > 0 and let ψλ be an orthonormal basis with regularityr > α. If there exists ε > 0 such that f ∈ Cε(Ω), then

(2.8) ∀x0, hf (x0) = lim infj→+∞

log dλj(x0)

log 2−j.

Hence, the pointwise Holder exponent can be computed from a dyadic family.This is also the case for the lower dimension of a measure μ. Indeed, it is easy tocheck that (1.3) and (1.4) can be rewritten as

hμ(x0) = lim infj→+∞

log μ(3λj(x0))

log 2−j.

3. A local multifractal formalism for a dyadic family

3.1. Multifractal analysis on a domain Ω.

Definition 6. Let Ω be a non-empty open subset of Rd. A collection ofnonnegative quantities E = (eλ) indexed by the set of dyadic cubes λ ⊂ Ω is calleda dyadic function on Ω.

The choice of the dyadic setting may seem arbitrary; however, it is justified bytwo reasons:

• It is the natural choice when dealing with orthonormal wavelet bases(though wavelets could be defined using other division rules, in practicethe dyadic one is the standard choice), and also the measure setting.

• When analyzing experimental data through regressions on log-log plots,for a given resolution, the dyadic splitting yields the largest number ofscales available in order to perform the regression.

Definition 7. The pointwise exponents associated with a dyadic function Eon Ω are the function h(x) and h(x) : Ω → R defined for x ∈ Ω as follows:

• The lower exponent of E is

(3.1) hE(x) = lim infj→+∞

log eλj(x)

log 2−j

• The upper exponent of E is

(3.2) hE(x) = lim supj→+∞

log eλj(x)

log 2−j.

By convention one sets hE(x) = hE(x) = +∞ if x /∈Supp(E).

We saw in the introduction the first example of scaling function which has beenused. We now define them in the abstract setting supplied by dyadic functions. Wedenote by ΛΩ

j the subset of Λj composed of the dyadic cubes contained in Ω.

Definition 8. Let Ω be a nonempty bounded open subset of Rd. The structurefunction of a dyadic function E on Ω is defined by

(3.3) ∀p ∈ R, Sj(Ω, p) =∑λ∈ΛΩ

j

(eλ)p.


The scaling function of E on Ω is defined by

(3.4) ∀p ∈ R, τΩE (p) = lim infj→+∞

logSj(Ω, p)

log 2−j.

If Ω is not bounded, one defines the scaling function as follows:

(3.5) if Ωn = Ω ∩B(0, n), ∀p ∈ R, τΩE (p) = limn→∞

τΩn

E (p).

Note that the limit exists because the sequence is decreasing. From now on, we willassume that the set Ω is bounded, so that, at each scale j, a finite number onlyof dyadic cubes λ satisfy λ ⊂ Ω. The corresponding results when Ω is unboundedfollow easily from (3.5).

Apart from the scaling function, an additional “global” parameter plays animportant role for classification in many applications; and, for multifractal analysis,checking its positivity is a prerequisite in the wavelet setting (see [2] and referencestherein): The uniform regularity exponent of E is defined by

(3.6) hΩE = lim inf

j→+∞

log

(supλ∈Λj

eλ

)log 2−j

.

The scaling function τΩE is concave (as a liminf of concave functions) taking

values in R. The following regularity assumption is often met in practice, andimplies that ηΩE is finite for any value of p.

Definition 9. A dyadic function E is regular in Ω if

(3.7) ∃C1, C2 > 0, ∃A,B ∈ R ∀λ ⊂ Ω : eλ �= 0 =⇒ C12−Bj ≤ eλ ≤ C22

−Aj .

The existence of A is equivalent to the condition hΩE > −∞. More precisely,

hΩE = sup{A : the right hand side of (3.7) holds}.

In the measure case and in the Holder exponent case, one can pick A = 0. Inthe Holder case, the uniform regularity assumption means that A > 0. When theeλ are wavelet leaders, the assumption on the lower bound implies that the functionf considered has no C∞ components.

Since the scaling function is concave, there is no loss of information in ratherconsidering its Legendre transform, defined by

(3.8) LΩE (H) := inf

p∈R

(Hp− τΩE (p)).

The function LΩE (H) is called the Legendre spectrum of E .

Though it is mathematically equivalent to consider LΩE (H) or τΩE (p), one often

prefers to work with the Legendre spectrum, because of its interpretation in termsof regularity exponents supplied by the multifractal formalism.

Definition 10. Let E be a dyadic function on Ω, and define, for H ∈ [−∞,+∞],the level set associated with E

EΩE (H) = {x ∈ Ω : hE(x) = H}.

The associated spectrum (on Ω) is defined by

dΩE : H ∈ R �→ dim EΩE (H).


Let us now show how a heuristic relationship can be drawn between the mul-tifractal and the Legendre spectra. The definition of the scaling function (3.4)

roughly means that, for j large, Sj(Ω, p) ∼ 2−τΩE (p)j . Let us estimate the contribu-

tion to Sj(Ω, p) of the dyadic cubes λ that cover the points of EE(H). By definitionof EE(H), they satisfy eλ ∼ 2−Hj ; by definition of dΩE (H), since we use cubes of

the same width 2−j to cover E, we need about 2dΩE (H)j such cubes; therefore the

corresponding contribution is ∼ 2dΩE (H)j2−Hpj = 2−j(Hp−dΩ

E (H)). When j → +∞,the smallest exponent brings an exponentially dominant contribution, so that

(3.9) τΩE (p) = infH

(Hp− dΩE (H)).

This formula can be interpreted as stating that the scaling function is the Legendretransform of the spectrum. Assuming that dΩE (H) is concave, it can be recoveredby an inverse Legendre transform, leading to

(3.10) dΩE (H) = infp∈R

(Hp− τΩE (p)).

When this equality holds, the dyadic function E satisfies the multifractal formalismon Ω, which therefore amounts to state that the Legendre spectrum coincides withthe multifractal spectrum.

Note that the derivation we sketched is not a mathematical proof, and thedetermination of the range of validity of (3.10) (and of its variants) is one of themain mathematical problems concerning multifractal analysis. The only resultswhich hold in all generality are upper bounds of dimensions of singularities.

Proposition 4. [20,35,45] Let E be a dyadic function on Ω. Then

(3.11) dΩE (H) ≤ LΩE (H).

An important consequence of this corollary is supplied by the only case wherethe knowledge of the scaling function is sufficient to deduce the multifractal spec-trum, and even the pointwise exponent hE everywhere.

Corollary 2. Let E be a dyadic function. If its scaling function τΩE satisfies

(3.12) ∃α > 0 such that ∀p ∈ R, τΩE (p) = τE(0) + αp,

then the multifractal formalism is satisfied on Ω, and the lower exponent of E sat-isfies

∀x ∈ Supp E , hE(x) = α.

Proof. (of Corollary 2) Assume that (3.12) is true. Then LE(H) = −∞except for H = α; Corollary 4 implies in this case that dE(H) ≤ −∞ for H �= α.Therefore only one Holder exponent is present, so that ∀x, h(x) = α; it followsthat dΩE (α) = 1, and the multifractal formalism therefore holds. �

This corollary has direct implications in modeling: Indeed, several experimentalsignals have a linear scaling function. In such situations, multifractal analysis yieldsthat the data have a constant pointwise exponent; therefore it supplies a non-parametric method which allows to conclude that modeling by, say, a fractionalBrownian motion, is appropriate (and the slope of the scaling function supplies theindex of the FBM), see e.g. [2] where one example of internet traffic data is shown.We will also see a local version of Corollary 2 which has implications in modeling:Corollary 5.


3.2. Local multifractal formalism.

Definition 11. Let E be a dyadic function on Ω. The local multifractal spec-trum of E is the function defined by

(3.13) ∀H, ∀x ∈ Ω, dE(x,H) = dim(EE(H), x)(= lim

r→0dB(x,r)E (H)

).

The following result, which is a direct consequence of Proposition 2, shows thatthe local spectrum allows to recover the spectrum of all possible restrictions of Eon a subset ω ∈ Ω.

Corollary 3. Let E be a dyadic function on Ω. Then for any open set ω ⊂ Ω,

(3.14) ∀H ∈ R, dωE (H) = supx∈ω

dE(x,H).

Definition 12. A dyadic family E is said to be homogeneously multifractalwhen the local multifractal spectrum dE(x, ·) does not depend on x, i.e.

∀x ∈ Ω, ∀H ∈ R, dE(x,H) = dΩE (H).

A local scaling function can also be defined by making the set Ω shrink downto the point x0.

Definition 13. Let E be a dyadic function on Ω. The local scaling of E is thefunction defined by

(3.15) ∀H, ∀x ∈ Ω, τE(x, p) = limr→0

τB(x,r)E (p).

Note that the right-hand side of (3.15) is a decreasing function of r, and there-fore it has a limit when r → 0. Similarly as in the multifractal spectrum case, astraightforward compacity argument yields that the scaling function on any domainω can be recovered from the local scaling function.

Corollary 4. Let E be a dyadic function on Ω. Then for any open set ω ⊂ Ω,

(3.16) ∀H ∈ R, τωE (p) = infx∈ω

τE(x, p).

Definition 14. The scaling function of a dyadic family E is said to be homo-geneous when the local scaling function τE(x, ·) does not depend on x.

The upper bound supplied by Corollary 4 holds for any given ball B(x, r).Fixing x ∈ Ω and making r → 0, we obtain a following local version of this result:

(3.17) ∀x ∈ Ω, ∀H, dE(x,H) ≤ infp∈R

(Hp− τE(x, p)) .

We will say that the multifractal formalism holds locally at x whenever (3.17) isan equality.

As above, this result has an important consequence: In some cases, it allowsto determine the regularity exponent at every point, even in situations where thisexponent is not constant.

Corollary 5. Let E be a dyadic function. If there exists a function α : R �→ Rsuch that the local scaling function τE satisfies

(3.18) ∀x ∈ Ω, ∀p ∈ R, τE(x, p) = τE(x, 0) + α(x)p,

then the multifractal formalism is locally satisfied on Ω, and the lower exponent ofE is

(3.19) ∀x ∈ Ω, hE(x) = α(x).


This result is a direct consequence of (3.17) and Corollary 2: Indeed, if (3.18)holds, then (3.17) implies that dE(x,H) = −∞ if H �= α(x). We pick now an

H �= α(x); recall that dE(x,H) = limr→0 dB(x,r)E (H); therefore ∃R > 0 such that

∀r ≤ r, dB(x,r)E (H) = −∞. In particular, H is not the pointwise exponent at x.

Since this argument holds for any H �= α(x), (3.19) holds, and Corollary 5 follows.

We will see an application of Corollary 5 concerning the multifractional Brown-ian Motion in Section 5.1. Combining (3.17) with Proposition 3, yields the followingupper bound.

Corollary 6. Let E be a dyadic function on Ω; for any open set ω ⊂ Ω,

(3.20) ∀H, dωE (H) ≤ supx∈ω

infp∈R

(Hp− τE(x, p)) .

It is remarkable that, though this result is a consequence of Corollary 4, itusually yields a sharper bound. Indeed, assume for example that the multifractalformalism holds for two separated regions ω1 and ω2 yielding two different spec-tra d1(H) and d2(H); then (3.20) yields max(d1(H), d2(H)) whereas the globalmultifractal formalism applied to Ω = ω1 ∪ ω2 only yields the concave hull ofmax(d1(H), d2(H)). More generally, each time (3.20) yields a non-concave upperbound, it will be strictly sharper than the result supplied by Corollary 4.

The uniform regularity exponent also has a local form:

Definition 15. The local exponent associated with E is the function

hE(x) = limr→0

hB(x,r)E .

Note that the most general possible local exponents are lower semi-continuousfunctions, see [44].

It would be interesting to obtain a similar characterization for the functions(x,H) �→ dE(x,H) and (x, p) �→ τE(x, p) (considered as as functions of two vari-ables) and determine their most general form.

3.3. An example from ergodic theory. Let Ω = (0, 1). Consider a 1-periodic function φ : R → R, as well as two continuous functions γ : [0, 1] → (0,∞)and θ : [0, 1] → R. Let T : x ∈ R �→ 2x. For x ∈ R and j ∈ N denote by Sjφ(x) thejth Birkhoff sum of φ at x, i.e.,

Sjφ(x) =

j−1∑k=0

φ(T kx).

Then, for any dyadic subinterval λ of Ω of generation j, let

eλ = supx∈λ

e−γ(x)Sjφ(x)−jθ(x).

When the functions γ and θ are constant, the multifractal analysis of the dyadicfamily E = (eλ)λ⊂Ω reduces to that of the Birkhoff averages of γφ + θ, since

lim infj→∞log eλj(x)

log 2−j = H if and only if lim infj→∞ Sj(x)/j = H log(2)−θγ . This is

a now classical problem in ergodic theory of hyperbolic dynamical systems, which


is well expressed through the thermodynamic formalism. The function log(2)τΩE isthe opposite of the pressure function of −(γφ + θ), that we denote by Pγ,θ(q), i.e.

− log(2)τΩE (p) = Pγ,θ(p) = limj→∞

1

jlog

∑λ∈ΛΩ

j

(supx∈λ

e−γSjφ(x)−jθ)p

(p ∈ R),

= P (−γp) − θp,

where P = P−1,0 is the pressure function of φ; and the following result follows forinstance from [29].

Theorem 1. Let H ∈ R; then EΩE (H) �= ∅ if and only if H belongs to the

interval [(τΩE )′(+∞), (τΩE )

′(−∞)] and in this case τΩE (H) = inf{Hp−τΩE (p) : p ∈ R}.

Continuing to assume that γ and θ are constant, and using the fact that Epossesses the same almost multiplicative properties as weak Gibbs measures (see[32, 42] for the multifractal analysis of these objects), i.e. some self-similarityproperty, it is easily seen that we also have τωE = τΩE and dωE = dΩE for all opensubsets of Ω.

Now suppose that γ or θ is not constant. Such a situation should be seen locallyas a small perturbation of the case where these functions are constant, and it isindeed rather easy using the continuity of γ and θ to get the following fact.

Proposition 5. ∀ x ∈ Ω, ∀q ∈ R,

(3.21) τE(x, p) = −Pγ(x),θ(x)(p)

log(2)=

−P (−γ(x)p) + θ(x)p

log(2).

Suppose also that φ is not cohomologous to a constant, i.e. the pressurefunction P of φ is not affine, which is also equivalent to saying that the intervalI = [P ′(−∞), P ′(+∞)] of possible values for lim infj→∞ Sj(y)/j, is non trivial.

For all H ∈ R, define

ξH : y ∈ (0, 1) �→ H log(2) − θ(y)

γ(y).

Notice that lim infj→∞log eλj(y)

log 2−j = H if and only if lim infj→∞ Sj(y)/j = h and

H = (γ(y)h + θ(y))/ log(2), i.e. h = ξH(y).Now fix x ∈ (0, 1). For every r > 0, we have

(3.22) EB(x,r)E (H) = {y ∈ B(x, r) : lim inf

j→∞Sj(y)/j = ξH(y)},

and due to Theorem 2.3 in [18], for all H > 0,

dimEB(x,r)E (H) ≥ sup{inf{P (p) − pα : p ∈ R} : α ∈ rg(ξH |B(x,r)) ∩ int(I)}.

Fix H ∈ (τ ′E(x,+∞), τ ′E(x,−∞)) = (γ(x)P ′(−∞) + θ(x), γ(x)P ′(+∞) + θ(x)). Byconstruction,

ξH |B(x,r)(x) = (H log(2) − θ(x))/γ(x) ∈ rg(ξH |B(x,r)) ∩ int(I).

Thus, due to (3.22),

dimEB(x,r)E (H) ≥ inf

{P (p) − p

(H log(2) − θ(x))/γ(x)

): p ∈ R

},

which, due to (3.21), is exactly inf{Hp − τE(x, p) : p ∈ R}. Since this estimateholds for all r > 0,

dE(x,H) ≥ inf{Hp− τE(x, p) : p ∈ R},


hence, by (3.17), it follows that

dE(x,H) = inf{Hp− τE(x, p) : p ∈ R}.For the case where H ∈ {τ ′E(x,+∞), τ ′E(x,−∞)}, it is difficult to conclude in

full generality. We thus have proved the following result.

Theorem 2. Suppose that φ is not cohomologous to a constant. Fix x ∈ Ω andH ∈ R. If H �∈ [τ ′E(x,+∞), τ ′E(x,−∞)] = [γ(x)P ′(−∞)+θ(x), γ(x)P ′(+∞)+θ(x)]

then EB(x,r)E (H) = ∅ for r small enough, and if H ∈ (τ ′E(x,+∞), τ ′E(x,−∞)) then

dE(x,H) = inf{Hp− τE(x, p) : p ∈ R}.Let us mention that, if the union of the sets of discontinuity points of γ and θ

has Hausdorff dimension 0, then the study achieved in [18] shows that the previousresult holds at any point x which is a point of continuity of both γ and ξ. Also,when φ and θ are positive, the family E can be used to build wavelet series whoselocal multifractal structure is the same as that of E .

4. Measures with varying local spectrum

4.1. General considerations. Let μ be a positive Borel measure supportedby [0, 1]d. Recall that one derives from μ the dyadic family Eμ = {eλ := μ(3λ)}λ∈Λ.

It is obvious that the definition (1.4) of the local dimension hμ(x0) is equivalentto (3.1) with the dyadic family Eμ. Similarly, the classical formalism for measureson [0, 1]d is the same as the one described in the previous section for the family Eμon Ω = [0, 1]d. Hence one can define a local multifractal spectrum for measures byDefinition 3.13.

In the measure setting, the following result shows that the mass distributionprinciple has a local version.

Proposition 6. Let μ be a Radon measure, A ⊂ Rd and x ∈ A ∩ supp (μ).Then

dim(x,A) ≥ hμ(x).

Proof. It follows from (7.3) applied on A ∪ B(x, r), remarking that the hy-pothesis x ∈ supp (μ) implies that μ(A ∪B(x, r)) > 0 and then letting r → 0. �

We introduced the local multifractal spectrum to study (in particular) non-homogeneous multifractal measures. therefore, it is relevant to recall the result of[21], where it is proved that homogeneous multifractal measures and non-homogeneousmultifractal measures do not exhibit the same multifractal properties.

Theorem 3. Consider a non-atomic homogeneous multifractal measure sup-ported on [0, 1]. Then the intersection of the support of the (homogeneous) multi-fractal spectrum of dμ with the interval [0, 1] is necessarily an interval of the form[α, 1], where 0 ≤ α ≤ 1.

This is absolutely not the case for non-homogeneously multifractal measures:consider for instance two uniform Cantor sets C0 and C1 of dimension 1/2 and1/4 on the intervals [0, 1/2) and [1/2, 1]. Then the barycenter of the two uniformmeasures naturally associated with C0 and C satisfies

dμ(h) =

⎧⎨⎩ 1/4 if h = 1/4,1/2 if h = 1/2,−∞ else.


Hence the local spectrum is the natural tool to study non-homogeneous multifractalmeasures.

4.2. A natural example where the notion of local spectrum is rele-vant. The Bernoulli (binomial) measure is perhaps the most natural and simplemultifractal object, and it is now folklore that is homogeneously multifractal (see[26] for another way to recover this result). We make a very natural modification inits construction, which will break homogeneity by making the Bernoulli parameterp depend on the interval which is split in the construction. Doing this, we obtaina ”localized” Bernoulli measure whose local spectrum depends on x. This exampleis closely related with the example developed in Section 3.3. See also [46] for ageneral construction of graph-directed Markov measures that can be compared insome way to our construction.

Let p : [0, 1] �→ (0, 1/2) be a continuous mapping. For n ≥ 1, (ε1, ε2, ..., εn) ∈{0, 1}n, we denote the dyadic number kε1ε2...εn =

∑ni=1 εi2

−i and the dyadic intervalIε1ε2...εn = [kε1ε2...εn , kε1ε2...εn + 2−n), where n ≥ 1, (ε1, ε2, ..., εn) ∈ {0, 1}n, and wewill use the natural tree structure of these intervals using the words (ε1ε2...εn).

Consider the sequence of measures (μn)n≥1 built as follows:

• μ1 is uniformly distributed on I0 and I1, and μ1(I0) = p(2−1) and μ1(I1) =1 − p(2−1).

• μ2 is uniformly distributed on each dyadic interval Iε1ε2 of second gener-ation, and

μ2(Iε10) = μ1(Iε1) · p(kε11) and μ2(Iε11) = μ1(Iε1) · (1 − p(kε11)).

• ...• μn is uniformly distributed on each dyadic interval Iε1ε2...εn of generationn, and

μn(Iε1ε2...εn−10) = μn−1(IIε1ε2...εn−1) · p(kIε1ε2...εn−11)

and μn(Iε1ε2...εn−11) = μn−1(Iε1 , ε2, ..., εn−1) · (1 − p(kε1ε2...εn−11)).

Observe that by construction, for every n, for every p ≥ n and every dyadicinterval I of generation n, one has μp(I) = μn(I).

Definition 16. The sequence of measures (μn)n≥1 converges weakly to a prob-ability measure μ that we call the “localized” Bernoulli measure associated withthe map p.

Obviously, if p is constant, one recovers the usual Bernoulli measure with pa-rameter p.

We indicate the sketch of the proof to obtain the local multifractal propertiesof μ. We do not use exactly the exponent hμ defined by (1.4), but, for simplicity,we rather work with the dyadic local exponent defined by

hdμ(x) = lim inf

n→+∞

log μ(In(x))

log 2−n,

where (as usual) In(x) stands for the unique dyadic interval of generation n con-taining x. The results we are going to prove also hold for the exponent hμ, butwould require longer technical developments. In particular, we would need an ex-tension of Corollary 2 of [18] on localized multifractal analysis of Gibbs measures.This exponent hd

μ can also be encompassed in the frame of Section 3 by using the


dyadic family E = {μ(λ)}λ∈Λ, thus all the “local” notions we introduced hold forthis exponent.

Theorem 4. For every x ∈ [0, 1], the local spectrum associated with the expo-nent hd

μ of μ at x is that of a Bernoulli measure of a parameter p(x), i.e.

∀ H ≥ 0, dμ(x,H) = dμp(x)(H).

For every x ∈ [0, 1], we consider its dyadic decomposition x = 0, ε1ε2....εn...,εn ∈ {0, 1}. Let N0,n(x) = #{1 ≤ k ≤ n : εk = 0} and N1,n(x) = #{1 ≤ k ≤ n :εk = 1} (= n−N0,n(x)). We consider the asymptotic frequencies of 0’s and 1’s inthe dyadic decomposition of x defined as

N0(x) = lim supn→+∞

1

nN0,n(x).

Proposition 7. For every x ∈ [0, 1], we have

hdμ(x) = −N0(x) log2 p(x) − (1 −N0(x)) log2(1 − p(x)).

Essentially, the localized binomial measure looks locally around x like the bi-nomial measure of parameter p(x).

Proof. Let us fix q ∈ (0, 1/2), and consider the classical Bernoulli measure μq

of parameter q on the whole interval [0, 1]. A standard argument yields that theHolder exponent of μq at every point x is

(4.1) hdμq

(x) = −N0(x) log2 q − (1 −N0(x)) log2(1 − q).

Inspired by this formula, a Caesaro argument gives the proposition. Indeed, byconstruction, the value of the μ-mass of the interval In(x) is given by

μ(In(x)) =

n∏i=1

p(kε1ε2...εi−11)∗,

where

p(kε1ε2...εi−11)∗ =

{p(kε1ε2...εi−11) if εi = 0

1 − p(kε1ε2...εi−11) if εi = 1.

Hence,

μ(In(x)) = 2

n∑i=1:εi=0

log2 p(kε1ε2...εi−11) +

n∑i=1:εi=0

log2(1 − p(kε1ε2...εi−11))

,

Since the sequence (p(kε1ε2...εi−11))i≥1 tends to p(x) when i tends to infinity,and since N0(x) is the asymptotic frequency of zeros in the dyadic expansion of x,it follows that

lim supn→+∞

1

n

(n∑

i=1:εi=0

log2 p(kε1ε2...εi−11) +n∑

i=1:εi=1

log2(1 − p(kε1ε2...εi−11))

)= N0(x)p(x) + (1 −N0(x))(1 − p(x)).(4.2)

Let α(x) = −N0(x) log2 p(x) − (1 − N0(x)) log2(1 − p(x)). The latter provesthat, given ε > 0:

• there exists an infinite number of integers N such that

2−N(α(x)+ε) ≤ μ(IN (x)) ≤ 2−N(α(x)−ε).


• for every n large enough,

μ(In(x)) ≤ 2−n(α(x)−ε).

This yields exactly that α(x)− ε ≤ hdμ(x) ≤ α(x)+ ε. Letting ε go to zero gives the

result. �

Consider an interval J ⊂ [0, 1], and the multifractal spectrum dμ(H, J) =dim {x ∈ J : hd

μ(x) = H}. The value of this spectrum is a non-trivial consequenceof the following theorem of Barral and Qu in [18] (who proved this result for anyGibbs measure μ).

Theorem 5. Fix q ∈ (0, 1/2), and consider the Bernoulli measure with param-eter q. Let us denote by Rq the support of the (homogeneous) multifractal spectrumof μq. Let h : [0, 1] → Rq be a continuous function. Then, for every intervalJ ⊂ [0, 1],

dim {x ∈ J : hdμq

(x) = h(x)} = sup{dμq(h(x)) : x ∈ J}.

We now prove Theorem 4.Fix H > 0. The upper bound for the spectrum follows from the general upper

bound obtained from the local multifractal formalism.Let x be such that hd

μ(x) = H, i.e.

H = −N0(x) log2 p(x) − (1 −N0(x)) log2(1 − p(x))).

For every q ∈ (0, 1/2), define the unique real number hq(H,x) such that

−N0(x) log2 q − (1 −N0(x)) log2(1 − q) = hq(H,x).

Let now x0 ∈ [0, 1] and consider the Bernoulli measure with parameter q =p(x0). We assume that x0 is not a local extremum of p (the reader can take careof the other case by adapting the following arguments). Consider the intervalI = B(x0, r) with r small.

Since p is continuous, both p(x) and q are strictly less than 1/2, and H is fixed,the mapping x ∈ IH �→ hq(H,x) is continuous with respect to x, where

IH := {x ∈ I : hdμ(x) = H}.

Since x0 is not a local extremum for p, this set IH is dense at least on one interval Jincluded in I (this is due to the very erratic character of the mapping x �→ N0(x)).Hence, the map x ∈ IH �→ hq(H,x) can be continuously extended to the interval J .Pay attention to the fact that hq(H,x) when viewed as a function of x ∈ I is notcontinuous with respect to x, but its restriction to x ∈ IH is, and this restrictionis the one we extend to J (this also explains the dependence on H we use in thenotation hq(H,x)).

Hence,

{x ∈ J : hμ(x) = H}= {x ∈ J :−N0(x) log2 p(x) − (1 −N0(x)) log2(1 − p(x)) = H}= {x ∈ J :−N0(x) log2 q − (1 −N0(x)) log2(1 − q) = hq(H,x)}.

But the Hausdorff dimension of this last set is exactly given by Theorem 5 sinceour mapping hq(H,x) is continuous, hence

dim{x ∈ J : hμ(x) = H} = sup{dμq(hq(H,x)) : x ∈ J}.


When r goes to zero, p(x) tends uniformly to q = p(x0). Hence hq(H,x) tends toH. In particular, the mapping dμq

being continuous (real analytic in fact), when rgoes to zero one finds

dim{x ∈ J : hμ(x) = H} → dμq(H).

Finally, since dim{x ∈ I : hμ(x) = H} ≥ dim{x ∈ J : hμ(x) = H}, one gets

dμ(x,H) ≥ dμq(H) = dμp(x)

(H).

This result can immediately be applied to the case where the mappingx0 �→ p(x0) is continuous by part (instead of simply continuous), and can cer-tainly be adapted when p is cadlag. It would be worth investigating the case wherep enjoys less regularity properties.

Remark 4.1. Many examples of Cantor set with varying local Hausdorff di-mensions have been constructed [8,54]; here the key point is that we perform the(global and local) multifractal analysis of measures sitting on these ”inhomoge-neous” Cantor sets.

5. Local spectrum of stochastic processes

Suppose now that f is a nowhere differentiable function defined on [0, 1]d; onecan associate with f the dyadic family Ef = {Oscf (3λ)}λ∈Λ, where the oscillationof f over a set ω ⊂ Ω is

Oscf (ω) = sup{f(x) : x ∈ ω} − inf{f(x) : x ∈ ω}.Then, it is obvious that the pointwise Holder exponent (1.6) of f at x is the sameas the one defined by (3.1) with the dyadic family Ef . Hence, the previous de-velopments performed in the abstract setting of dyadic functions family holds fornon-differentiable functions.

We start by giving a simple general probabilistic setting which naturally leadsto a weak, probabilistic form of homogeneity. Let X be a random field on Rd; Xhas stationary increments if ∀s ∈ Rd, the two processes

x �→ Ys(x) := X(s + x) −X(s) and x �→ X(x)

share the same law. Indeed, this equality in law implies the equality in law of thelinear forms applied to the two processes Ys and X, hence of iterated differencesand wavelet coefficients. It follows that local suprema of iterated differences andof wavelet coefficients computed on dyadic cubes also share the same laws, andProposition 3 implies that, if X has locally bounded sample paths, then the Holderexponent has a stationary law. Therefore, the Holder spectra on dyadic intervalsof the same width also share the same law almost surely. As a result, the Holderspectra on all dyadic intervals share the same law. This leads to the followingresult.

Proposition 8. Let X be a random field on Rd with stationary increments.If X has locally bounded sample paths, then

∀s a.s. ∀H dX(s,H) = dX(0, H).

Note that this result does not imply that a given sample path necessarily ishomogeneously multifractal: For instance, the local spectrum of a sample pathof a Poisson process differs depending whether the interval where it is computedincludes a jump or not.


5.1. Local analysis of the multifractional Brownian motion. Let Hdenote a function defined on Rd with values in a fixed compact subinterval [a, b]of (0, 1). We assume that H satisfies locally a uniform Holder condition of orderβ ∈ (b, 1), that is, H ∈ Cβ(Ω) for every open subset Ω of Rd. Now, recall thatthe multifractional Brownian motion (MBM) with functional parameter H hasbeen introduced in [19,52] as the continuous and nowhere differentiable Gaussianrandom field BH = {BH(x), x ∈ Rd} that can be represented as the followingstochastic integral

BH(x) =

∫Rd

eıx·ξ − 1

|ξ|H(x)+d/22

dW (ξ),

where x·ξ denotes the standard inner product, |ξ|2 is the usual Euclidean norm, and

dW stands for the “Fourier transform” of the real-valued white noise dW , meaningthat for any square-integrable function f , one has∫

Rd

f(ξ) dW (ξ) =

∫Rd

f(x) dW (x).

In particular, the MBM reduces to a fractional Brownian motion when the functionH is chosen to be constant. The pointwise regularity of the MBM is well known;as a matter of fact, it has been shown in [7] that

(5.1) a.s. ∀x ∈ Rd hBH(x) = H(x).

Thus, the Holder exponent of the MBM is completely prescribed by the functionH. Our purpose is now to give an illustration to Corollary 5 above by showingthat the multifractal formalism is locally satisfied by almost every sample path ofthe MBM. To be specific, we shall establish in the remainder of this section thefollowing result which, with the help of Corollary 5, enables one to recover (5.1).

Proposition 9. Let EH denote the dyadic function that is obtained by consid-ering the wavelet leaders of the multifractional Brownian motion BH , and assumethat the wavelets belong to the Schwartz class. Then, the local scaling function τEH

satisfies

a.s. ∀x ∈ Rd ∀p ∈ R τEH(x, p) = H(x)p− d.

In order to establish Proposition 9, we shall work with a Lemarie-Meyer waveletbasis of L2(Rd) formed by the functions 2dj/2ψ(i)(2jx − k), see [43], and moregenerally with the biorthogonal systems generated by the fractional integrals of thebasis functions ψ(i), namely, the functions ψ(i),h defined by

ψ(i),h(ξ) =ψ(i)(ξ)

|ξ|h+d/22

.

It will also be convenient to consider the Gaussian field Y = {Y (x, h), (x, h) ∈Rd × (0, 1)} given by

Y (x, h) =

∫Rd

eıx·ξ − 1

|ξ|h+d/22

dW (ξ).

Note, in particular, that BH(x) = Y (x,H(x)) for all x ∈ Rd, and that the randomfield {Y (x, h), x ∈ Rd} is merely a fractional Brownian motion with Hurst param-eter h. By expanding its kernel in the orthonormal basis of L2(Rd) formed by the


Fourier transforms of the functions 2dj/2ψ(i)(2jx−k), and by virtue of the isometryproperty, the stochastic integral defining Y (x, h) may be rewritten in the form

Y (x, h) =∑i

∑j∈Z

∑k∈Zd

εij,k2−hj

(ψ(i),h(2jx− k) − ψ(i),h(−k)

),

where the εij,k form a collection of independent standard Gaussian random variables.It is possible to show that the above series converges uniformly on any compactsubset of Rd×(0, 1), see [6]. Moreover, the above decomposition yields the followingnatural wavelet-like expansion of the field BH :

(5.2) BH(x) =∑i

∑j∈Z

∑k∈Zd

εij,k2−H(x)j

(ψ(i),H(x)(2jx− k) − ψ(i),H(x)(−k)

).

Furthermore, it is shown in [6] that the low-frequency component of Y , that is,∑i

−1∑j=−∞

∑k∈Zd

εij,k2−hj

(ψ(i),h(2jx− k) − ψ(i),h(−k)

),

is almost surely a C∞ function in the two variables x and h. Hence, the low-frequency component of the MBM, which is obtained by summing only over thenegative values of j in (5.2), is in Cβ(Ω) for any open subset Ω of Rd, just as thefunctional parameter H. As β is larger than all the values taken by the functionH, it follows that the pointwise regularity of the MBM is merely given by that ofits high-frequency component, that is,

BH(x) =∑i

∞∑j=0

∑k∈Zd

εij,k2−H(x)j

(ψ(i),H(x)(2jx− k) − ψ(i),H(x)(−k)

).

As a consequence, we may consider in what follows the high-frequency component

BH instead of the whole field BH . In addition, in view of the regularity of H,it follows from standard results on Calderon-Zygmund operators (see [48]) androbustness properties of the local scaling functions, that τEH

coincides with the

local scaling function of the dyadic family EH which is obtained by considering thewavelet leaders associated with the wavelet coefficients

cij,k = εij,k2−H(k2−j)j .

(Recall that in [35], it is proved that the scaling function is “robust”, i.e. does notdepend on the smooth enough wavelet basis chosen; furthermore, the arguments ofthe proof clearly are local, so that the local scaling function also is robust.)

Letting λ denote the cube corresponding to the indices i, j and k as in Sec-tion 2.3, these coefficients may naturally be rewritten in the form

cλ = ελ2−H(xλ)〈λ〉,

where ελ is the standard Gaussian random variable εij,k, xλ is the basis point k2−j of

the cube λ and 〈λ〉 is its scale j. Recall that the wavelet leaders dλ are then definedin terms of the wavelet coefficients through (2.7). Finally, for the sake of simplicityand without loss of generality, we shall study the local scaling function τEH

only on

the open set (0, 1)d, so that we only have to consider the dyadic subcubes of [0, 1)d.Let us now establish a crucial lemma concerning the behavior on the subcubes

of [0, 1)d of the new dyadic family EH .


Lemma 1. With probability one, for any dyadic cube λ ⊂ [0, 1)d with scale 〈λ〉large enough,

1

〈λ〉3H(xλ)≤ 2H(xλ)〈λ〉dλ ≤ 2〈λ〉.

Proof. We begin by the proving the lower bound. For any proper dyadicsubcube λ of [0, 1)d with scale 〈λ〉 = j, we have

P(dλ ≤ 〈λ〉−3H(xλ)2−H(xλ)〈λ〉) =∏

λ′⊂3λ

P(|ελ′ | ≤ 〈λ〉−3H(xλ)2H(xλ′)〈λ′〉−H(xλ)〈λ〉).

Let l(j) = j + �(2/d) log2 j , where � · denotes the ceiling function and log2 thebase two logarithm. Considering in the above product only the subcubes λ′ ⊂ 3λwith scale 〈λ′〉 equal to l(j), and using the elementary fact that the modulus of astandard Gaussian random variable is bounded above by t with probability at mostt, we deduce that

P(dλ ≤ 〈λ〉−3H(xλ)2−H(xλ)〈λ〉) ≤∏

λ′⊂3λ〈λ′〉=l(j)

〈λ〉−3H(xλ)2H(xλ′)〈λ′〉−H(xλ)〈λ〉.

Moreover, the function H satisfies locally a uniform Holder condition of order β,so there exists a real C > 0 that does not depend on λ such that

(5.3) ∀λ′ ⊂ 3λ |H(xλ′) −H(xλ)| ≤ C2−βj .

Combined with the observation that there are at least j2 subcubes λ′ ⊂ 3λ suchthat 〈λ′〉 = l(j), this implies that

P(dλ ≤ 〈λ〉−3H(xλ)2−H(xλ)〈λ〉) ≤(j−3H(xλ)2H(xλ)(l(j)−j)+Cl(j)2−βj

)j2.

Given that the function H is valued in the interval [a, b], we infer that

P(dλ ≤ 〈λ〉−3H(xλ)2−H(xλ)〈λ〉) ≤(j(2/d−3)a2b+Cl(j)2−βj

)j2.

The right-hand side is clearly bounded above by e−j2 when j is larger than someinteger j0, so that∑

λ⊂[0,1)d

〈Λ〉≥j0

P(dλ ≤ 〈λ〉−3H(xλ)2−H(xλ)〈λ〉) ≤∑j≥j0

2dje−j2 < ∞,

and we deduce the required lower bound from the Borel-Cantelli lemma.In order to establish the upper bound, let us begin by observing that with

probability one, for any dyadic cube λ ⊂ [0, 1)d with scale 〈λ〉 = j large enough,|ελ| ≤ j. This follows again from the Borel-Cantelli lemma, together with the factthat

P(|ελ| > j) = 2(1 − Φ(j)) ≤ e−j2/2

j

√2

π,

which itself follows from standard estimates on the asymptotic behavior of thecumulative distribution function Φ of the standard Gaussian distribution. Now,along with (5.3), this implies that for 〈λ〉 = j large enough,

dλ ≤ supλ′⊂3λ

〈λ′〉2−(H(xλ)−C2−βj)〈λ′〉 = j2−(H(xλ)−C2−βj)j ≤ 2j2−H(xλ)j ,

and the required upper bound follows. �


We may now finish the proof of Proposition 9. To this end, let x ∈ (0, 1)d andr > 0 such that Ω = B(x, r) ⊂ (0, 1)d. Then, owing to Lemma 1, the structure

function of the dyadic function EH on Ω, which is defined by (3.3), satisfies

(5.4)∑λ∈ΛΩ

j

(2−H(xλ)j

j3H(xλ)

)p

≤ Sj(Ω, p) ≤∑λ∈ΛΩ

j

(2j2−H(xλ)j

)pfor j large enough and p ≥ 0. Given that H satisfies locally a uniform Holdercondition of order β, there exists a real C > 0 that depends on neither x nor r suchthat |H(xλ)−H(x)| ≤ Crβ for all dyadic cubes λ ⊂ Ω. In addition, the cardinalityof ΛΩ

j is comparable with rd2dj . Thus, there is a constant C ′ > 0 such that

rd2dj

C ′

(2−(H(x)+Crβ)j

j3(H(x)+Crβ)

)p

≤ Sj(Ω, p) ≤ C ′rd2dj(2j2−(H(x)−Crβ)j

)p.

It follows that the scaling function of EH on Ω satisfies

(H(x) − Crβ)p− d ≤ τΩEH(p) ≤ (H(x) + Crβ)p− d.

Letting r go to zero, we may finally conclude that τEH(x, p) = H(x)p − d for all

p ≥ 0 and x ∈ (0, 1)d. The same approach still holds for the negative values ofp except that the inequalities have to be reversed in (5.4) and in the subsequentestimates as well. Proposition 9 follows.

5.2. A Markov process with a varying local multifractal spectrum.In this section we reinterpret the results of [9] in terms of local spectrum. A quitegeneral class of one-dimensional Markov processes consists of stochastic differentialequations (S.D.E.) with jumps. Recall that such a process is the sum of a Brow-nian motion and a pure jump process. We will assume in the following that theprocess has no Brownian part; indeed, since Brownian motion is mono-Holder, itsconsequence on the spectrum is straightforward to handle: It eliminates Holder ex-ponents larger than 1/2 and, eventually adds a point at (1/2, 1). Thus the Markovprocesses that will be studied are jumping S.D.E. without Brownian and drift part,starting e.g. from 0, and with jump measure ν(y, du) (meaning that, when locatedat y, the process jumps to y + u at rate ν(y, du)). Again, since this is a ”toy”model, we will make additional simplifying assumptions: Namely that the processis increasing (that is, ν(y, (−∞, 0)) = 0 for all y ∈ R). Classically, a necessarycondition for the process to be well-defined is that

∫∞0

u ν(y, du) < ∞.

If ν is chosen so that the index βν(y,.) is constant with respect to y, then oneexpects that the local multifractal spectrum dM (t, h) of the process M = (Mt)t≥0

will be deterministic and independent of t. Hence, the index of the jump measurewill depend on the value y of the process. The most natural example of such asituation consists in choosing

νγ(y, du) := γ(y)u−1−γ(y)1[0,1](u)du,

for some function γ : R �→ (0, 1). The lower exponent of this family of measures is

∀ y ≥ 0, βνγ(y,.) = γ(y).

In [9], the following assumption is made


(H)

{There exists ε > 0 such that γ : [0,∞) �−→ [ε, 1 − ε]

is a Lipschitz-continuous strictly increasing function.

It is relatively clear that the assumptions can be relaxed, and that many classesof Markov processes could be further studied. An interesting subject to investigateis the range of functions γ that could be used in the construction. For a process,M = (Mt)t≥0, one sets ΔMt = Mt −Mt−, where Mt− = lim

s→t, s<tMs

Proposition 10. [9] Assume that (H) holds. There exists a strong Markovprocess M = (Mt)t≥0 starting from 0, increasing and cadlag (i.e. right-continuous,with a left limit), and with generator L defined for all y ∈ [0,∞) and for anyfunction φ : [0,∞) �→ R Lipschitz-continuous by

(5.5) Lφ(y) =

∫ 1

0

[φ(y + u) − φ(y)]νγ(y, du).

Almost surely, this process is continuous except on a countable number of jumptimes. Denote by J the set of its jump times, that is J = {t > 0 : ΔM(t) �= 0}.Finally, J is dense in [0,∞).

This representation of M is useful for its local regularity analysis.The following theorem of [9] summarizes the multifractal features of M .

Theorem 6. Assume (H) and consider the process M constructed in Proposi-tion 10. Then, the following properties hold almost surely:

(i) For every t ∈ (0,∞)\J , the local spectrum of M at t is given by

(5.6) dM (t, h) =

{h · γ(Mt) if 0 ≤ h ≤ 1/γ(Mt),

−∞ if h > 1/γ(Mt),

while for t ∈ J ,

(5.7) dM (t, h) =

⎧⎪⎨⎪⎩h · γ(Mt) if 0 ≤ h < 1/γ(Mt),

h · γ(Mt−) if h ∈ [1/γ(Mt), 1/γ(Mt−)],

−∞ if h > 1/γ(Mt−).

(ii) The spectrum of M on any interval I = (a, b) ⊂ (0,+∞) is

∀h ≥ 0, dM (h) = sup{h · γ(Mt) : t ∈ I, h · γ(Mt) < 1

}(5.8)

= sup{h · γ(Ms−) : s ∈ J ∩ I, h · γ(Ms−) < 1

}.(5.9)

In (5.8) and (5.9), we adopt the convention that sup ∅ = −∞.

As can be seen from the definition of the local multifractal spectrum, in orderto prove Theorem 6, it is enough to show (5.9). Indeed, (5.6) simply follows fromconsidering the limit of (5.9) when the interval I is the centered ball B(t, r) andletting r tend to zero.

Formula (5.9) is better understood when plotted: for every s ∈ I ∩ J , plot asegment whose endpoints are (0, 0) and (1/γ(Ms−), 1) (open on the right), and takethe supremum to get DM (I, .). Sample paths of the process M and their associatedspectra are given in Figure 1.


0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

7

8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

1

2

3

4

5

6

7

8

9

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 1. Two sample paths of the stochastic process M builtusing the function γ(y) := min(1/2+y/4, 0.9). On the right hand-side are plotted the theoretical spectra DM ([0, 3], .).

The formulae giving the local and global spectra are based on the computationof the pointwise Holder exponents at all times t. The value of the pointwise Holderexponent of M at t depends on two parameters: The value of the process M inthe neighborhood of t, and the approximation rate of t by the set of jumps J . Inparticular, the following properties holds a.s.,

for every t ≥ 0, hM (t) ≤ 1/γ(Mt),

for Lebesgue-almost every t, hM (t) = 1/γ(Mt),

for every κ ∈ (0, 1), dimH{t ≥ 0 : hM (t) = κ/γ(Mt)} = κ.

The relevance of the local spectrum in this context is thus obvious: Dependingon the local value of M , the pointwise Holder exponents change, and so is the(local) multifractal spectrum.

It is worth emphasizing that, as expected from the construction of the processM , the local spectrum (5.6) at any point t > 0 essentially coincides with that ofa stable Levy subordinator of index γ(Mt). This local comparison is strengthenedby the following theorem, which proves the existence of tangent processes for M(which are Levy stable subordinators).

Proposition 11. We denote by Ft := σ({N(A), A ∈ B([0, t] × [0,∞))}). Let

t0 ≥ 0 be fixed. Conditionally on Ft0 , the family of processes(Mt0+αt −Mt0

α1/γ(Mt0)

)t∈[0,1]

converges in law, as α → 0+, to a stable Levy subordinator with Levy measureγ(Mt0)u

−1−γ(Mt0)du. Here the Skorokhod space of cadlag functions on [0, 1] is

endowed with the uniform convergence topology.

Observe that for all s ∈ J , all h ∈ (1/γ(Ms), 1/γ(Ms−)], dM (h) = h · γ(Ms−).Thus the spectrum dM of M on an interval I is a straight line on all segments ofthe form (1/γ(Ms), 1/γ(Ms−)], s ∈ J ∩ I. By the way, this spectrum, when viewed


as a map from R+ to R+, is very irregular, and certainly multifractal itself. Thisis in sharp contrast with the spectra usually obtained, which are most of the timeconcave or (piecewise) real-analytic. Hence, the difference between the global andthe local multifractal spectra is stunning: While dM is very irregular, dM (t, ·) is astraight line.

This example naturally leads to the following open problem, which would ex-press that a natural compatibility holds for local multifractal analysis: Find generalconditions under which a stochastic process X which has a tangent process at apoint x0 satisfies that the multifractal spectrum of the tangent process coincideswith the local spectrum of X at x0.

6. Other regularity exponents characterized by dyadic families

Other exponents than those already mentioned fit in the general frameworkgiven by Definition 7 and therefore the results supplied by multifractal analysis canbe applied to them. We now list a few of them.

Pointwise Holder regularity is pertinent only if applied to locally bounded func-tions. An extension of pointwise regularity fitted to functions that are only assumedto belong to Lp

loc is sometimes required: The corresponding notion was introducedby Calderon and Zygmund in 1961, see [22], in order to obtain pointwise regularityresults for elliptic PDEs.

Definition 17. Let p ∈ [1,+∞) and α > −d/p. Let f ∈ Lploc(Ω), and x0 ∈ Ω;

f belongs to T pα(x0) if there exist C > 0 and a polynomial P of degree less than α

such that, for r small enough,

(6.1)

(1

rd

∫B(x0,r)

|f(x) − P (x− x0)|pdx)1/p

≤ Crα.

The p-exponent of f at x0 is

hpf (x0) = sup{α : f ∈ T p

α(x0)}.

Remarks:

• The normalization chosen in (6.1) is such that cusps |x − x0|α (whenα /∈ 2N) have an Holder and a p-exponent which take the same value α atx0.

• The Holder exponent corresponds to the case p = +∞.• We only define lower exponents here: Upper exponents could also be de-

fined in this context, by considering local Lp norms of iterated differences.• Definition 17 is a natural substitute for pointwise Holder regularity when

functions in Lploc are considered. In particular, the p-exponent can take

negative values down to −d/p, and typically allows to take into accountbehaviors which are locally of the form

(6.2)1

|x− x0|γfor γ < d/p,

A pointwise regularity exponent associated with tempered distributions hasbeen introduced by Y. Meyer: The weak scaling exponent (see [49], and also [1]for a multifractal formalism based on this exponent). It coincides with the Holderexponent for cusps like |x−x0|α and can also be interpreted as a limit case of other


exponents for distributions, which can be related with the Holder exponent; let usbriefly recall how this can be done.

Let f be a tempered distribution defined over Rd. One can define fractionalprimitives of order s of f in the Fourier domain by

f (−s)(ξ) = (1 + |ξ|2)s/2f(ξ).

Since f is of finite order, for s large enough, f (−s) locally belongs to Lp (or L∞).It follows that one can define regularity exponents of distributions through p-exponents (or Holder exponents) of a fractional primitives of large enough order. Iff is only defined on a domain Ω, one can still define the same exponents at x0 ∈ Ωby using a function g ∈ D(Rd) such that g is supported inside Ω and g(x) = 1 ina neighborhood of x0; then fg is a tempered distribution defined on Rd and theexponents of (fg)(−s) at x0 clearly do not depend on the choice of g.

Let f be a tempered distribution defined on a open domain. Denote by hsf (x)

the Holder exponent of f (−s) (which is thus canonically well defined for s largeenough). By definition, the weak scaling exponent of f at x is

Wf (x) = lims→+∞

(hsf (x) − s

)(note that the limit always exists because the quantity considered is an increasingfunction of s). We will not deal directly with this exponent because it does notdirectly fit in the framework given by Definition 7. But we will rather consider thefollowing intermediate framework.

Definition 18. Let f be a tempered distribution defined on a non-empty openset Ω ⊂ Rd. Let p ≥ 1 and s be large enough so that f (−s) belongs to Lp in aneighborhood of x0. The fractional p-exponent of order s of f at x0 is defined by

hp,sf (x0) = hp

f(−s)(x0)

(using the convention h∞f = hf ).

Note that, in practice, the standard way to perform the multifractal analysisof data that are not locally bounded is to deal with the exponent h∞,s

f , where s

is chosen large enough so that f (−s) ∈ L∞loc, i.e. it consists in first performing

a fractional integration, and then a standard multifractal analysis based on theHolder exponent, see [2] and references therein.

Similarly, in the function case, if the pointwise regularity exponents are smallenough, they can be recovered for the oscillation of f . Recall that the oscillation off of order l on a convex set A is defined through conditions on the finite differencesof the function f , denoted by ΔM

h f : The first order difference of f is

(Δ1hf)(x) = f(x + h) − f(x).

If l > 1, the differences of order l are defined recursively by

(Δlhf)(x) = (Δl−1

h f)(x + h) − (Δl−1h f)(x).

Then

Osclf (A) = supx,x+lh∈A

∣∣(Δlhf)(x)

∣∣ .One easily checks that the Holder exponent can be derived for the oscillation onthe cubes 3λ. Let f be locally bounded on an open set Ω.


If l > hf (x0), then

(6.3) ∀x0 ∈ Ω, hf (x0) = lim infj→+∞

logOsclf (3λj(x0))

log 2−j.

Recall also Proposition 3 which allows to derive numerically the Holder expo-nent by a log-log plot regression bearing on the the dλj(x0) when j → +∞, see[35].

However, in contradistinction with the measure case, a similar formula does nothold for the upper Holder exponent, see [23] where partial results in this directionand counterexamples are worked out.

We now turn to the wavelet characterization of the p-exponent. We will assumethat f locally belongs to Lp, with slow Lp-increase, i.e. satisfies

∃C,N > 0

∫Ω∩B(0,R)

|f(x)|pdx ≤ C(1 + |R|)N .

In the following, when dealing with the T pα regularity of a function f , we will

always assume that, if f is defined on an unbounded set Ω, then it has slow Lp-increase, and, if Ω �= Rd, then the wavelet basis used is compactly supported.

Definition 19. Let f ∈ Lploc(Ω), and let ψλ be a given wavelet basis. The

local square function of f is

Sf,λ(x) =

( ∑λ′⊂3λ

|cλ′ |21λ′(x)

)1/2

,

and the p-leaders are defined by dpλ = 2dj/p ‖ Sf,λ ‖p .

The following result of [37] yields a wavelet characterization of the p-exponentwhich is similar to (2.8).

Proposition 12. Let p ∈ (1,∞) and f ∈ Lp. Assume that the wavelet basisused is r-smooth with r > hp

f (x0) + 1. Then

(6.4) hpf (x0) = lim inf

j→+∞

log dpλj(x0)

log 2−j.

Recall that the “almost-diagonalization” principle for fractional integrals onwavelet bases states that, as regards Holder regularity, function spaces or scalingfunctions, one can consider that a fractional integration just acts as if it werediagonal on a wavelet basis, with coefficients 2−sj on ψλ. This rule of thumb isjustified by the fact that a fractional integration actually is the product of sucha diagonal operator and of an invertible Calderon-Zygmund operator A such thatA and A−1 both belong to the Lemarie algebras Mγ , for a γ arbitrarily large(and which depends only on the smoothness of the wavelet basis) see [47,48] forthe definition of the Lemarie algebras and for the result concerning function spacesand [35] and references therein for Holder regularity, function spaces or scalingfunctions.

It follows from Proposition 12 , and the “almost-diagonalization” principle forfractional integrals on wavelet bases, that the exponent hp,s

f (x0) can be obtainedas follows.


Corollary 7. Let p ∈ (1,∞) and f ∈ Lp. Let

Ssf,λ(x) =

( ∑λ′⊂3λ

|2−sj′cλ′ |21λ′(x)

)1/2

and dp,sλ = 2dj/p ‖ Ssf,λ ‖p .

Then, if the wavelet basis is r-smooth with r > hpf (x0) + s + 1, then

(6.5) hp,sf (x0) = lim inf

j→+∞

log dp,sλj(x0)

log 2−j.

7. A functional analysis point of view

7.1. Function space interpretation: Constant regularity. If p > 0, thescaling function has a function space interpretation, in terms of discrete Besovspaces which we now define. Recall that the elements of a dyadic family are alwaysnon-negative.

Definition 20. Let s ∈ R and p ∈ R. A dyadic function E belongs to bs,∞p (Ω)

if

(7.1) ∃C, ∀j, 2−dj∑λ∈ΛΩ

j

∗(eλ)p ≤ C · 2−spj .

If p = +∞, a dyadic function E belongs to bs,∞∞ (Ω) if

(7.2) ∃C ∀λ : eλ ≤ C · 2−sj .

Note that, if p > 0, this condition (if applied to the moduli of the coefficients)defines a vector space. It is a Banach space if p ≥ 1, and a quasi-Banach space if0 < p < 1; recall that, in a quasi-Banach space, the triangular inequality is replacedby the weaker condition :

∃C, ∀x, y, ‖ x + y ‖≤ C(‖ x ‖ + ‖ y ‖).Definition 20 yields a function space interpretation to the scaling function whenp > 0. It is classical in this context to rather consider the scaling function

ηE(p) = τE(p) − d.

Then, if Ω is a bounded set,

∀p ∈ R, ηΩE (p) = sup{s : E ∈ bs/p,∞p (Ω)};

and, if Ω is unbounded, then the function space interpretation is the same, usingthe precaution supplied by (3.5). Additionally,

hΩE = sup {s : E ∈ bs,∞

∞ (Ω)} .The terminology of ”discrete Besov spaces” is justified by the fact that, if the

eλ are wavelet coefficients, then (7.1) and (7.2) are the wavelet characterization ofthe ”classical” Besov spaces Bs,∞

p (Rd) of functions (or distributions) defined on Rd;therefore each wavelet decomposition establishes an isomorphism between the spacebs,∞p (Rd) and the space Bs,∞

p (Rd), see [47]. Note that, when p = ∞, these Besov

spaces coincide with the Holder spaces Cs(Rd), so that, when the (eλ) are waveletcoefficients, then the uniform regularity exponent has the following interpretation

hΩE = sup{s : E ∈ Cs(Rd)}.


In the measure case, H. Triebel showed that the discrete Besov conditionsbearing on the μ(3λ) can also be related with the Besov regularity of the measureμ, see [62]:

If s < d, (μ(3λ)) ∈ bs,∞p (Rd) ⇐⇒ μ ∈ Bs−d,∞

p (Rd).

In the case where p = +∞, uniform regularity gives an important informationconcerning the sets A such that μ(A) > 0, as a consequence of the mass distributionprinciple, see Section 2: Since this estimate precisely means that the sequence(eλ) = (μ(3λ)) belongs to bs,∞

∞ (Ω), it follows that, if A ⊂ Ω and if a measure μsatisfies μ(A) > 0, then

(7.3) dim(A) ≥ hΩμ .

When the sequence E is composed of wavelet leaders, or of p-leaders, the cor-responding function spaces are no more Besov spaces, but alternative families offunction spaces, the Oscillation Spaces, see [34,36].

hΩE = sup{A : (3.7) holds}.

The following upper bounds for dimensions are classical for measures, see [20],and are stated in the general setting of dyadic functions in [38].

Proposition 13. Let E be a dyadic function, and let

JΩH = {x ∈ Ω : hE(x) ≥ H}, GΩ

H = {x ∈ Ω : hE(x) ≥ H},

FΩH = {x ∈ Ω : hE(x) ≤ H)}, KΩ

H = {x ∈ Ω : hE(x) ≤ H)}.• If E ∈ bs,∞

p (Ω) with p > 0, then dim(GΩH) ≤ d− sp + Hp.

• If E ∈ bs,∞p (Ω) with p > 0, then dimp(F

ΩH) ≤ d− sp + Hp.

• If E ∈ bs,∞p (Ω) with p < 0, then dim(KΩ

H) ≤ d− sp + Hp.

• If E ∈ bs,∞p (Ω) with p < 0, then dimp(J

ΩH) ≤ d− sp + Hp.

7.2. Function space interpretation: Varying regularity. Recall that theglobal scaling function has a function space interpretation in terms of Besov spaceswhich contain the dyadic function E . Similarly, the local scaling function can begiven two functional interpretations; one is local, and in terms of germ spaces ata point, and the second is global, and is in terms of function spaces with varyingsmoothness. We now recall these notions, starting with germ spaces in a general,abstract setting.

Definition 21. Let E be a Banach space (or a quasi-Banach space) of dis-tributions satisfying D ↪→ E ↪→ D′. Let x ∈ Rd; a distribution f belongs to Elocally at x if there exists ϕ ∈ D such that ϕ(x) = 1 in a neighborhood of x andfϕ ∈ E. We also say that f belongs to the germ space of E at x, denoted by Ex.

Let us draw the relationship between the local scaling function and germ spaces:If the (eλ) are the wavelet coefficients of a function f , then

∀p > 0, ηf (x, p) = sup{s : f ∈ Bs/p,∞

p,x

}.

Note that, in the wavelet case, these local Besov regularity indices have been inves-tigated by H. Triebel, see Theorem 4 of [63] where their wavelet characterization isderived, (the reader should be careful that what is referred to as “pointwise regular-ity” in the terminology introduced by H. Triebel is called here “local regularity”).


The uniform exponent can also be reformulated in terms of Holder spaces:

Hf (x) = sup {s : f ∈ Csx} .

In that case, the function Hf (x) is called the Local Holder exponent of f . Itsproperties have been investigated by S. Seuret and his collaborators, see e.g. [44].

Note that the definition of germ spaces can be adapted to the dyadic functionssetting.

Definition 22. Let E be a Banach space (or a quasi-Banach space) definedon dyadic functions over Ω ; a dyadic function (eλ) belongs to Ex if there exists aneighborhood ω of x such that the dyadic function (eλ) restricted to ω belongs toE.

It is clear that this definition, when restricted to the case of Besov spaces andwavelet coefficients coincides with Definition 21.

We now turn to function spaces with varying smoothness. Such spaces wereinitially introduced by Unterberger and and Bokobza in [64,65], followed by manyauthors (see [55] for an extensive review on the subject). A general way to introducesuch spaces is to remark that the classical Sobolev spaces Hs,p(Rd) can be definedby the condition

‖ T (f) ‖p< ∞,

where T is the pseudo-differential operator defined by

(Tf)(x) =1

(2π)d

∫Rd

eixξ(1 + |ξ|2)s/2f(ξ)dξ.

This definition leads to operators with constant order s because the symbol (1 +|ξ|2)s/2 is independent of x. However, one can define more general spaces, withpossibly varying order if replacing (1 + |ξ|2)s/2 by a symbol σ(x, ξ). In particularthe symbols (1+ |ξ|2)a(x)/2 will lead to Sobolev spaces of varying order Ha,p wherewe can expect that, if a is a smooth enough function (say continuous), then thelocal order of smoothness at x will be a(x). This particular case, and its extensionsin the Besov setting, has been studied by H.G. Leopold, followed by J. Schneider,Besov, H. Triebel, A. Almeida, P. Hasto, J. Vybıral, and several other authors,who gave alternative characterizations of these space in terms of finite differencesor Littlewood-Paley decomposition. They also studied their mutual embeddings(and also in the case where both the order of smoothness and the order of integra-bility p vary) and their interpolation properties, see [56,57] and references therein,and also [55] for an historical account. The reader can also consult [3, 66] forrecent extensions in particular when both the order of smoothness and the orderof integrability p vary. We follow here the presentation of J. Schneider, since thisauthor obtained Littlewood-Paley characterizations, which are clearly equivalent tothe wavelet characterization that we now give. For the sake of simplicity, we as-sume form now on that the distributions considered are defined on Rd and that thewavelet basis used belongs to the Schwartz class (the usual adaptations are stan-dard in the case of functions on a domain, or for wavelets with limited regularity).We additionally assume that the function a is uniformly continuous and satisfies

(7.4) ∃c, C > 0, ∀x ∈ Rd, c ≤ a(x) ≤ C.

Then the Besov space Ba,qp (for p, q ∈ (0,∞]) can be characterized by the following

wavelet condition, which is independent of the wavelet basis used.


Proposition 14. Let a be a uniformly continuous function satisfying ( 7.4),and let p, q ∈ (0,∞]. The Besov space of varying order Ba,q

p is characterized by thefollowing condition:

Let cλ denote the wavelet coefficients of a distribution f , and let

aj =

⎛⎝2−dj∑λ∈Λj

(cλ2a(λ)j)p

⎞⎠1/p

,

where a(λ) denotes the average of the function a on the cube λ; then f ∈ Ba,qp if

(aj) ∈ lq.

Note that when p = q = 2 one recovers the Sobolev space Ha,2 defined above,and when a is a constant equal to s, then one recovers the standard Besov spaceBs,q

p . Furthermore, the embeddings

Ba,1p ↪→ Ha,p ↪→ Ba,∞

p

yield easy to handle “almost characterizations” of Sobolev spaces of varying order.The following result, which follows directly from the definition of the local

scaling function (Definition 8) and the characterization supplied by Proposition14, gives the interpretation of the local scaling function in terms Besov spaces ofvarying order.

Proposition 15. Let f be a distribution defined on Rd. Then, for p > 0, thelocal wavelet scaling function of f can be recovered by

∀p > 0, ηf (p, x) = p · sup{a : f ∈ Ba,∞p }.

Acknowledgement

This work is supported by the ANR grants AMATIS and MUTADIS and bythe LaBeX Bezout. The last two authors thank M. Lapidus for his kind invitationto the fractal geometry session in the AMS conference, March 2012, which allowedthem to discuss the topics of the paper with several participants.

References

[1] Stephane Jaffard, Bruno Lashermes, and Patrice Abry, Wavelet leaders in multifractal analy-sis, Wavelet analysis and applications, Appl. Numer. Harmon. Anal., Birkhauser, Basel, 2007,pp. 201–246, DOI 10.1007/978-3-7643-7778-6 17. MR2297921 (2008e:42067)

[2] P. Abry, S. Jaffard and H. Wendt, Irregularities and scaling in signal and Image process-ing: Multifractal analysis, to appear in: M. Frame Ed., Benoit Mandelbrot: A Life in ManyDimensions, World Scientific, to appear, 2013.

[3] Alexandre Almeida and Peter Hasto, Besov spaces with variable smoothness and integrability,J. Funct. Anal. 258 (2010), no. 5, 1628–1655, DOI 10.1016/j.jfa.2009.09.012. MR2566313(2011e:46045)

[4] A. Arneodo, B. Audit, N. Decoster, J.-F. Muzy, C. Vaillant, Wavelet-based multifractalformalism: applications to DNA sequences, satellite images of the cloud structure and stockmarket data, in: The Science of Disasters; A. Bunde, J. Kropp, H. J. Schellnhuber Eds.,Springer pp. 27–102 (2002).

[5] Jean-Marie Aubry and Stephane Jaffard, Random wavelet series, Comm. Math. Phys. 227(2002), no. 3, 483–514, DOI 10.1007/s002200200630. MR1910828 (2003g:42054)

[6] Antoine Ayache and Murad S. Taqqu, Multifractional processes with random exponent,Publ. Mat. 49 (2005), no. 2, 459–486, DOI 10.5565/PUBLMAT 49205 11. MR2177638(2006m:60057)


[7] Antoine Ayache, Stephane Jaffard, and Murad S. Taqqu, Wavelet construction of gen-eralized multifractional processes, Rev. Mat. Iberoam. 23 (2007), no. 1, 327–370, DOI10.4171/RMI/497. MR2351137 (2008h:60142)

[8] In-Soo Baek, Hausdorff dimension of deranged Cantor set without some bound-edness condition, Commun. Korean Math. Soc. 19 (2004), no. 1, 113–117, DOI10.4134/CKMS.2004.19.1.113. MR2029113 (2004j:28013)

[9] Julien Barral, Nicolas Fournier, Stephane Jaffard, and Stephane Seuret, A pure jump Markov

process with a random singularity spectrum, Ann. Probab. 38 (2010), no. 5, 1924–1946, DOI10.1214/10-AOP533. MR2722790 (2011m:60253)

[10] J. Barral, J. Berestycki, J. Bertoin, A. H. Fan, B. Haas, S. Jaffard, G. Miermont, and J.Peyriere,Quelques interactions entre analyse, probabilites et fractals, Panoramas et Syntheses[Panoramas and Syntheses], vol. 32, Societe Mathematique de France, Paris, 2010 (French).A Benoıt Mandelbrot, in memoriam. [To Benoıt Mandelbrot, in memoriam]. MR2920320(2012m:28001)

[11] Julien Barral, Fathi Ben Nasr, and Jacques Peyriere, Comparing multifractal formalisms:the neighboring boxes condition, Asian J. Math. 7 (2003), no. 2, 149–165. MR2014962(2004i:28015)

[12] Julien Barral and Benoıt B. Mandelbrot, Non-degeneracy, moments, dimension, and mul-tifractal analysis for random multiplicative measures (Random multiplicative multifractalmeasures. II), Fractal geometry and applications: a jubilee of Benoıt Mandelbrot, Part 2,Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 17–52.MR2112120 (2005i:28014)

[13] Julien Barral and Stephane Seuret, Function series with multifractal variations, Math. Nachr.274/275 (2004), 3–18, DOI 10.1002/mana.200410199. MR2092321 (2005f:28014)

[14] Julien Barral and Stephane Seuret, Combining multifractal additive and multiplicativechaos, Comm. Math. Phys. 257 (2005), no. 2, 473–497, DOI 10.1007/s00220-005-1328-3.MR2164606 (2006e:28007)

[15] Julien Barral and Stephane Seuret, The singularity spectrum of Levy processes in multifractaltime, Adv. Math. 214 (2007), no. 1, 437–468, DOI 10.1016/j.aim.2007.02.007. MR2348038(2008k:60086)

[16] Julien Barral and Stephane Seuret, The multifractal nature of heterogeneous sums ofDirac masses, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 707–727, DOI10.1017/S0305004107000953. MR2418713 (2009c:28010)

[17] Julien Barral and Stephane Seuret, Information parameters and large deviation spectrumof discontinuous measures, Real Anal. Exchange 32 (2007), no. 2, 429–454. MR2369854(2009m:60062)

[18] Julien Barral and Yan-Hui Qu, Localized asymptotic behavior for almost additive poten-tials, Discrete Contin. Dyn. Syst. 32 (2012), no. 3, 717–751, DOI 10.3934/dcds.2012.32.717.MR2851877 (2012i:37045)

[19] Albert Benassi, Stephane Jaffard, and Daniel Roux, Elliptic Gaussian random processes, Rev.Mat. Iberoamericana 13 (1997), no. 1, 19–90, DOI 10.4171/RMI/217 (English, with Englishand French summaries). MR1462329 (98k:60056)

[20] G. Brown, G. Michon, and J. Peyriere, On the multifractal analysis of measures, J. Statist.Phys. 66 (1992), no. 3-4, 775–790, DOI 10.1007/BF01055700. MR1151978 (93c:58120)

[21] Z. Buczolich, S. Seuret, Measures and functions with prescribed homogeneous multifractalspectrum, Preprint, 2012.

[22] A.-P. Calderon and A. Zygmund, Local properties of solutions of elliptic partial differentialequations, Studia Math. 20 (1961), 171–225. MR0136849 (25 #310)

[23] Marianne Clausel and Samuel Nicolay, Wavelets techniques for pointwise anti-Holderianirregularity, Constr. Approx. 33 (2011), no. 1, 41–75, DOI 10.1007/s00365-010-9120-9.MR2747056 (2011k:42072)

[24] Arnaud Durand, Singularity sets of Levy processes, Probab. Theory Related Fields 143

(2009), no. 3-4, 517–544, DOI 10.1007/s00440-007-0134-6. MR2475671 (2010e:60018)[25] Arnaud Durand and Stephane Jaffard, Multifractal analysis of Levy fields, Probab. Theory

Related Fields 153 (2012), no. 1-2, 45–96, DOI 10.1007/s00440-011-0340-0. MR2925570[26] K. E. Ellis, M. L. Lapidus, M. C. Mackenzie, and J. A. Rock, Partition zeta functions, multi-

fractal spectra, and tapestries of complex dimensions, in Benoit Mandelbrot: A Life in Many


Dimensions (M. Frame, Ed.), World Scientific, Singapore, in press, 2012. (arXiv:1007.1467v2[math-ph], 2011).

[27] K. J. Falconer, Sets with large intersection properties, J. London Math. Soc. (2) 49 (1994),no. 2, 267–280, DOI 10.1112/jlms/49.2.267. MR1260112 (95f:28004)

[28] Kenneth Falconer, Fractal geometry: Mathematical foundations and applications, 2nd ed.,John Wiley & Sons Inc., Hoboken, NJ, 2003. MR2118797 (2006b:28001)

[29] Ai-Hua Fan and De-Jun Feng, On the distribution of long-term time averages on sym-

bolic space, J. Statist. Phys. 99 (2000), no. 3-4, 813–856, DOI 10.1023/A:1018643512559.MR1766907 (2002d:82003)

[30] De-Jun Feng, Lyapunov exponents for products of matrices and multifractal analysis. II.General matrices, Israel J. Math. 170 (2009), 355–394, DOI 10.1007/s11856-009-0033-x.MR2506331 (2010e:37068)

[31] De-Jun Feng, Ka-Sing Lau, and Xiang-Yang Wang, Some exceptional phenomena in multi-fractal formalism. II, Asian J. Math. 9 (2005), no. 4, 473–488. MR2216241 (2006m:28009)

[32] De-Jun Feng and Eric Olivier, Multifractal analysis of weak Gibbs measures and phasetransition—application to some Bernoulli convolutions, Ergodic Theory Dynam. Systems23 (2003), no. 6, 1751–1784, DOI 10.1017/S0143385703000051. MR2032487 (2004m:37053)

[33] Stephane Jaffard, The multifractal nature of Levy processes, Probab. Theory Related Fields114 (1999), no. 2, 207–227, DOI 10.1007/s004400050224. MR1701520 (2000g:60079)

[34] Stephane Jaffard, Oscillation spaces: properties and applications to fractal and multifractralfunctions, J. Math. Phys. 39 (1998), no. 8, 4129–4141, DOI 10.1063/1.532488. MR1633152(2000c:28011)

[35] Stephane Jaffard, Wavelet techniques in multifractal analysis, Fractal geometry and appli-cations: a jubilee of Benoıt Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, Amer.Math. Soc., Providence, RI, 2004, pp. 91–151. MR2112122 (2006c:28005)

[36] Stephane Jaffard, Beyond Besov spaces. II. Oscillation spaces, Constr. Approx. 21 (2005),no. 1, 29–61. MR2105390 (2005g:42055)

[37] Stephane Jaffard, Pointwise regularity associated with function spaces and multifractal analy-sis, Approximation and probability, Banach Center Publ., vol. 72, Polish Acad. Sci., Warsaw,2006, pp. 93–110, DOI 10.4064/bc72-0-7. MR2325740 (2008g:28044)

[38] Stephane Jaffard, Patrice Abry, Stephane G. Roux, Beatrice Vedel, and Herwig Wendt, Thecontribution of wavelets in multifractal analysis, Wavelet methods in mathematical analysisand engineering, Ser. Contemp. Appl. Math. CAM, vol. 14, Higher Ed. Press, Beijing, 2010,pp. 51–98, DOI 10.1142/9789814322874 0003. MR2757646 (2012j:42086)

[39] H. Jurgensen and L. Staiger, Local Hausdorff dimension, Acta Inform. 32 (1995), no. 5,491–507, DOI 10.1007/s002360050025. MR1348439 (96j:68097)

[40] A. Kaenmaki, T. Rajala and N. Suomala, Local homogeneity and dimensions of measures,preprint, arXiv:1003.2895, 2012.

[41] A. Kaenmaki, T. Rajala and N. Suomala, Local multifractal analysis in metric spaces,preprint, arXiv:1209.3648, 2012.

[42] Marc Kessebohmer, Large deviation for weak Gibbs measures and multifractal spectra,Nonlinearity 14 (2001), no. 2, 395–409, DOI 10.1088/0951-7715/14/2/312. MR1819804(2002a:60037)

[43] P. G. Lemarie and Y. Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2(1986), no. 1-2, 1–18, DOI 10.4171/RMI/22 (French). MR864650

[44] Stephane Seuret and Jacques Levy Vehel, The local Holder function of a continuous function,Appl. Comput. Harmon. Anal. 13 (2002), no. 3, 263–276, DOI 10.1016/S1063-5203(02)00508-0. MR1942744 (2003m:42065)

[45] Jacques Levy Vehel and Robert Vojak, Multifractal analysis of Choquet capacities, Adv. inAppl. Math. 20 (1998), no. 1, 1–43, DOI 10.1006/aama.1996.0517. MR1488230 (98j:28003)

[46] R. Daniel Mauldin and Mariusz Urbanski, Graph directed Markov systems, Cambridge Tractsin Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003. Geometry and dy-

namics of limit sets. MR2003772 (2006e:37036)[47] Yves Meyer, Ondelettes et operateurs. I, Actualites Mathematiques. [Current Mathematical

Topics], Hermann, Paris, 1990 (French). Ondelettes. [Wavelets]. MR1085487 (93i:42002)[48] Yves Meyer and Ronald Coifman, Wavelets, Cambridge Studies in Advanced Mathematics,

vol. 48, Cambridge University Press, Cambridge, 1997. Calderon-Zygmund and multilinear


operators; Translated from the 1990 and 1991 French originals by David Salinger. MR1456993(98e:42001)

[49] Yves Meyer, Wavelets, vibrations and scalings, CRM Monograph Series, vol. 9, Ameri-can Mathematical Society, Providence, RI, 1998. With a preface in French by the author.MR1483896 (99i:42051)

[50] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), no. 1, 82–196, DOI10.1006/aima.1995.1066. MR1361481 (97a:28006)

[51] Norbert Patzschke, Self-conformal multifractal measures, Adv. in Appl. Math. 19 (1997),no. 4, 486–513, DOI 10.1006/aama.1997.0557. MR1479016 (99c:28020)

[52] R. Peltier and J. Levy-Vehel, Multifractional Brownian motion: definition and preliminaryresults, Tech. Rep. RR 2645, INRIA Le Chesnay, France, 1995.

[53] Yakov B. Pesin, Dimension theory in dynamical systems: Contemporary views and appli-cations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.MR1489237 (99b:58003)

[54] J. Schmeling, S. Seuret, On measures resisting multifractal analysis, Book chapter in “Non-linear Dynamics: New Directions” (in honor of V. Afraimovich), Springer 2012 (in press).

[55] J. Schneider, Function spaces with varying smoothness, PhD Dissertation available onhttp://personal-homepages.mis.mpg.de/jschneid/Public.html 2005.

[56] Jan Schneider, Function spaces of varying smoothness. I, Math. Nachr. 280 (2007), no. 16,1801–1826, DOI 10.1002/mana.200610580. MR2365019 (2009a:46032)

[57] Jan Schneider, Some results on function spaces of varying smoothness, Function spaces VIII,Banach Center Publ., vol. 79, Polish Acad. Sci. Inst. Math., Warsaw, 2008, pp. 187–195.MR2404993 (2009e:46030)

[58] Stephane Seuret, On multifractality and time subordination for continuous functions, Adv.Math. 220 (2009), no. 3, 936–963, DOI 10.1016/j.aim.2008.10.009. MR2483233 (2010a:28016)

[59] Pablo Shmerkin, A modified multifractal formalism for a class of self-similar measures withoverlap, Asian J. Math. 9 (2005), no. 3, 323–348. MR2214956 (2007b:28009)

[60] B. Testud, Phase transitions for the multifractal analysis of self-similar measures, Non-linearity 19 (2006), no. 5, 1201–1217, DOI 10.1088/0951-7715/19/5/009. MR2222365(2007h:28014)

[61] Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos.Soc. 91 (1982), no. 1, 57–74, DOI 10.1017/S0305004100059119. MR633256 (84d:28013)

[62] Hans Triebel, Fractal characteristics of measures; an approach via function spaces, J.Fourier Anal. Appl. 9 (2003), no. 4, 411–430, DOI 10.1007/s00041-003-0020-2. MR1999566(2004g:28013)

[63] Hans Triebel, Wavelet frames for distributions; local and pointwise regularity, Studia Math.154 (2003), no. 1, 59–88, DOI 10.4064/sm154-1-5. MR1949049 (2004e:42063)

[64] Andre Unterberger and Juliane Bokobza, Sur une generalisation des operateurs de Calderon-Zygmund et des espaces Hs, C. R. Acad. Sci. Paris 260 (1965), 3265–3267 (French).MR0188832 (32 #6264)

[65] Andre Unterberger and Juliane Bokobza, Les operateurs pseudo-differentiels d’ordre variable,C. R. Acad. Sci. Paris 261 (1965), 2271–2273 (French). MR0187113 (32 #4567)

[66] Jan Vybıral, Sobolev and Jawerth embeddings for spaces with variable smoothness and inte-grability, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 529–544. MR2553811 (2010m:46053)

Julien BARRAL, LAGA, Institut Galilee, Universite Paris 13, 99 avenue Jean-

Baptiste Clement, 93430 - Villetaneuse France


Arnaud DURAND, Laboratoire de Mathematiques d’Orsay, UMR 8628, Universite

Paris-Sud, 91405 Orsay Cedex France


Stephane JAFFARD, Universite Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC,

CNRS, F-94010, Creteil, France


Stephane SEURET, Universite Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS,

F-94010, Creteil, France



Extreme Risk and Fractal Regularity in Finance

Laurent E. Calvet and Adlai J. Fisher

Abstract. As the Great Financial Crisis reminds us, extreme movements inthe level and volatility of asset prices are key features of financial markets.These phenomena are difficult to quantify using traditional approaches thatspecify extreme risk as a singular rare event detached from ordinary dynamics.Multifractal analysis, whose use in finance has considerably expanded over the

past fifteen years, reveals that price series observed at different time horizonsexhibit several major forms of scale-invariance. Building on these regularities,researchers have developed a new class of multifractal processes that permitthe extrapolation from high-frequency to low-frequency events and generateaccurate forecasts of asset volatility. The new models provide a structuredframework for studying the likely size and price impact of events that aremore extreme than the ones historically observed.

1. Introduction

Fractal modeling uses invariance principles to parsimoniously specify complexobjects at multiple scales. It has proven to be of major importance in mathemat-ics and the natural sciences, as this issue illustrates. Fractals also offer enormousbenefits for the field of finance, in particular for modeling the price of traded se-curities, for computing the risk of financial portfolios, for managing the exposureof institutions, or for pricing derivative securities. These benefits should becomemore apparent as the adoption of fractal methods by the financial industry contin-ues to gain ground. The fields of finance and economics also play a singular rolein the intellectual history of fractals. Benoıt Mandelbrot first discovered evidenceof fractal behavior in financial returns, household income and household wealth inthe late 1950’s and early 1960’s, and subsequently found similar patterns in coast-lines, earthquakes and other natural phenomena. These observations prompted thedevelopment of the fractal and multifractal geometry of nature ([M82]).1

2010 Mathematics Subject Classification. Primary 60G18, 60G22, 60G51, 62M05, 62M20,91G70.

The authors thank the Editors and two anonymous referees for insightful and constructive

suggestions. The paper also benefited from helpful comments from Robert Barro, Mico Loretan,as well as seminar participants at numerous institutions. Charles Martineau provided outstandingresearch assistance.

1Benoıt Mandelbrot eventually returned to finance in the mid-1990’s, when he taught aFractals in Finance course at Yale University. We attended this course and went on to developwith Benoıt Mandelbrot the first applications of multifractals to financial series.


65

66 LAURENT E. CALVET AND ADLAI J. FISHER

In financial markets, the distribution of price changes is of key importance be-cause it determines the risk, as well as the potential gains, of a position or a portfolioof assets. Different investors may measure price changes at different horizons. Forinstance, a high-frequency trader may look at price changes over microseconds,while a pension fund or a university endowment may have horizons of months,years, or decades. Researchers have correspondingly investigated invariance prop-erties in the distribution of price changes observed over different time increments.The French economist Jules Regnault (1863) may have been the first to observe thatthe standard deviation of a price change over a time interval of length Δt scales asthe square root of Δt ([R]). This observation motivated Louis Bachelier ([Ba]) toformalize the definition of the Brownian motion and propose it as a possible modelof a stock price. That is, Bachelier postulated that price changes are Gaussian,identically distributed and independent. While these assumptions have importantlimitations, Bachelier opened up the field of financial statistics, which has remainedvibrant ever since.

In the early 1960’s, Benoıt Mandelbrot discovered that price changes have muchthicker tails than the Gaussian distribution permits ([M63]). He proposed to re-place the Brownian motion with another family of scale-invariant processes withindependent increments – the stable processes of Paul Levy ([L24]). Let p(t) de-note the logarithm of a stock price or an exchange rate. If p(t) follows a Brownianor a Levy process, the distributions of the returns p(t + Δt) − p(t) observed overvarious horizons Δt can be obtained from each other by linear rescaling. Thisform of linear invariance turns out to be a rather crude approximation of finan-cial series. Another shortcoming of Brownian motion and Levy processes is theassumption that price changes are independent. As Benoıt Mandelbrot ([M63])himself pointed out,2 the size of price changes, |p(t + Δt) − p(t)|, is persistent infinancial data ([E82], [Bo87]). In fact for many series, the size of price changes isa long-memory process characterized by a hyperbolically declining autocorrelationat long lags ([DGE],[D]).

Since the mid–1990’s, researchers have uncovered alternative forms of scale in-variance in financial returns, based on multifractal moment-scaling. [G], [CFM],and [CF02] found evidence that the moments of the absolute value of price changes,E(|p(t + Δt) − p(t)|q), scale as power functions of the horizon Δt. Multifrac-tal moment-scaling had until then been observed in natural phenomena such asthe distribution of energy in turbulent flows and the distribution of minerals inEarth’s crust. These physical regularities can be modeled with multifractal mea-sures ([M74]). The observation of moment-scaling in financial returns motivatedresearchers to construct the first family of multifractal diffusions ([CFM], [CF01],[BDM]). These processes are parsimonious and capture well the fat tails, long mem-ory in volatility and moment scaling of financial series ([CF01], [CF04], [CF]).

The leading example considered in the survey, the Markov-Switching Multifrac-tal (MSM), assumes that the size of a price change is driven by components thathave identical distributions but different degrees of persistence. The dynamics ofevery component are specified by a Markov chain with its own transition probabil-ities. MSM thus constructs a multifractal measure stochastically over time, whichimproves over earlier multifractal measures with predetermined switching dates.

2Benoıt Mandelbrot noted that: “[...] large changes tend to be followed by large changes –of either sign – and small changes tend to be followed by small changes, [...].”

EXTREME RISK AND FRACTAL REGULARITY IN FINANCE 67

The dynamic definition of MSM permits the adoption of powerful estimation andfiltering methods. MSM generates accurate forecasts of the conditional distributionof returns and therefore of the upside potential and downside risk of a position.

Fractals provide a natural mathematical structure for modeling large risks. Acommon approach in finance is to represent an asset price as the sum of an Ito dif-fusion and a jump process. The diffusion describes “ordinary” fluctuations, whilejumps are meant to capture “rare events.” Difficulties in the empirical implemen-tation of such approaches are readily apparent. Because rare events are modeledas intrinsically different from regular variations, inference on rare events must beconducted on a small set of observations and is therefore highly imprecise. Ofincreasing importance, researchers would like to understand the implications forasset prices of events that have never been previously observed (“peso effects,”[R88], [B06], [G12], [W], [IM]). Statistical inference on an empty set, however, isa notoriously challenging exercise!

Fractal modeling offers scale invariance as a solution to this quandary. It istherefore a promising tool to a profession that is becoming increasingly aware of theimportance of rare events. Scale-invariant properties permit researchers to modelall price variations using a single data-generating mechanism. As a consequence,models constructed using fractal principles are extremely parsimonious. A smallnumber of well-identified parameters, combined with testable assumptions on scale-invariance, specify price dynamics at all timescales. The tight specification of rareevents, even those more extreme than have been observed in existing data, is anatural outcome of a fractal approach to modeling financial prices.

The organization of the paper is as follows. Section 2 discusses early fractalmodels and reviews fractal regularities in financial markets. Section 3 presentsthe Markov-Switching multifractal model and its empirical applications. Section 4analyzes the pricing implications of multifractal risk. Section 5 concludes.

2. Fractal Regularities in Financial Markets

2.1. Self-Similar Proposals. Let P (t) denote the price at date t of a finan-cial asset, such as a stock or a currency, and let p(t) denote its logarithm. Theasset’s logarithmic return between dates t and t + Δt is given by:

p(t + Δt) − p(t).

For over a century, one of the leading themes in finance has been to understand thedynamics of asset returns.

In his 1900 doctoral dissertation, the French mathematician Bachelier intro-duced an early definition of the Brownian motion as a model of the stock price([Ba]). If p(t) follows a Brownian motion with drift, the return p(t+Δt)−p(t) hasa Gaussian distribution with mean μΔt and variance σ2Δt, or more concisely

p(t + Δt) − p(t)d= N (μΔt;σ2Δt).

The Brownian motion now pervades modern financial theory and notably the Black–Merton–Scholes approach to continuous–time valuation ([BS], [M]). Its lasting suc-cess arises from its tractability and consistency with the financial concepts of no-arbitrage and market efficiency. However, empirical difficulties with the Brownianmotion have become apparent over time.

In the late 1950’s and early 1960’s, advances in computing technology madeit possible to conduct more precise tests of Bachelier’s hypotheses. In a series of


pathbreaking papers, Benoıt Mandelbrot ([M63], [M67]) uncovered major depar-tures from the Brownian motion in commodity, stock and currency series. His mainobservation was that the tails of return distributions are thicker than the Brow-nian motion permits. Benoıt Mandelbrot understood that this phenomenon wasnot a mere statistical curiosity, as some researchers suggested at the time, but amajor failure of the Brownian paradigm. In layman’s terms, extreme price changesare key features of financial markets that the Brownian motion cannot capture.Since the purpose of risk management is to weather financial institutions againststorms, underestimating the size of these storms, as the thin-tailed Brownian modeldoes, is a recipe for financial disaster, panic and bankruptcy. The Great FinancialCrisis reminds us of the severity of the shocks that can be unleashed on financialinstitutions, especially those who took on excessive risk as a result of poor riskmanagement models and practices.

Benoıt Mandelbrot advocated that financial prices should be modeled by abroader class of stochastic processes.

Definition 1. (Self-similar process) The real-valued process {p(t); t ∈ R+}is said to be self-similar with index H if the vector (p(ct1), ..., p(ctn)) has the samedistribution as (cHp(t1), ..., c

Hp(tn)), or more concisely

(2.1) (p(ct1), ..., p(ctn))d= (cHp(t1), ..., c

Hp(tn)),

for every c > 0, n > 0, and t1, ..., tn ∈ R+.

The Brownian motion satisfies (2.1) with the index H = 1/2. Two other families ofself-similar processes have also been influential in finance (e.g., [M97]), as we nowdiscuss.3

The stable processes of Paul Levy ([L24]) are self-similar with H ∈ (1/2,+∞).Their increments are independent and have Paretian tails:

(2.2) P{|p(t + Δt) − p(t)| > x} ∼ Kα Δt x−α

as x → +∞, where α = 1/H ∈ (0; 2) and Kα is a positive constant. As (2.2) shows,Levy processes have thicker tails than the Brownian motion and are therefore morelikely to accommodate large price changes ([M63], [M67]). One difficulty withLevy-stable processes, however, is that they have infinite variances, which is atodds with the empirical evidence available for a large number of price series ([BG],[FR], [AB]). Furthermore, infinite variances pose major difficulties for financialtheory because much of the asset pricing literature uses a security return’s varianceor its covariance with other securities or factors as the main quantitative measuresof risk (see, e.g, [CV], [MK], [S], [T], [M]).

Fractional Brownian motions represent another important class of self-similarprocesses ([K], [M65], [MV68]). A fractional Brownian motion with initial valueBH(0) = 0 can be defined as:

BH(t) =1

Γ(H + 1

2

) {∫ 0

−∞[(t− s)H−1/2 − (−s)H−1/2]dZs +

∫ t

0

(t− s)H−1/2dZs

},

3We refer the reader to [ST] for a detailed treatment of self-similar processes.


where Γ denotes Euler’s gamma function, Z is a standard Brownian motion, andthe index H is between 0 and unity. A fractional Brownian motion is therefore con-structed by assigning hyperbolic weights to the increments of a standard Brownianmotion, which generates strong persistence.

Let

(2.3) rt = p(t) − p(t− 1)

denote the return on a time interval of unit length, such as one day. If the pricep(t) follows a fractional Brownian motion with self-similarity index H �= 1/2, thereturn autocorrelation declines at the hyperbolic rate:

(2.4) Corr(rt; rt+n) ∼ H(2H − 1)n2H−2 as n → ∞.

The strong dependence of returns implied by the fractional Brownian motion isat odds with empirical evidence. Indeed, a large body of research (e.g., [K53],[GM63], [F65]) shows that over a wide range of sampling frequencies, asset returnsexhibit either zero or weak autocorrelation: Corr(rt, rt+n) ≈ 0 for all n �= 0, as thesimplest forms of market efficiency suggest. Furthermore, long memory in returns(2.4) is theoretically inconsistent with arbitrage-pricing in continuous time ([MS]),which makes it an unappealing model of financial prices. Fractional integrationcan be, however, useful for modeling persistence in the size of price changes (e.g.,[BBM], [HMS]), as will be further discussed below.

Besides the aforementioned shortcomings of stable processes and fractionalBrownian motions, self-similar processes with stationary increments face another,common difficulty. By (2.1), the returns at various horizons should have identicaldistributions up to a scalar renormalization:

(2.5) p(t + Δt) − p(t)d= (Δt)Hp(1).

Most financial series, however, are not exactly self-similar, but have thicker tailsand are more peaked in the bell at shorter horizons than the self-similarity condition(2.5) predicts. This empirical observation is consistent with the economic intuitionthat higher frequency returns are either large if new information has arrived, or closeto zero otherwise. For this reason, self-similar processes cannot be fully satisfactorymodels of asset returns.

2.2. Empirical Evidence on Fat Tails and Long Memory. Following[M63] and [M67], a number of researchers have measured the return tail indexesα and α′ defined by

P(rt > x) ∼ Kx−α,

P(rt < −x) ∼ K ′x−α′

as x → +∞, where K and K ′ are fixed elements of R+. Early studies on financialreturn tails were mainly parametric ([F63], [F65], [BG], [FR], [AB]). In the 1970’s,statisticians developed precise techniques for the nonparametric estimation of thetail indexes of a distribution ([Hi], [CDM]), and subsequent empirical analysesconfirmed that tail indexes are finite in financial series (e.g., [KK], [KSV], [PMM],[JD], [LP], [G09]). In most studies, α and α′ are also measured to be larger than2. Asset returns therefore have a finite variance, consistent with the assumptionsof financial theory.

In the early 1990’s, researchers uncovered evidence of strong persistence inthe absolute value of returns ([D], [DGE]). Long memory is often defined by a


06/73 03/81 12/88 09/96 06/04 03/12

−6

−4

−2

0

2

4

6

Japanese Yen (JPY/USD) Log Returns

01/26 09/33 04/41 01/49 09/57 09/66 11/75 11/84 11/93 12/02 12/11

−20

−15

−10

−5

0

5

10

15CRSP Value Weighted Index Log Returns

Figure 1. Financial Return Series. This figure shows daily log-arithmic returns for the Japanese yen / U.S. dollar exchange rateseries, and for the value weighted U.S. stock index compiled by theCenter for Research in Securities Prices (CRSP). The yen seriesbegins on June 1, 1973 and ends on March 30, 2012. The stockseries begins on January 2, 1926 and ends on December 30, 2011.

hyperbolic decline in the autocorrelation function as the lag goes to infinity. Forevery moment q ≥ 0 and every integer n, let

(2.6) ρq(n) = Corr(|rt|q , |rt+n|q)denote the autocorrelation in levels. We say that the asset exhibits long memoryin the size of returns if ρq(n) is hyperbolic in n :

(2.7) ρq(n) ∝ cqn−δ(q)

as n → +∞.These important features of financial data can be seen by casual observation of

standard asset returns. Figure 1, Panel A, shows the Japanese Yen / U.S. dollarexchange rate series from 1973, following the demise of the Bretton-Woods systemof fixed exchange rates, to the present day. The yen series contains 9751 returnobservations. The series shows both fat tails and volatility clustering at differenttime scales, including over periods as long as several years, as occurs in the presence


100

101

102

10−2

10−1

Japanese Yen/U.S. Dollar

Lag

Aut

ocor

rela

tion

100

101

102

10−1

CRSP Value Weighted Index

Lag

Aut

ocor

rela

tion

Figure 2. Long Memory in Squared Returns. This figure illus-trates the autocorrelations of squared logarithmic returns for theyen (Panel A) and the CRSP stock index return series (Panel B).The autocorrelations are plotted on a log-log scale, so that a hy-perbolic decay in autocorrelations, as occurs under long-memory,will appear as an approximately straight line in the figure.

of long memory. Panel B shows the same features in a long time series of 22,780daily U.S. stock index returns obtained from the University of Chicago’s Center forResearch in Security Prices (CRSP).

In Figure 2, we display the autocorrelation of squared returns ρ2(n) on thevertical axis versus the lag length n on the horizontal axis. For both series theplots are approximately linear on a double logarithmic scale, indicating that ρ2(n)is hyperbolic in n. The yen/dollar exchange rate and the U.S. stock index seriesthus both exhibit long memory in the size of price changes.

2.3. Multifractal scaling. In the mid-1990’s, the observation that asset re-turns exhibit both fat tails and long memory in volatility led researchers to considerthat asset prices may exhibit multifractal moment-scaling:

(2.8) E (|p(t + Δt) − p(t)|q) = cq(Δt)τ(q)+1


d w m y

10−1

100

101

102

103

Japanese Yen (JPY/USD)

Δt

Sq(T

, Δt)

q=1

q=2

q=3

q=5

d w m y

10−1

100

101

102

103

104

CRSP Value Weighted Index

Δt

Sq(T

, Δt)

q=1

q=2

q=3

q=5

Figure 3. Moment Scaling of Financial Returns. For every sam-pling interval Δt, we partition the total observation period [0, T ]into N = T/Δt subintervals and compute the partition function

Sq(T,Δt) ≡∑N−1

i=0 |p(iΔt + Δt) −X(iΔt)|q . The figure provideslog-log plots of Sq(T,Δt) against values of Δt ranging from 1day (“d”) to 1 year (“y”). If p(t) is multifractal: E [|p(Δt)|q] =cq(Δt)τ(q)+1, then logE[Sq(T,Δt)] = τ (q) log(Δt) + log(Tcq) andthe plots should be approximately linear.

for every (finite) moment q and time interval Δt. A self-similar process satisfies(2.8), with τ (q) = Hq − 1. The process p is said to be strictly multifractal if (2.8)holds for a strictly concave function τ (q).

Strict multifractality has been observed in fields as diverse as fluid mechanics,geology and astronomy. We now also have strong evidence of strict multifractalmoment-scaling in a variety of financial series, including currencies and equities([CF02], [CFM], [G], [VA]). As an example, we illustrate in Figure 3 the moment-scaling properties of the Yen / U.S. dollar exchange rate and the CRSP stockindex. The panels of the figure plot the partition interval Δt on the horizontal axisversus an empirical estimate of E (|p(t + Δt) − p(t)|q) on the vertical axis. Theempirical estimate is obtained by taking the sample analogue of (2.8), as explainedin the figure caption, for a variety of moments q. The dotted lines in the figurerepresent the scaling implied by Brownian motion, which satisfies self-similaritywith H = 1/2. The panels both show evidence of moment-scaling that is linear in


Δt, but the scaling coefficients τ (q) cannot be captured as a linear function of asingle index H. These empirical facts are characteristics of multifractal scaling.

The mathematical modeling of multifractal objects first focused on randommeasures, constructed by iterative reallocation of mass over a domain (e.g., [M74]).One of the simplest examples is the binomial measure4 on the unit interval [0, 1],which we derive as the limit of a multiplicative cascade. Consider a fixed m0 ∈[1/2, 1] and the Bernoulli (also called binomial) distribution taking the high valuem0 or the low value 1−m0 with equal probability. In the first step of the cascade,we draw two independent values M0 and M1 from the binomial distribution. Wedefine a measure μ1 by uniformly spreading the mass M0 on the left subinterval[0, 1/2], and the mass M1 on the right subinterval [1/2, 1]. The density of μ1 is astep function.

In the second stage of the cascade, we draw four independent binomials M0,0,M0,1, M1,0 and M1,1. We split the interval [0, 1/2] into two subintervals of equallength; the left subinterval [0, 1/4] is allocated a fraction M0,0 of μ1[0, 1/2], while theright subinterval [1/4, 1/2] receives a fraction M0,1. Applying a similar procedureto [1/2, 1], we obtain a measure μ2 such that:

μ2[0, 1/4] = M0M0,0, μ2[1/4, 1/2] = M0M0,1,μ2[1/2, 3/4] = M1M1,0, μ2[3/4, 1] = M1M1,1.

Iteration of this procedure generates an infinite sequence of random measures (μk)that weakly converges to the binomial measure μ.

Consider a dyadic interval5 [t, t + 2−k], where t =∑k

i=1 ηi2−i and η1, ..., ηk ∈

{0, 1}. The measure of the interval is

μ[t, t + 2−k] = Mη1Mη1,η2

. . .Mη1,...,ηkΩ,

where Ω is a random variable determined by the change in the mass generated bystages k + 1, . . . ,∞ of the cascade. Equation (2.3) implies that

E(μ[t, t + 2−k]q) = cq [E(Mq)]k = cq (Δt)τμ(q)+1,

where cq = E(Ωq), Δt = 2−k, and τμ(q) = − log2[E(Mq)] − 1. The moments of thelimiting measure of a dyadic interval is therefore a power of its length Δt, similarto the scaling relation (2.8).

The extension of multifractality from random measures to stochastic processeswas first achieved in the Multifractal Model of Asset Returns (“MMAR”, [CFM],[CF02]). The MMAR provides a class of diffusions consistent with the multifractalscaling relation (2.8). In the MMAR, an asset price is specified by compounding aBrownian motion with an independent random time-deformation:

p(t) = p(0) + B[θ(t)],

where θ is the cumulative distribution of the multifractal measure μ. The definingfeature of the MMAR is the use of a multifractal time deformation. The MMARis thus related to subordination, a concept introduced in harmonic analysis byBochner ([B55]) and used for the first time by Clark ([Cl]) in the finance litera-ture. In the original formulation of [B55] and [Cl], a subordinator θ(t) is a right-continuous increasing process that has independent and homogenous increments.

4The binomial is sometimes called the Bernoulli or Besicovitch measure.5A number t ∈ [0, 1] is called dyadic if t = 1 or t = η12−1 + ... + ηk2

−k for a finite k andη1, .., ηk ∈ {0, 1}. A dyadic interval has dyadic endpoints.


While the original assumptions of independence and homogeneity have proven toorestrictive for financial applications, stochastic time changes are generally appealingfor modeling financial prices (see, eg., [AG]). By specifying the evolution of ran-dom trading time as multifractal, the MMAR provides an empirically realistic classof models that parsimoniously captures the fat tails and long-memory volatilitydependence of financial series.

The MMAR price process inherits the moment-scaling properties of the mea-sure, in the sense that E(|p(t+Δt)− p(t)|q) = (Δt)τμ(q/2)+1 on any dyadic interval[t, t+ Δt]. These moment restrictions represent the basis of estimation and testing([CFM], [CF02], [L08]). The MMAR provides a well-defined stochastic frameworkfor the analysis of moment-scaling. In [CF02], we have verified that the moment-scaling properties of financial returns, such as the ones exhibited in Figure 3, areconsistent with the range of variations predicted by the MMAR. Consistent withits ability to explain return moments at various frequencies, the MMAR capturesnonlinear variations in the unconditional density of returns observed at various timehorizons ([L01]).

The moment-scaling properties of the MMAR have generated extensive interestin econophysics (for example, [LB]). They are also related to recent econometricresearch on power variation, which interprets return moments at various frequen-cies in the context of traditional jump-diffusions (for example, [ABDL], [BNS]).Furthermore, recent research confirms that the arrival of transactions in financialmarkets is well described by a multifractal driving process ([CDS]), which confirmsthe economic motivation of the time deformation θ(t) as a multifractal “tradingtime.”

Despite its appealing properties, the MMAR is unwieldy for econometric ap-plications because of two features of the underlying measure: (a) the recursivereallocation of mass on an entire time-interval does not fit well with standard timeseries tools; and (b) the limiting measure contains a residual grid of instants thatmakes it non-stationary. A solution to these problems is proposed in the nextsection.

3. The Markov-Switching Multifractal (MSM)

The Markov-switching multifractal (MSM) is a fully stationary multifractaldiffusion ([CF01], [CF04], [CF]), which parsimoniously incorporates arbitrarilymany components of heterogeneous durations. MSM builds a bridge between mul-tifractality and Markov-switching and therefore permits the application of powerfulstatistical methods.

3.1. Definition in Discrete Time. We assume that time is defined on thegrid t = 0, 1, . . . ,∞. We consider:

– a first-order Markov state vector Mt = (Mk,t)1≤k≤k ∈ Rk+, and

– a random variable M ≥ 0 with a unit mean: E(M) = 1.

In this survey, we consider for simplicity that M has a Bernoulli distribution tak-ing either a high value m0 or a low value 2 − m0 with equal probability, wherem0 is fixed element of the interval [1, 2]. We also assume that the componentsM1,t,M2,t, . . . ,Mk,t are mutually independent across k.

Each component {Mk,t}t≥0 is a Markov process in its own right, which is con-structed through time as follows. Given Mk,t−1, the next-period component Mk,t


isdrawn from the distribution of M with probability γkset equal to its current level Mk,t−1 with probability 1 − γk.

The transition probabilities are tightly specified by

(3.1) γk = 1 − (1 − γ1)(bk−1) ,

where γ1 ∈ (0, 1) and b ∈ (1,∞). The definition implies that γ1 < ... < γk, so thatcomponents with a low index k are more persistent than higher-k components. If theparameter γ1 is small compared to unity, the transition probabilities γk ∼ γ1b

k−1

grow approximately at geometric rate b for low values of k; the growth rate of γkeventually slows down for high values of k so that γk remains lower than unity.

We assume that returns rt = pt − pt−1 are given by

(3.2) rt = μ + σ(Mt)εt,

where μ ∈ R and σ ∈ R++ are constants, {εt}t≥0 are independent standard Gaus-sians, and volatility at date t is

(3.3) σ(Mt) = σ

⎛⎝ k∏k=1

Mk,t

⎞⎠1/2

.

We call this construct the Markov-Switching Multifractal (MSM). We observe thatthe MSM process rt is stationary, with unconditional mean E(rt) = μ and uncon-ditional standard deviation {E[(rt − μ)2]}1/2 = σ.

The multiplicative structure (3.3) is appealing to model the high variability andhigh volatility persistence exhibited by financial time series. The components havethe same marginal distribution M but differ in their transition probabilities γk.When a low–k multiplier changes, volatility varies discontinuously and has strongpersistence. In addition, high frequency multipliers produce substantial outliers.

Figure 4 illustrates the construction of binomial MSM. The top three panelsrepresent the sample path of the volatility components Mk,t for k varying from 1to 3. We see that the number of switches increases with k, as implied by (3.1). Thefourth panel represents the variance σ2(Mt) ≡ σ2 M1,t...Mk,t, where k = 8 andσ = 1. The construction generates cycles of different frequencies, consistent withthe empirical observation that there are volatile decades and less volatile decades,volatile years and less volatile years, and so on. MSM thus provides a tight modelfor the behavior of financial returns at various horizons documented in [DG] and[LZ]. The panel also shows pronounced peaks and intermittent bursts of volatility,which produce fat tails in returns. The last panel illustrates the impact of thevarious frequencies on the return series.

In empirical applications, it is numerically convenient to estimate parametersof the same magnitude. Since γ1 < · · · < γk < 1 < b, we choose γk and b to specifythe set of transition probabilities. Overall, an MSM process with k components isfully parameterized by

ψ ≡ (m0, σ, b, γk, μ) ∈ [1, 2] × R++ × (1,+∞) × (0, 1) × R,

where m0 characterizes the distribution of the multipliers, σ is the unconditionalstandard deviation of returns, b and γk define the set of switching probabilities, andμ the unconditional mean of returns. The number of components k ∈ N∗ can also


m1

m0

M1,t

m1

m0

M2,t

m1

m0

M3,t

Volatility (σ2)

Logarithmic Returns

↓ ↓

Figure 4. Construction of the Binomial Markov-Switching Mul-tifractal. The figure illustrates construction of binomial MSMwith k = 8 components over T = 10, 000 periods with parame-ters m0 = 1.4, b = 2, and γ1 = b/T . The first three panels showthe components M1,t, M2,t, and M3,t. The fourth panel shows thevariance σ2

t , which is the product of all eight multipliers. The finalpanel shows the return series.

be viewed as a discrete parameter of MSM, and we discuss below how to estimateit along with the continuous vector ψ.

3.2. Filtering and Estimation. Since the components Mk,t have binomial

distributions, the state vector Mt takes d = 2k possible values m1, . . . ,md ∈ Rk+.

The transition matrix of Mt is by definition the d×d matrix A = (ai,j)1≤i,j≤d with


components

aij = P(Mt+1 = mj∣∣Mt = mi).

For a general Markov chain with d states, the transition matrix contains d2 ele-ments. So for example if d = 28 states, the transition matrix contains 216 = 65, 536parameters and estimation is generally unfeasible with current numerical methods.By comparison, an MSM return process with k = 8 components and 28 = 256states is fully defined by only five parameters.6 MSM thus offers a parsimoniousspecification of a high-dimensional state space, which paves the way for statisticalestimation and inference.

The financial statistician observes the returns rt but not the state vectorMt. She therefore seeks to compute the conditional probability distribution Πt =(Π1

t , . . . ,Πdt

)∈ Rd

+, where for every j ∈ {1, . . . , d},

Πjt ≡ P

(Mt = mj |r1, . . . , rt

).

Conditional on the volatility state, the return has Gaussian density ωj(rt) = n[(rt−μ)/σ(mj)]/σ(mj), where n (·) denotes the density of a standard normal. Bayes’ ruleimplies that

(3.4) Πjt ∝ ωj(rt)P

(Mt = mj |r1, . . . , rt−1

),

or Πjt ∝ ωj(rt)

∑di=1 ai,jP

(Mt−1 = mi|r1, . . . , rt−1

). The vector Πt is therefore a

function of its lagged value and the contemporaneous return rt :

(3.5) Πt =ω(rt) ◦ (Πt−1A)

[ω(rt) ◦ (Πt−1A)] ι′,

where ω(rt) = [ω1(rt), . . . , ωd(rt)] , ι = (1, . . . , 1) ∈ Rd, and x ◦ y denotes theHadamard product (x1y1, . . . , xdyd) for any x, y ∈ Rd. The vector Πt can there-fore be computed recursively, as is familiar in regime-switching models ([H]). Inempirical applications, the initial vector Π0 is set equal to the ergodic distributionΠ∞ = ι/d of the Markov chain Mt.

Let L(r1, ..., rT ;ψ) denote the probability density function of the time seriesr1, ..., rT under the MSM model with parameter vector ψ. We easily check that:

(3.6) logL (r1, ..., rT ;ψ) =T∑

t=1

log[ω(rt) · (Πt−1A)].

For a fixed k, the maximum likelihood estimator (ML),

ψ = argmaxψ logL (r1, ..., rT ;ψ) ,

is consistent and asymptotically normal:√T (ψ − ψ)

d−→ N (0, V ). The ML esti-mator is asymptotically efficient, in the sense that no other estimator has a smaller

asymptotic variance-covariance matrix V (see, e.g., [C]). In the case of MSM, ψalso performs well in finite samples ([CF04]).

6The transition probabilities of MSM are given by:

aij =k∏

k=1

[(1− γk) 1{mi

k=m

jk

} + γkP(M = mjk)

],

where mik denotes the mth component of vector mi, and 1{mi

k=m

jk} is the dummy variable equal

to 1 if mik = mj

k, and 0 otherwise.


MSM specifications with different values of k are non-nested but are specifiedby the same number of parameters for every k ≥ 2. Comparing their likelihoodstherefore provides meaningful information about the goodness-of-fit. The standardVuong test ([V]), or alternatively a version adjusted for heteroskedasticity, is asimple and appropriate model selection criterion, as illustrated in [CF04] and [CF].Filtering and parameter estimation are therefore remarkably convenient with MSM.

3.3. Empirical Estimation and Forecasting. We apply MSM to the Japan-ese yen / U.S. dollar exchange rate series illustrated in Figure 1. The daily logarith-mic returns are calculated from exchange rates beginning in June 1973, extendingto the end of our sample at the end of March, 2012. Overall, the series contains9751 observations.

Table 1 reports the maximum likelihood estimates. For convenience, we set thedrift parameter equal to zero: μ = 0. In Panel A, following [CF04] and [CFT],we estimate the four remaining parameters m0, σ, γk, and b, for a number ofcomponents k varying from 1 to 12. The first column corresponds to a standardMarkov-switching model with only two volatility states. As k increases, the numberof states increases geometrically as 2k. There are over four thousand states whenk = 12.

The estimate of m0 declines monotonically with k: as more components areadded, less variability is required in each Mk,t to match the fluctuations in volatil-ity exhibited by the data. The estimates of σ vary across k with no particularpattern; their standard errors increase with k, consistent with the fact that long-run averages are difficult to identify in models permitting long volatility cycles. Wenext examine the frequency parameters γk and b. When k = 1, the single multiplierhas a duration 1/γ1 = 1/0.192 of about 5 business days, which corresponds to onecalendar week. As k increases, the switching probability of the highest frequencymultiplier increases until a switch occurs almost once a day for large k. At thesame time, the estimate of b decreases steadily with k. The increasing number offrequencies permits durations to fan out to both very short and very long values,ranging from 1 day to decades, while the spacing of durations becomes tighter.

We finally examine the behavior of the log-likelihood function as the numberof frequencies k increases from 1 to 12. The likelihood rises substantially as kincreases from low to moderate values, and continues to rise at a decreasing rate aswe add components. The likelihood function eventually flattens out when k ≥ 10.The monotonic relationship between the likelihood and k confirms one of the mainpremises of MSM: fluctuations in volatility occur with heterogeneous degrees ofpersistence, and explicitly incorporating a larger number of frequencies results in abetter fit.

In Panel B, we restrict two of the MSM parameters. Consistent with the ideathat the long-run mean of volatility is poorly identified, we set the unconditionalvolatility σ equal to the sample standard deviation of returns. Since the lowest-frequency volatility component is difficult to identify even in a long data sample,we set γ1 = 1/(4T ), so that a switch in this component is expected to occur once ina sample containing four times as many observations as the available sample. Withthese restrictions, we only need to estimate the remaining parameters m0 and b.Empirically, these restrictions reduce the likelihood substantially when k is small,but for large values of k the restricted likelihood is very close to the unrestrictedlikelihoods shown in Panel A. These results suggest that restricting the values of σ


TABLE1.

–Maximum

Likelihood

Estimation

k=

12

34

56

78

910

1112

JapaneseYen

/USDollar

A.Fou

rPar

am

eter

sm

01.

732

1.73

01.

663

1.62

51.

563

1.54

11.

493

1.48

81.

435

1.40

51.

400

1.37

6(0

.012)

(0.0

14)

(0.0

10)

(0.0

11)

(0.0

09)

(0.0

12)

(0.0

09)

(0.0

10)

(0.0

10)

(0.0

09)

(0.0

09)

(0.0

09)

σ0.

658

0.56

30.

564

0.48

30.

493

0.59

20.

599

0.49

70.

519

0.51

60.

448

0.44

2(0

.009)

(0.0

14)

(0.0

15)

(0.0

24)

(0.0

16)

(0.0

25)

(0.0

20)

(0.0

16)

(0.0

22)

(0.0

09)

(0.0

12)

(0.0

24)

γk

0.19

20.

352

0.30

90.

719

0.85

60.

932

0.99

20.

989

1.00

01.

000

1.00

01.

000

(0.0

22)

(0.0

41)

(0.0

59)

(0.0

80)

(0.0

53)

(0.0

50)

(0.0

15)

(0.0

13)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

b-

53.9

613

.57

16.8

111

.14

8.86

7.07

6.48

4.80

4.03

3.81

3.20

(23.6

5)

(2.0

8)

(3.3

4)

(1.3

9)

(0.9

1)

(0.8

0)

(0.6

0)

(0.4

8)

(0.4

6)

(0.0

2)

(0.6

7)

lnL

-888

7.13

-852

0.07

-833

9.72

-826

9.27

-823

3.90

-821

7.42

-820

8.84

-820

3.64

-820

0.27

-819

9.34

-819

7.46

-819

6.81

B.T

wo

Para

met

ers

m0

-1.

716

1.71

01.

640

1.61

21.

552

1.50

61.

471

1.45

31.

427

1.40

31.

382

(0.0

08)

(0.0

12)

(0.0

10)

(0.0

08)

(0.0

09)

(0.0

08)

(0.0

08)

(0.0

10)

(0.0

09)

(0.0

08)

(0.0

08)

b-

5460

86.8

442

.32

15.4

010

.12

7.73

6.53

4.43

3.94

3.61

3.33

(522)

(7.9

7)

(3.8

8)

(0.6

7)

(0.4

2)

(0.3

8)

(0.3

1)

(0.1

7)

(0.1

3)

(0.1

1)

(0.1

1)

lnL

-857

3.37

-842

0.14

-830

8.86

-823

6.95

-822

2.57

-821

3.22

-821

4.04

-820

8.38

-820

2.43

-820

0.51

-819

9.23

Notes:

Thistable

reportsthemaxim

um

likelihoodestimationofbinomialMSM

ontheyen

/U.S.dollardatasetcontainingT

=9751dailyreturns.

Panel

Ashow

sunrestricted

estimationofthefourparametersm

0,σ,γk,andb.

InPanel

B,wesetσeq

ualto

thesample

standard

dev

iationand

γ1equalto

1/(4T),

whichcorrespondsto

alowestfrequen

cyarrivaloccurringonav

erageonce

inaperiodfourtimes

thesample

size.Columns

correspondto

thenumber

offrequen

cies

kin

theestimatedmodel.Asymptoticstandard

errors

are

inparenthesis.


and γ1 can be a pragmatic empirical approach that further simplifies the estimationof MSM.

It is natural to compare the MSM maximum likelihood results with estimatesfrom a standard volatility process. Generalized Auto-Regressive Conditional Het-eroskedasticity (“GARCH”, [E82], [Bo87]) assumes that returns are of the form

rt = h1/2t et, where ht is the conditional variance of rt at date t−1. The innovations

{et}t≥1 are independent and identically distributed as centered Student variableswith a unit variance and ν degrees of freedom. In GARCH(1,1), the conditionalvariance satisfies the recursion ht+1 = ω + αr2t + βht, and the return process isoverall defined by the four parameters ν, ω, α, and β. We estimate GARCH(1,1)on the Yen / U.S. dollar exchange rate data, and find a likelihood of -8299.20,almost 100 points lower than MSM.

The MSM model produces accurate out-of-sample forecasts, as we now show.For both MSM and GARCH we estimate the models in-sample using returns fromthe beginning of the sample until the end of 1995. We then use returns from thebeginning of 1996 to the end of the sample to evaluate out of sample performance.Each model is used to predict the realized volatility

RVt,n =t∑

s=t−n+1

r2s

for forecasting horizons n ranging from 1 to 100 days. Let the out-of-sample periodbegin on date T0 and assume a forecasting horizon n. The N = T − (n − 1) −T0 realized volatility observations in the out-of-sample period have mean RV n =

N−1∑T−(n−1)

t=T0RVt,n. The out-of-sample forecasting R2 is given by R2 = 1 −

MSE/TSS, where the total sum of squares (TSS) is the out-of-sample variance of

realized volatility: TSS = N−1∑T−(n−1)

t=T0(RVt,n −RV n)2, the mean squared error

(MSE) quantifies forecast errors: MSE = N−1∑T−(n−1)

t=T0[RVt,n−Et−1RVt,n]2, and

the conditional expectation is taken under the assumption that the model holds.Table 2 reports summary forecasting results for horizons of 1, 5, 10, 20, 50,

and 100 days. In addition to the yen / dollar series, we consider three additionalcurrencies: the euro, the British pound and the Canadian dollar, all against theU.S. dollar.7 MSM shows robust good performance at all horizons and for allcurrencies, with particular strength appearing over longer horizons of 50 and 100days. [CF04], [CFT], [L08], [BKM], [CDS], [BSZ], and [I] confirm the excellentin- and out-of-sample performance of the multifractal approach applied to a varietyof financial series. [C09] obtains similar results with a reduced-form version of amultifractal model.

3.4. Long Memory in Volatility and Moment-Scaling. MSM generatesa hyperbolic decline in the autocorrelation ρq(n) defined in (2.6) for a range of lagsn. Consider two arbitrary numbers α1 and α2 in the open interval (0, 1). The set

of integers Ik = {n : α1 logb(bk) ≤ logb n ≤ α2 logb(b

k)} contains a large range ofintermediate lags. We show in [CF04]:

7The Euro / U.S. dollar series is obtained by splicing the Deutschemark exchange rate withthe Euro exchange rate, using the official Deutschemark / Euro exchange rate instituted at theend of 1998.


TABLE 2. – Volatility Forecasts

Horizon (Days)1 5 10 20 50 100

Forecasting R2

Euro/U.S. DollarBinomial MSM 0.036 0.205 0.298 0.347 0.280 0.088GARCH 0.045 0.197 0.285 0.328 0.174 -0.396

Japanese Yen/U.S. DollarBinomial MSM 0.052 0.120 0.166 0.206 0.166 0.103GARCH 0.051 0.094 0.101 0.071 -0.172 -0.384

British Pound/U.S. DollarBinomial MSM 0.085 0.352 0.414 0.418 0.343 0.181GARCH 0.117 0.410 0.485 0.489 0.452 0.118

Canadian Dollar/U.S. DollarBinomial MSM 0.100 0.270 0.324 0.316 0.257 0.219GARCH 0.142 0.430 0.574 0.574 0.378 0.170

Notes: This table summarizes out-of-sample forecasting performance for MSM and GARCH acrossmultiple forecasting horizons. We estimate the models in-sample using data from the beginningof the sample until the end of 1995. We then use the data from the beginning of 1996 to March30, 2012 to evaluate out of sample performance. For each model we evaluate ability to forecastrealized volatility RVt,n =

∑ts=t−n+1 r

2s , for forecasting horizons n ranging from 1 to 100 days as

described in the text. To obtain the Euro series, we splice the Deutsche Mark / U.S. Dollar seriesfrom the beginning of the sample to December 31, 1998, with the Euro / U.S. Dollar series fromJanuary 1, 1999 onwards, using the official Deutsche Mark / Euro conversion rate on January 1,1999 to convert the Deutsche Mark series to Euros.

Theorem 1 (Hyperbolic autocorrelation in volatility). Consider a fixed vectorψ and let q > 0. The autocorrelation in levels satisfies

limk→+∞

(supn∈Ik

∣∣∣∣ log ρq(n)

log n−δ(q)− 1

∣∣∣∣)

= 0,

where δ(q) = logb E(Mq) − 2 logb E(Mq/2).

MSM mimics the hyperbolic autocorrelograms log ρq(n) ∼ −δ(q) log n exhibited bymany financial series (e.g., [D], [DGE], [BBM]).

MSM illustrates that a Markov-chain regime-switching model can theoreti-cally exhibit one of the defining features of long memory, a hyperbolic declineof the autocorrelogram at long lags. Fractional Brownian motions ([K], [M65])and their discrete-time equivalents ([MV68], [GJ], [B]) generate hyperbolic au-tocorrelograms by assuming that an innovation linearly affects future periods at ahyperbolically declining weight; as a result, fractional integration tends to produce


0 1 2 3 −2

0

2

4

log(Δt)

q=1

q=2

q=3

q=5

Figure 5. Moment Scaling of the Markov-Switching Multifrac-tal. The figure illustrates moment scaling in binomial MSM. Wesimulate 500 independent samples of length T = 20, 000 of thebinomial MSM process with parameters m0 = 1.4, b = 2, andγ1 = b/T . For each simulated path, we calculate the partition

function Sq(T,Δt) ≡∑N−1

i=0 |p(iΔt + Δt) −X(iΔt)|q for a set ofinterval lengths Δt. The solid lines plot averages across the ran-dom samples of the logarithm of the sample moments Sq(T,Δt)against the logarithm of Δt, for moments q = 1, 2, 3, 5. For conve-nience, the lines are vertically displaced to begin at zero. Dottedlines show the 20th and 80th percentiles for each moment. Theplots show approximate moment scaling, consistent with (2.8).

smooth processes. By contrast, MSM generates long cycles with a switching mech-anism that also gives abrupt volatility changes. The combination of long-memorybehavior with sudden volatility movements has a natural appeal for financial mod-eling.

MSM captures the moment-scaling properties of financial series. Intuitively,MSM is a randomized version of the MMAR, and therefore inherits the moment-scaling properties of its precursor. Figure 5 shows moment scaling in binomialMSM. We simulate 500 random paths of length T = 20, 000, and for each samplecalculate an empirical estimate of E (|p(t + Δt) − p(t)|q), as explained in the figurecaption, for a variety of moments q. We take the averages across the random samplesof the logarithm of the sample moments and plot these against the logarithm of the


interval length Δt. The plots are approximately linear, consistent with the scalingrelation (2.8). We refer the reader to [CF] for theoretical results on the asymptoticscaling of MSM and statistical tests of the ability of MSM to replicate scaling inempirical data.

3.5. Continuous-Time MSM. The MSM construction works just as wellin continuous time. We now assume that time is defined on the interval [0,+∞).Given the Markov state vector

Mt = (M1,t;M2,t; . . . ;Mk,t) ∈ Rk+

the dynamics over an infinitesimal interval are defined as follows. For each k ∈{1, . . . , k}, a change in Mk,t may be triggered by a Poisson arrival with intensity λk.The component Mk,t+dt is drawn from a fixed distribution M if there is an arrival,and otherwise remains at its current value: Mk,t+dt = Mk,t. The construction canbe summarized as:

Mk,t+dt drawn from the distribution of M with probability λkdtMk,t+dt = Mk,t with probability 1 − λkdt.

The Poisson arrivals and new draws from M are independent across k and t. Thesample paths of a component Mk,t are cadlag, i.e. are right-continuous and have alimit point to the left of any instant.8

The arrival intensities are specified by

(3.7) λk = λ1bk−1, k ∈ {1, . . . , k}.

The parameter λ1 determines the persistence of the lowest frequency component,and b the spacing between component frequencies.

Finally, we assume that the log price process p(t) satisfies the stochastic differ-ential equation diffusion

(3.8) dp(t) = μdt + σ(Mt)dZt,

where Zt is a standard Brownian motion and σ(Mt) follows the maintained equation(3.3). The price

(3.9) p(t) = p(0) + μt +

∫ t

0

σ(Ms)dZs

is a continuous Ito diffusion with constant drift μ and time-varying multifrequencyvolatility σ(Mt). [CF01] and [CF08] investigate the tight link between the dis-crete and continuous time constructions of MSM, and show that the transitionprobabilities (3.1) are discretized versions of the geometric intensities (3.7). Mul-tifrequency switches in the drift μ can also be useful for asset pricing, permittingthe construction of multifrequency long-run risk models ([BY04]), as in [CF07].

3.6. Limiting Process with Countably Many Frequencies. The MSMconstruction can accommodate an infinity of frequencies, as we now show. Forgiven parameters (μ, σ,m0, λ1, b), let Mt = (Mk,t)

∞k=1 ∈ R∞

+ denote an MSMMarkov state process with countably many components. Each component Mk,t

is characterized by the arrival intensity λk = λ1bk−1. For any finite k, stochastic

volatility is defined as the product of the first k components of the state vector:

σk(Mt) ≡ σ(∏k

k=1Mk,t

)1/2.

8Cadlag is a French acronym for continue a droite, limite a gauche.


Since instantaneous volatility σk(Mt) depends on an increasing number of com-ponents, the differential representation (3.8) becomes unwieldy as k → ∞. In fact,the instantaneous volatility σk(Mt) converges almost surely to zero as k → ∞.Since volatility is unbounded, however, the Lebesgue dominated convergence doesnot apply. We consider instead the time deformation

(3.10) θk(t) ≡∫ t

0

σ2k(Ms)ds.

At any given instant t, the sequence {θk(t)}∞k=1is a positive martingale with

bounded expectation. By the martingale convergence theorem, the random variableθk(t) converges to a limit distribution when k → ∞. A similar argument applies toany vector sequence {θk(t1); . . . ; θk(td)}, guaranteeing that the stochastic processθk has at most one limit point. As shown in [CF01], the sequence {θk}k is tight9

under the following sufficient condition.

Condition 1 (Tightness). E(M2) < b.

Intuitively, tightness prevents the time deformation θk from oscillating too wildlyas k → ∞. Correspondingly, Condition 1 imposes that the volatility shocks aresufficiently small or that their durations λ−1

k decrease sufficiently fast to guaranteeconvergence.10 Let D[0,∞) denote the space of cadlag functions defined on [0,∞),and let d◦∞ denote the Skohorod distance.

Theorem 2 (Time deformation with countably many frequencies). Under Con-dition 1, the sequence (θk)k weakly converges as k → ∞ to a measure θ∞ definedon the metric space (D[0,∞), d◦∞). Furthermore, the sample paths of θ∞ are con-tinuous almost surely.

The limiting time deformation θ∞ is driven by the state vector Mt = (Mk,t)∞k=1

and therefore has a Markov structure analogous to MSM with a finite k.The limiting price process

(3.11) p∞(t)d= p(0) + μt + B[θ∞(t)]

has sample paths that are continuous but can be more irregular than a Brownianmotion at some instants. Specifically, the local variability of a sample path at agiven date t is characterized by the local Holder exponent

α(t) = sup{β ≥ 0 : |p∞(t + Δt) − p∞(t)| = O(|Δt|β) as Δt → 0}.Heuristically, we can express the infinitesimal variations of the price process asbeing of order (dt)α(t) around instant t. Lower values of α(t) correspond to moreabrupt variations. Traditional jump-diffusions impose that α(t) be equal to 0 atpoints of discontinuity, and to 1/2 otherwise.11 In a multifractal diffusion such asp∞, however, the exponent α(t) takes a continuum of values in any time interval.

9We refer the reader to [Bi] for a detailed exposition of tightness and weak convergence infunction spaces.

10Because volatility exhibits increasingly extreme behavior as k goes up, the time deforma-tion θ∞ cannot be computed by taking the pointwise limit of the integrand σ2

k(Mt) in equation

(3.10). Specifically, σ2k(Ms) converges almost surely to zero as k → ∞ (by the Law of Large

Numbers), suggesting that θ∞ ≡ 0. This conclusion would of course be misleading. For everyfixed t, Condition 1 implies that supk E

[θ2k(t)

]< ∞ ([CF]), and the sequence {θk(t)}k is therefore

uniformly integrable. Hence Eθ∞(t) = Eθk(t) = σ2 t > 0.11See [Ka] for further discussion.


3.7. Extensions. MSM has been extended along several directions. [CFT]considers a multivariate version of MSM that captures both the correlation in lev-els and the correlation in volatility of the returns on several financial assets. Thelikelihood function and the Bayesian filter of multivariate MSM are available an-alytically, as in the univariate case. Multivariate MSM captures well the jointdynamics of asset returns and provides accurate forecasts of the value at risk of aportfolio of assets.

[I] develops an extension of bivariate MSM that incorporates dynamic cor-relation in the Gaussian innovations. The new model, which the author coinsMSMDCC, combines the multifrequency structure of bivariate MSM with the flex-ible correlation of Engle’s Dynamic Conditional Correlation model [E02]. Thelikelihood and Bayesian filter of MSMDCC are available analytically. MSMDCCoutperforms its two building blocks – MSM and DCC – both in and out of sample.

[CDS] and [BSZ] introduce Markov-switching multifractal models of inter-trade duration, that is the time interval between two consecutive trades on a givensecurity. Inter-trade durations play an important role in the financial econometricsand microstructure literatures (e.g., [ER]) and can help design algorithmic tradingstrategies. The MSM duration models capture the key features of financial marketinter-trade durations: long-memory dynamics and highly dispersed distributions.They also outperform their short-memory competitors in and out of sample.

4. Pricing Multifractal Risk

The integration of multifractal risk into asset pricing is now at the forefront ofcurrent research. We begin with an illustrative example drawn from [CF08].

4.1. An Equilibrium Model of Stock Prices. We consider an infinitely-lived asset, such as the stock of a corporation, that pays off a random cash flowDt every period. Since the profitability of the company is impacted by multipleshocks that each have their own degrees of persistence, the cash flow process hasmultifractal characteristics and is therefore a source of multifractal risk. We knowfrom financial theory that in the absence of arbitrage, the stock price at a given datet is the present value of expected future dividends, where the discount rate takesinto account the risk aversion of investors ([M],[DD]). In the following example,we assume that the discount rates are obtained from the classic Lucas valuationmodel ([Lu]), as we now explain.

The model is formally defined as follows on the continuous time interval [0,∞).

Let Z(t) ∈ R denote a standard Brownian motion, let k ∈ N∗, and let Mt ∈ Rk+

denote an MSM state vector with k components. The processes Z and M aremutually independent. The stock pays off the continuous stream of cash flows Dt,which includes dividends and the proceeds from stock repurchases. For simplicity,we will simply refer to Dt as the dividend process.

Condition 2 (Dividends). The dividend process satisfies

log(Dt) ≡ log(D0) +

∫ t

0

[gD − σ2

D(Ms)

2

]ds +

∫ t

0

σD(Ms)dZD(s)

at every instant t ∈ [0,∞), where gD and σD are strictly positive elements of the

real line and σD(Mt) = σD(∏k

k=1Mk,t)1/2.


The stock is priced by a collection of identical risk-averse agents, who observethe realization of the processes Z and M. Risk aversion is defined as follows. Anagent ranks the desirability of a random consumption stream {Ct}t≥0 according tothe utility index

U({Ct}) = E

[∫ +∞

0

e−δtu(Ct)dt

∣∣∣∣ I0] ,where δ is a strictly positive constant, I0 denotes the agent’s information set att = 0, and u is the Bernoulli utility:

u(C) ≡{

C1−α/(1 − α) if α �= 1,log(C) if α = 1.

The agent strictly prefers the consumption stream {Ct} to the consumptionstream {C ′

t} if and only if U({Ct}) > U({C ′t}). We let ρ = δ− (1−α)gD, which we

assume to be strictly positive. We use lower cases for the logarithms of all variables.

Theorem 3 (Equilibrium stock price). The stock price is in logs the sum ofthe continuous dividend process and the price:dividend ratio:

pt = dt + q(Mt),

where

(4.1) q(Mt) = logE

(∫ +∞

0

e−ρs−α(1−α)2

∫ s0σ2D(Mt+h)dhds

∣∣∣∣Mt

).

The price process therefore follows a jump-diffusion. A price jump occurs whenthere is a discontinuous change in the Markov state Mt driving the continuousdividend process.

The price jumps are endogenous implications of market pricing, and the dis-continuities of the price p(t) contrast with the continuous behavior of the dividendprocess d(t). Over an infinitesimal time interval, the stock price changes by

d(pt) = d(dt) + Δ(qt),

where Δ(qt) ≡ q(Mt) − q(Mt−) denotes the finite variation of the price:dividendratio triggered by a Markov switch. If α < 1, a switch that increases the volatility ofcurrent and future dividends induces a negative realization of Δ(qt). Market pricingthus generates an endogenous negative correlation between volatility changes andprice jumps.

The size of a jump Δ(qt) = q(Mt) − q(Mt−) depends on the persistence of thecomponent that changes. Low-frequency multipliers deliver persistent and discreteswitches, which by (4.1) have a large price impact. By contrast, higher frequencycomponents have no noticeable effect on prices, but give additional outliers in re-turns through their direct effect on the tails of the dividend process. The priceprocess is therefore characterized by a large number of small jumps (high frequencyMk,t), a moderate number of moderate jumps (intermediate frequency Mk,t), anda small number of very large jumps. Earlier empirical research suggests that this isa good characterization of the dynamics of stock returns. The multifractal modelavoids the difficult choice of a unique frequency and size for rare events, which is acommon issue with traditional jump-diffusions.12

12In the simplest exogenously specified jump-diffusions, it is often possible that discontinuitiesof heterogeneous but fixed sizes and different frequencies can be aggregated into a single collectivejump process with an intensity equal to the sum of all the individual jumps, and a random


Figure 6 illustrates the dynamics of the pricing model. The top two panelspresent a simulated dividend process, in growth rates and in logarithms of the levelrespectively. The middle two panels display the corresponding stock returns andlog prices. The price series exhibits much larger movements than dividends, dueto the presence of endogenous jumps in the price-dividend ratio, eq(Mt). To seethis clearly, the bottom two panels show consecutively: 1) the “feedback” effects,defined as the difference between log stock returns and log dividend growth, and 2)the price:dividend ratio. Consistent with Theorem 3, we observe a few infrequentbut large jumps in prices, with smaller but more numerous small discontinuities.The simulation demonstrates that the difference between stock returns and dividendgrowth can be large even when the price-dividend ratio varies in a plausible and rel-atively modest range. Overall, the equilibrium pricing model captures endogenousmultifrequency price jumps, multifrequency stochastic volatility, and endogenouscorrelation between volatility and returns.

4.2. Convergence to a Multifractal Jump-Diffusion. We now investigatehow the price diffusion evolves as k → ∞, i.e. as components of increasingly highfrequency are added into the state vector. By Condition 2 and Theorem 2, thedividend process dk(t) converges in distribution to

d∞(t) = d0 + gDt− θ∞(t)/2 + B[θ∞(t)]

as k → ∞. By (4.1), the process qk(t) is a positive submartingale, which alsoconverges to a limit as k → ∞.

Theorem 4 (Jump-diffusion with countably many frequencies). We assumethat α < 1 and that the maintained conditions 1 and 2 hold. When the number offrequencies goes to infinity, the log-price process weakly converges to

p∞(t) ≡ d∞(t) + q∞(t),

where

q∞(t) = logE

[∫ +∞

0

e−ρs−α(1−α)2 [θ∞(t+s)−θ∞(t)]ds

∣∣∣∣ (Mk,t)∞k=1

]is a pure jump process. The limiting price is thus a jump diffusion with countablymany frequencies.

The limiting log-price process p∞(t) is the sum of: (i) the continuous multifractaldiffusion d∞(t); and (ii) the pure jump process q∞(t). We correspondingly callp∞(t) a multifractal jump-diffusion.

When k = ∞, the state space is a continuum and the multifractal jump-diffusion is tightly specified by the seven parameters (gD, σD,m0, γ1, b, α, ρ). Thelimiting process q∞(t) exhibits rich dynamic properties. Within any bounded timeinterval, there exists almost surely a multiplier Mk,t that switches and triggers ajump in the stock price. Hence a jump in price occurs almost surely in the neigh-borhood of any instant. Furthermore, the number of switches is countable almostsurely within any bounded time interval, implying that the process q∞(t) has infi-nite activity and is continuous almost everywhere.

distribution of sizes. A comparable analogy can be made for the state vector Mt in our model,but due to the equilibrium linkages between jump size and the duration of volatility shocks, andthe state dependence of price jumps, no such reduction to a single aggregated frequency is possiblefor the equilibrium stock price.


0 500 1000 1500 2000−0.04

−0.02

0

0.02

0.04Dividend Growth

0 500 1000 1500 2000−0.3

−0.2

−0.1

0

0.1Returns

0 500 1000 1500 2000−0.2

−0.1

0

0.1

0.2Dividends (Log)

0 500 1000 1500 20004

4.2

4.4

4.6Prices (Log)

0 500 1000 1500 2000−0.3

−0.2

−0.1

0

0.1feedback

0 500 1000 1500 200028

30

32

34

36p/d ratio

Figure 6. The Multifractal Jump-Diffusion. The figure illustratesthe dynamics of the pricing model. The top two panels showdividend growth and dividend levels. The middle panel showsthe equilibrium returns and price process, which are considerablymore variable. The bottom panels isolate the endogenous portionof returns and prices. The bottom left panel displays the pricejumps Δ(qt) = q(Mt)− q(Mt−). The bottom right panel shows theprice:dividend ratio exp[q(Mt)].

The convergence results provide useful guidance on the choice of the number offrequencies in theoretical and empirical applications. On the one hand, the conver-gence of the price process implies that when k is large, the marginal contributionof additional components is likely to be small in applications concerned with fittingthe price or return series. It is then convenient to consider a number of frequenciesk that is sufficiently large to capture the heteroskedasticity of financial series, butsufficiently small to remain tractable. On the other hand, countably many frequen-cies might prove useful in more theoretical contexts, in which the local behaviorof the price process needs to be carefully understood. Examples could include theconstruction of learning models or the design of dynamic hedging strategies.

4.3. Other Work. Several other papers derive the pricing implications ofmultifractal risk. [CF07] develops a discrete-time model of stock returns in whichthe volatility of dividend news follows an MSM. The resulting variance of stockreturns is substantially higher than the variance of dividends, as is the case with


the data. The MSM dividend specification improves on the classic [CH] model,which generates more modest amplification effects with a GARCH dividend pro-cess. [CF07] also investigates the dynamics of returns when the agent is not fullyinformed about the state Mt but must sequentially learn about it from dividendsand other signals; the implied return process exhibits substantial negative skewness,which is again consistent with the data. [Ki] builds on [CF07] to explain a rangeof empirical findings.

Multifractal volatility has direct implications for option pricing. [CFFL] in-troduces an extension of MSM that can account for the variation in skewness andterm-structure of option data. Jumps to the return process are triggered by changesin lower-frequency volatility components, and the “leverage effect” is generated bya negative correlation of high-frequency innovations to returns and volatility. UsingS&P 500 index returns and a panel of options with multiple maturities and strikes,the latent volatility components enable the model to dynamically fit a wide rangeof option surfaces both in and out of sample.

Parsimonious models with multiple components have a natural use in interestrate modeling. [CFW] develops a class of dynamic term structure models thataccommodates arbitrarily many interest-rate factors with a fixed number of pa-rameters. The approach builds on a short-rate cascade, a parsimonious recursiveconstruction that ranks the state variables by their rates of mean reversion, eachrevolving around the preceding lower-frequency factor. The cascade accommodatesa wide range of volatility and risk premium specifications, and the forward curvesimplied by absence of arbitrage are smooth, dynamically consistent, and availablein closed form. [CFW] provides conditions under which, as the number of factorsgoes to infinity, the construction converges to a well-defined, infinite-dimensionaldynamic term structure. The cascade overcomes the curse of dimensionality as-sociated with general affine models. Using a panel of 15 LIBOR and swap rates,[CFW] estimates specifications with a number of factors ranging from one to 15, allspecified by only five parameters. In sample, the implied yield curve fits the dataalmost perfectly. Out of sample, interest rate forecasts significantly outperformprior benchmarks.

Overall, the results presented in this section show that multifractal risk hasrich pricing implications that have already allowed researchers to overcome keyshortcomings of standard financial models based on smaller state spaces. Theseearly successes suggest that multifractals are promising powerful tools for assetpricing.

5. Conclusion

Fifty years ago, Benoıt Mandelbrot discovered that financial returns exhibitstrong departures from Gaussianity and advocated the use of self-similar Levy-stable processes for modeling market fluctuations. These two insights sparked theintroduction of fractal methods in finance. Since then, fat tails, fractional integra-tion, and multifractal scaling have become familiar tools to financial practitioners,econometricians, statisticians and econophysicists. Fractal methods are now rou-tinely combined with more traditional approaches, and have given rise to popularhybrid models such as fractionally integrated GARCH ([BBM]) or long-memorystochastic volatility ([HMS]). These advances are testimony to the successful inte-gration of fractal methods into mainstream finance.


In the past fifteen years, fractal research in finance has centered on the devel-opment of multifractal models of returns, which can jointly capture fat tails, long-memory volatility persistence, multifractal moment scaling, and nonlinear changesin the distribution of returns observed over various horizons. Multifractal modelscapture these empirical regularities with a remarkably small number of parametersand are strong performers both in- and out-of-sample, as the empirical section ofthis article illustrates.

These developments in financial research have led to advances in multifractalmethodology itself. Multifractal measures can now be constructed dynamicallythrough time ([CF01], [BDM]), and several classes of multifractal diffusions arenow available ([CFM], [CF01], [BDM]). These innovations provide new intuitionsabout the emergence of multifractal behavior in economic and natural phenomena.For instance, MSM shows that multifractality can be generated by a Markov processwith multiple components, each of which has its own degree of persistence. MSMpermits the application of efficient statistical methods, such as likelihood estimationand Bayesian filtering, to a multifractal process. These developments are new tothe multifractal literature and are now spreading outside the field of finance (e.g.,[RR]). Furthermore, incorporating multifractal risk into a pricing model generatesmultifractal jump-diffusions, an entirely new mathematical object that deservesfurther investigation.

Despite these successes, multifractal finance remains a young field and manychallenges remain. The statistical methodology can be improved to incorporate finerfeatures of financial returns, for instance along the lines of [CFFL]. Improvementsin statistical inference are undoubtedly possible, for instance by using differentdistributions M , by exploring different transition probability specifications or bysimplifying the estimation method. Last but not least, the integration of fractal riskinto asset pricing offers considerable potential for financial economics, as illustratedby recent work on options and the term structure of interest rates.

References

[AB] V. Akgiray and G. G. Booth, The stable-law model of stock returns, Journal of Businessand Economic Statistics 6 (1988), 51–57.

[ABDL] Torben G. Andersen, Tim Bollerslev, Francis X. Diebold, and Paul Labys, The distri-bution of realized exchange rate volatility, J. Amer. Statist. Assoc. 96 (2001), no. 453,42–55, DOI 10.1198/016214501750332965. MR1952727

[AG] T. Ane, and H. Geman, Order flow, transaction clock, and normality of asset returns,Journal of Finance 55 (2000), 2259–84.

[Ba] L. Bachelier, Theorie de la speculation, Ann. Sci. Ecole Norm. Sup. (3) 17 (1900), 21–86(French). MR1508978

[BDM] E. Bacry, J. Delour, and J.-F. Muzy, Multifractal random walks, Physical Review E 64(2001), 026103–06.

[BKM] E. Bacry, A. Kozhemyak, and Jean-Francois Muzy, Continuous cascade modelsfor asset returns, J. Econom. Dynam. Control 32 (2008), no. 1, 156–199, DOI10.1016/j.jedc.2007.01.024. MR2381693

[B] Richard T. Baillie, Long memory processes and fractional integration in econometrics,J. Econometrics 73 (1996), no. 1, 5–59, DOI 10.1016/0304-4076(95)01732-1. MR1410000(97d:62194)

[BBM] Richard T. Baillie, Tim Bollerslev, and Hans Ole Mikkelsen, Fractionally integrated gen-eralized autoregressive conditional heteroskedasticity, J. Econometrics 74 (1996), no. 1,3–30, DOI 10.1016/S0304-4076(95)01749-6. MR1409033

[BY04] R. Bansal and A. Yaron, Risks for the long run: a potential resolution of asset pricingpuzzles, Journal of Finance 49 (2004), 1481–509.


[BNS] O. Barndorff-Nielsen and N. Shephard, Power and bipower variation with stochasticvolatility and jumps, Journal of Financial Econometrics 2 (2004), 1–37.

[B06] R. Barro, Rare disasters and asset markets in the twentieth century, Quarterly Journalof Economics 121 (2006), 823–866.

[BSZ] J. Barunık, N. Shenaiz, and F. Zikes, Modeling and forecasting persistent financial dura-tions, Working paper, Academy of Sciences of the Czech Republic and Imperial CollegeLondon Business School.

[Bi] Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Proba-bility and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999.A Wiley-Interscience Publication. MR1700749 (2000e:60008)

[BS] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of

Political Economy 81 (1973), 637–654.[BG] R. C. Blattberg and N. J. Gonedes, A comparison of the Student and stable distributions

as statistical models of stock prices, Journal of Business 47 (1974), 244–280.[B55] Salomon Bochner, Harmonic analysis and the theory of probability, University of Cali-

fornia Press, Berkeley and Los Angeles, 1955. MR0072370 (17,273d)[Bo87] T. Bollerslev, A conditionally heteroskedastic time series model for speculative prices

and rates of return, Review of Economics and Statistics 69 (1987), 542–547.[CF01] Laurent Calvet and Adlai Fisher, Forecasting multifractal volatility, J. Econometrics 105

(2001), no. 1, 27–58, DOI 10.1016/S0304-4076(01)00069-0. Forecasting and empiricalmethods in finance and macroeconomics. MR1863146

[CF02] Laurent Calvet and Adlai Fisher, Multifractality in asset returns: theory and evidence,Review of Economics and Statistics 84 (2002a), 381–406.

[CF04] Laurent Calvet and Adlai Fisher, How to forecast long-run volatility: Regime-switchingand the estimation of multifractal processes, Journal of Financial Econometrics 2 (2004),49–83.

[CF07] Laurent Calvet and Adlai Fisher, Multifrequency news and stock returns, Journal ofFinancial Economics 86 (2007), 178–212.

[CF08] Laurent E. Calvet and Adlai J. Fisher,Multifrequency jump-diffusions: an equilibrium ap-proach, J. Math. Econom. 44 (2008), no. 2, 207–226, DOI 10.1016/j.jmateco.2007.06.001.MR2376592 (2009a:91035)

[CF] Laurent E. Calvet and Adlai J. Fisher, Multifractal volatility: Theory, forecasting, andpricing, Burlington, MA, Academic Press, 2008.

[CFFL] L. E. Calvet, M. Fearnley, A. J. Fisher, and M. Leippold, What’s beneath the surface?Option pricing with multifrequency latent states, Working paper, HEC Paris, Universityof British Columbia, and University of Zurich, 2012.

[CFM] L. E. Calvet, A. J. Fisher, and B. Mandelbrot, A multifractal model of asset returns,Cowles Foundation Discussion Papers 1164–1166, Yale University, 1997, available athttp://www.ssrn.com.

[CFT] Laurent E. Calvet, Adlai J. Fisher, and Samuel B. Thompson, Volatility comove-ment: a multifrequency approach, J. Econometrics 131 (2006), no. 1-2, 179–215, DOI10.1016/j.jeconom.2005.01.008. MR2275999

[CFW] L. E. Calvet, A. J. Fisher, and L. Wu, Dimension-invariant dynamic term structures,Working paper, HEC Paris, University of British Columbia and Baruch College, 2011.

[CH] J. Y. Campbell and L. Hentschel, No news is good news: an asymmetric model of chang-ing volatility in stock returns, Journal of Financial Economics 31 (1992), 281–318.

[CV] J. Y. Campbell and L. Viceira, Strategic Asset Allocation, Oxford University Press, 2002.[CDS] F. Chen, F. X. Diebold, and F. Schorfheide, A Markov-switching multi-fractal inter-trade

duration model, with application to U.S. equities, manuscript, Huazhong University andUniversity of Pennsylvania, 2012.

[Cl] Peter K. Clark, A subordinated stochastic process model with finite variance for specu-lative prices, Econometrica 41 (1973), no. 1, 135–155. MR0415944 (54 #4022a)

[C09] F. Corsi, A simple approximate long-memory model of realized volatility, Journal ofFinancial Econometrics 7 (2009), 174–196.

[C] Harald Cramer, Mathematical Methods of Statistics, Princeton Mathematical Series, vol.9, Princeton University Press, Princeton, N. J., 1946. MR0016588 (8,39f)


[CDM] Sandor Csorgo, Paul Deheuvels, and David Mason, Kernel estimates of the tail index ofa distribution, Ann. Statist. 13 (1985), no. 3, 1050–1077, DOI 10.1214/aos/1176349656.MR803758 (87b:62043)

[DG] M. Dacorogna, R. Gencay, U. Muller, R. Olsen, and O. Pictet, An Introduction to High-Frequency Finance, Academic Press, 2001.

[D] M. Dacorogna, U. Muller, R. Nagler, R. Olsen, and O. Pictet, A geographical modelfor the daily and weekly seasonal volatility in the foreign exchange market, Journal of

International Money and Finance 12 (1993), 413–38.[DGE] Z. Ding, C. Granger, and R. Engle, A long memory property of stock returns and a new

model, Journal of Empirical Finance 1 (1993), 83–106.[DD] D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press, Third Edition,

2001.[E82] Robert F. Engle, Autoregressive conditional heteroscedasticity with estimates of the

variance of United Kingdom inflation, Econometrica 50 (1982), no. 4, 987–1007, DOI10.2307/1912773. MR666121 (83j:62158)

[E02] Robert Engle, Dynamic conditional correlation: a simple class of multivariate generalizedautoregressive conditional heteroskedasticity models, J. Bus. Econom. Statist. 20 (2002),no. 3, 339–350, DOI 10.1198/073500102288618487. MR1939905

[ER] Robert F. Engle and Jeffrey R. Russell, Autoregressive conditional duration: a new modelfor irregularly spaced transaction data, Econometrica 66 (1998), no. 5, 1127–1162, DOI10.2307/2999632. MR1639411

[F63] E. Fama, Mandelbrot and the stable Paretian hypothesis, Journal of Business 36 (1963),420–29.

[F65] E. Fama, The behavior of stock market prices, Journal of Business 38 (1965), 34–105.[FR] B. D. Fielitz and J. P. Rozelle, Stable distributions and mixtures of stable distributions

hypotheses for common stock returns, Journal of the American Statistical Association 78(1982), 28–36.

[G09] X. Gabaix, Power laws in economics and finance, Annual Review of Economics andFinance 1 (2009), 255–294.

[G12] X. Gabaix, An exactly solved framework for ten puzzles in macro-finance, Quarterly

Journal of Economics 127 (2012), 645–700.[G] S. Ghashgaie, W. Breymann, J. Peinke, P. Talkner, and Y. Dodge, Turbulent cascades

in foreign exchange markets, Nature 381 (1996), 767–70.[GJ] C. W. J. Granger and Roselyne Joyeux, An introduction to long-memory time series

models and fractional differencing, J. Time Ser. Anal. 1 (1980), no. 1, 15–29, DOI10.1111/j.1467-9892.1980.tb00297.x. MR605572 (82d:62140)

[GM63] C. Granger and O. Morgenstern, Spectral analysis of New York stock market prices,Kyklos 16 (1963), 1–27.

[H] James D. Hamilton, A new approach to the economic analysis of nonstationarytime series and the business cycle, Econometrica 57 (1989), no. 2, 357–384, DOI10.2307/1912559. MR996941

[Hi] Bruce M. Hill, A simple general approach to inference about the tail of a distribution,Ann. Statist. 3 (1975), no. 5, 1163–1174. MR0378204 (51 #14373)

[HMS] Clifford M. Hurvich, Eric Moulines, and Philippe Soulier, Estimating long mem-ory in volatility, Econometrica 73 (2005), no. 4, 1283–1328, DOI 10.1111/j.1468-0262.2005.00616.x. MR2149248 (2006i:62037)

[I] J. Idier, Correlation: A Markov-switching multifractal model with time-varying correla-tions, Working paper, Bank of France, 2010.

[JD] D. W. Jansen and C. G. DeVries, On the frequency of large stock returns: Putting boomsand busts into perspective, Review of Economics and Statistics 73 (1991), 18–24.

[Ka] Jean-Pierre Kahane, A century of interplay between Taylor series, Fourier seriesand Brownian motion, Bull. London Math. Soc. 29 (1997), no. 3, 257–279, DOI

10.1112/S0024609396002913. MR1435557 (98a:01015)[K53] M. G. Kendall, The analysis of economic time series Part I: Prices, Journal of the Royal

Statistical Society Series A 96 (1953), 11–25.[Ki] S. Kim, Asset prices in turbulent markets with rare disasters, manuscript, Northwestern

University, 2012.


[KK] K. G. Koedijk and C. J. M. Kool, Tail estimates of East European exchange rates, Journalof Business and Economic Statistics 10 (1992), 83–96.

[KSV] K. G. Koedijk, M. M. A. Schafgans, and C. G. deVries, The tail index of exchange ratereturns, Journal of International Economics 29 (1990), 93–108.

[K] A. N. Kolmogoroff, Wienersche Spiralen und einige andere interessante Kurven imHilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 (1940), 115–118(German). MR0003441 (2,220c)

[LB] Blake LeBaron, Stochastic volatility as a simple generator of apparent financial powerlaws and long memory, Quant. Finance 1 (2001), no. 6, 621–631, DOI 10.1088/1469-7688/1/6/304. MR1870018

[L24] P. Levy, Theorie des erreurs: la loi de Gauss et les lois exceptionnelles, Bulletin de laSociete Mathematique de France 52 (1924), 49–85.

[LP] M. Loretan and P. C. B. Phillips, Testing covariance stationarity of heavy-tailed timeseries: An overview of the theory with applications to several financial datasets, Journalof Empirical Finance 1 (1994), 211–248.

[Lu] Robert E. Lucas Jr., Asset prices in an exchange economy, Econometrica 46 (1978),no. 6, 1429–1445, DOI 10.2307/1913837. MR513698 (82h:90019)

[L01] Thomas Lux, Power laws and long memory. Comment on: “Stochastic volatility as asimple generator of apparent financial power laws and long memory” [Quant. Finance1 (2001), no. 6, 621–631; 1 870 018] by B. LeBaron, Quant. Finance 1 (2001), no. 6,560–562, DOI 10.1088/1469-7688/1/6/604. MR1870014

[L08] Thomas Lux, The Markov-switching multifractal model of asset returns: GMM estima-tion and linear forecasting of volatility, J. Bus. Econom. Statist. 26 (2008), no. 2, 194–210,DOI 10.1198/073500107000000403. MR2420147

[LZ] P. Lynch and G. Zumbach, Market heterogeneities and the causal structure of volatility,Quantitative Finance 3 (2003), 320–331.

[MS] S. Maheswaran and C. Sims, Empirical implications of arbitrage-free asset markets, inModels, Methods and Applications of Econometrics, ed. P. Phillips, Basil Blackwell,Oxford, 1993.

[M63] B. B. Mandelbrot, The variation of certain speculative prices, Journal of Business 36

(1963), 394–419.[M65] , Une classe de processus stochastiques homothetiques a soi, Comptes Rendus de

l’Academie des Sciences de Paris 260 (1965), 3274–77.[M67] , The variation of some other speculative prices, Journal of Business 40 (1967),

393–413.[M74] , Intermittent turbulence in self-similar cascades: divergence of high moments and

dimension of the carrier, Journal of Fluid Mechanics 62 (1974), 331–58.[M82] Benoit B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Co., San Fran-

cisco, Calif., 1982. Schriftenreihe fur den Referenten. [Series for the Referee]. MR665254(84h:00021)

[M97] Benoit B. Mandelbrot, Fractals and scaling in finance, Selected Works of Benoit B.Mandelbrot, Springer-Verlag, New York, 1997. Discontinuity, concentration, risk; SelectaVolume E; With a foreword by R. E. Gomory. MR1475217 (98i:90019)

[MV68] Benoit B. Mandelbrot and John W. Van Ness, Fractional Brownian motions, fractionalnoises and applications, SIAM Rev. 10 (1968), 422–437. MR0242239 (39 #3572)

[MK] H. Markowitz, Portfolio selection, Journal of Finance 7 (1952), 77–91.[IM] I. Martin, Consumption-based asset pricing with higher cumulants, NBERWorking Paper

No. 16153 (2010), forthcoming in Review of Economic Studies.[M] R. C. Merton, Continuous-Time Finance, Basil Blackwell, Oxford, 1990.[PMM] P. C. B. Phillips, J. W. McFarland, and P. C. McMahon, Robust tests of forward ex-

change market efficiency with empirical evidence from the 1920’s, Journal of AppliedEconometrics 11 (1996), 1–22.

[R] J. Regnault, Calcul des Chances et Philosophie de la Bourse, Mallet-Bachelier and Castel,Paris, 1863.

[R88] T. A. Rietz, The equity risk premium: A solution, Journal of Monetary Economics 22(1988), 117–131.


[RR] M. Rypdal and K. Rypdal, Discerning a linkage between solar wind turbulence and iono-spheric dissipation by a method of confined multifractal motions, Journal of GeophysicalResearch 116, A02202.

[ST] Gennady Samorodnitsky and Murad S. Taqqu, Stable non-Gaussian random processes,Stochastic Modeling, Chapman & Hall, New York, 1994. Stochastic models with infinitevariance. MR1280932 (95f:60024)

[S] W. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk,

Journal of Finance 19 (1964), 425–42.[T] J. Tobin, Liquidity preference as behavior towards risk, Review of Economic Studies 25

(1958), 68–85.[VA] N. Vandewalle and M. Ausloos, Multi-affine analysis of typical currency exchange rates,

European Physical Journal B 4 (1998), 257–61.[V] Quang H. Vuong, Likelihood ratio tests for model selection and nonnested hypotheses,

Econometrica 57 (1989), no. 2, 307–333, DOI 10.2307/1912557. MR996939 (90g:62048)[W] J. Wachter, Can time-varying risk of rare disasters explain aggregate stock market volatil-

ity?, Journal of Finance (2012), forthcoming.

Department of Finance, HEC Paris, 1 rue de la Liberation, 78351 Jouy-en-Josas,

France


Department of Finance, Sauder School of Business, Vancouver, BC, V6T 1Z2, Canada



An Algorithm for Dynamical Games with Fractal-LikeTrajectories

David Carfı and Angela Ricciardello

Abstract. In this paper, we propose an algorithm to represent the payoff tra-

jectory of two-player discrete-time dynamical games. Specifically, we considerdiscrete dynamical games which can be modeled as sequences of normal-formgames (the states of the dynamical game) with payoff functions of class C1.In this context, the payoff evolution of such type of dynamical games is thesequence of the payoff spaces of their game-states and the payoff trajectory ofsuch games is the union of the members of the evolution. The formulation ofthe algorithm is motivated - especially in several applicative contexts such asEconomics, Finance, Politics, Management Sciences, Medicine and so on ... -by the need of a complete knowledge of the payoff evolution (problem whichis still open in the most part of the cases), when the real problem requiresa Complete Analysis of the interactions, beyond the study of just the Nashequilibria. We consider, to prove the efficiency and strength of our method,the development (by the algorithm itself) of some non-linear dynamical gamestaken from applications to Microeconomics and Finance. The dynamical gamesthat we shall examine are already deeply studied and represented, at least attheir initial state - by the application of the topological method presented byCarfı in [7] - in several applicative papers by Carfı, Musolino and Perrone (see[10], [11–20] by a long, quite indirect and step by step implementations ofother standard computational softwares (such as AutoCad, Derive, Grapher,Graph and Maxima) or following a pure mathematical way (see for example[8]): on the contrary, our algorithm provides the direct and one shot graphicalrepresentation of the entire evolution of those games (by movies) and conse-quently of the entire trajectory. Moreover, the applicative games we consider in

the paper (inspired and suggested by Economics and Finance) have a naturaldynamics having fractal-like trajectories.

1. Introduction

1.1. Brief history of the related past researches. In 2009, D. Carfı andA. Ricciardello (see [25]) presented a new computational procedure to determinethe payoff spaces of non-parametric differentiable normal form games. Then, theauthors applied a new procedure (see [1]) to numerically determine an original typeof 3-dimensional representation of the payoff space of a normal-form C1 parametricgame, with two players. Moreover, the method in [25] has been pointed out in

2010 Mathematics Subject Classification. Primary 91A05, 91A80, 28A80, 91A25, 65Y04, 68W30.Key words and phrases. Discrete dynamical games, payoff evolution, payoff trajectories, economicgames, financial games, Pareto boundary, Complete study of a game, fractal-like geometry.


95

96 DAVID CARFI AND ANGELA RICCIARDELLO

[26] and assumed with the aim of realizing a numerical procedure providing thegeometrical representation of the payoff scenarios of C1-families of normal-formgames, with two players.

1.2. Aims of the paper. Our study pertains discrete families of normal-formC1-games with 2 players, whose payoff functions are defined on intervals of the realEuclidean 2-space. This study includes also games whose payoff functions present aparameter varying in a discrete set. In [23,24,27,28] David Carfı et al analyze alsoparametric games where the parameter set is interpreted as a coopetitive strategyspace. Our analysis of discrete parametric games allows us, also, to pass from thepayoff representation of standard normal-form games (see, for this classic games,[2,3,49,50]) to some types of coopetitive extensions.

1.3. Structure of the paper. To ease the reader, in the first section of thepaper we bring to mind terminology and some definitions, while in the second part,the method proposed in [7] and applied in the development of our algorithm, ispresented. The application of our algorithm to several examples concludes thepaper.

1.4. Motivation of the paper: the complete study of a game. Gametheory has proved a powerful tool to suggest strategies that must be employed byrational individuals in competitive and cooperative environments. Nevertheless,in the great part of current applied literature about the subject, the methodolo-gies used are essentially taken from the finite Game Theory and devoted to thestudy of Nash equilibria; this precludes several more deep applications, studies anddevelopments. On the contrary, we want to concentrate our attention on infinitedifferentiable games, which are models more complex and much more adherent tothe real human, economic and financial interactions: this is the final task of theComplete Analysis of a Differentiable Game. Its first goal is the precise knowledgeof the Pareto boundaries (maximal and minimal) of the payoff space, this knowl-edge will allow us to evaluate the quality of the different Nash equilibria (by thedistances from the Nash equilibria themselves to Pareto boundaries, with respectto appropriate metrics), in order to determine some focal equilibrium points (in thesense of Meyerson) collectively more satisfactory than each other. Moreover, thecomplete knowledge of the payoff-space will allow to develop explicitly the coopera-tive phase of the game and the various bargaining problems rising from the strategicinteraction of the tourist firms (Nash bargaining problem, Kalai-Smorodinski bar-gaining problem and so on). The complete study of an infinite differentiable gamef , introduced in [2] and [6] by D. Carfı , consists of the following points of investi-gation:

0. Structure analysis of the game

0.1) classify the game (linearity, symmetries, invertibility, ...);0.2) find the critical zone of the game and its image by f ;0.3) determine the biloss space im(f);0.4) determine inf and sup of the game f and see if they are shadow optima;

1. Pareto analysis of the game

1.1) determine the Pareto boundaries ∂∗f and ∂∗f of f ;1.2) determine the inverse images by f of the Pareto boundaries;1.3) specify the control of each player upon the boundaries;1.4) specify the noncooperative reachability of the Pareto boundaries;

AN ALGORITHM FOR DYNAMICAL GAMES WITH FRACTAL-LIKE TRAJECTORIES 97

1.5) find possible Pareto solutions and crosses;

2. Nash (Selfish) analysis

2.1) find best reply correspondences and Nash equilibria;2.2) study the existence of Nash equilibria (Brouwer and Kakutany);2.3) find Nash equilibria, if any;2.4) evaluate non-cooperative reachability of Nash equilibria;2.5) evaluate the position of Nash equilibria with respect to ∂∗f and ∂∗f ;

3. Devotion analysis

3.1) find devotion correspondences and devotion equilibria;3.2) specify the efficiency and noncooperative reachability of devotion equilibria;3.3) confrontation of the devotion equilibrium with the Nash equilibrium;

4. Dominant analysis

4.1) find dominant strategies, if any;4.2) find strict and dominant Nash equilibria;4.3) reduce the game by elimination of dominated strategies;

5. Conservative analysis

5.1) find conservative values and worst loss functions of the players;5.2) find conservative strategies and crosses;5.3) find all the conservative parts of the game (in bistrategy and biloss spaces);5.4) find core of the game and conservative knots;5.5) evaluate Nash equilibria by the core and the conservative bivalue;

6. Offensive analysis

6.1) find worst offensive correspondences and offensive equilibria;6.2) evaluate non-cooperative reachability of offensive equilibria;6.3) evaluate the position of offensive equilibria with respect to ∂∗f and ∂∗f ;6.4) find worst offensive strategies of any player against the other player;6.5) find possible dominant offensive strategies;6.6) confront Nash equilibria with offensive equilibria;

7. Cooperative analysis

7.1) find the best compromises (Kalai-Smorodinsky solutions) and their bilosses;7.2) find the elementary core best compromise and corresponding biloss;7.3) find the Nash bargaining solutions and corresponding bilosses;7.4) find the solutions with closest bilosses to the shadow minimum;7.5) find the maximum collective utility solutions;7.6) study the transferable utility case.

8. Solution analysis

8.1) confront the possible non-cooperative solutions among them;8.2) confront the possible cooperative solutions among them;8.3) confront noncooperative and cooperative solution.

1.5. Confrontation with other papers in Game Theory literature. For whatconcerns the confrontation with other papers in the Game Theory literature we observethat:

• the dynamical games that we shall examine are already deeply studied and rep-resented, at least at their initial state - by the application of the topologicalmethod presented by Carfı in [7] - in several applicative papers by Carfı, Mu-solino and Perrone (see [10], [11], [12], [13], [14], [15] [16], [20], [21], [22]) bya long, quite indirect and step by step implementations of other standard com-putational softwares (such as AutoCad, Derive, Grapher, Graph and Maxima)or following a pure mathematical way (see for example [8]): on the contrary, our


algorithm provides the direct and one shot graphical representation of the entireevolution of those games (by movies) and consequently of the entire trajectory.

• the standard literature on game theory does not present algorithms devoted tothe graphical representation and computation of the payoff spaces, but essen-tially devoted to the determination of Nash equilibria, their stabilities and theirapproximations, see for example [4–6], [29–45,47,48] and [51–58].

2. Preliminaries and notations

In order to help the reader and increase the level of readability of the paper, we recallsome notations and definitions about n-player games in normal-form, presented yet in[1,7]. Although the below definition seems, at a first sight, different from the standardone (presented, for example, in [49]), we desire to note that it is substantially the same;on the other hand, the definition in this new form underlines that a normal-form game isnothing but a vector-valued function and that any possible exam or solution of a normal-form games attains, indeed, to this functional nature. After the new definition, we shallcomment the equivalence of the two forms of the definition.

Definition 1 (of game in normal-form). Let E = (Ei)ni=1 be an ordered family of

non-empty sets. We call n-person game in normal-form, upon the support E, eachfunction f : ×E → Rn, where ×E denotes the Cartesian product ×n

i=1Ei of the family E.The set Ei is called the strategy set of player i, for every index i of the family E, andthe product ×E is called the strategy profile space, or the n-strategy space, of the game.

Remark. First of all we recall a standard form definition of normal-form game:

Definition. A strategic game consists of a system (N,E, f), where:

1. a finite set N (the set of players) of cardinality n is canonically identified with theset of the first n positive integers;

2. E is an ordered family of nonempty sets, E = (Ei)i∈N , where, for each player i inN, the nonempty set Ei is the set of actions available to player i;

3. f is an ordered family of real functions f = (fi)i∈N , where, for each player i in N ,the function fi :

×E → R is the utility function of player i (inducing a preference relationon the Cartesian product ×E := ×j∈NEj (the preference relation of player i on the wholestrategy space). �

Well, it is quite clear that the above system (N,E, f) is nothing but a redundant formof the family f itself, which we prefer to consider in its vector-valued functional nature

f : ×j∈NEj → Rn : x �→ (fi(x))i∈N .

Terminology. Together with the previous definition of game in normal form, wehave to introduce some terminologies:

• the set {i}ni=1 of the first n positive integers is said the set of players of thegame;

• each element of the Cartesian product ×E is said a strategy profile, or n-strategy,of the game;

• the image of the function f , i.e., the set f(×E) of all real n-vectors of type f(x),with x in the strategy profile space ×E, is called the n-payoff space, or simplythe payoff space, of the game f .

Moreover, we recall the definition of Pareto boundary whose main properties havebeen presented in [9]. By the way, the maximal boundary of a subset T of the Euclideanspace Rn is the set of those s ∈ T which are not strictly less than any other element of T .

Definition 2 (of Pareto boundary). The Pareto maximal boundary of a gamef is the subset of the n-strategy space of those n-strategies x such that the correspondingpayoff f(x) is maximal in the n-payoff space, with respect to the usual order of the euclideann-space Rn. If S denotes the strategy space ×E, we shall denote the maximal boundary of


the n-payoff space by ∂f(S) and the maximal boundary of the game by ∂f (S) or by ∂(f)

. In other terms, the maximal boundary ∂f (S) of the game is the reciprocal image (by thefunction f) of the maximal boundary of the payoff space f(S). We shall use analogousterminologies and notations for the minimal Pareto boundary.

Remark (on the definition of Pareto boundary). Also in the case of thisdefinition, essentially the definition of maximal (Pareto) boundary is the standard one,unless perhaps the name Pareto: it is nothing more that the set of maximal elements in thestandard pre-order set sense, that is the set of all elements that are not strictly less thanother elements of the set itself. The only circumstance to point out is that the naturalpre-order of the strategy set ×E is that induced by the standard point-wise order of theimage f(S) by means of the function f , that is the reciprocal image of the point-wise orderon f(S) via f .

3. The method for C1 games

In this paper, we deal with normal-form game f defined on the product of n compactand non-degenerate intervals of the real line, and such that f is the restriction to then-strategy space of a C1 function defined on an open set of Rn containing the n-strategyspace S (which, in this case, is a compact infinite part of the n-space Rn). Details are in[7,25], but in the following we recall some basic notions.

3.1. Topological boundary. For easy of the not-specialized reader, we recall thatthe topological boundary of a subset S of a topological space (X, τ) is the set of thosepoints x of the space X such that every neighborhood of x contains at least one point ofS and at least one point in the complement of S. Observe that the topological boundaryof the support X of the topological space (X, τ) is empty (in the topological space itself).

The key theorem of our method is the following one, we invite the reader to paymuch attention to the topologies used below.

Theorem 1. Let f be a C1 function defined upon an open set O of the euclideanspace Rn and with values in Rn. Then, for every part S of the open set O, the topologicalboundary of the image of S by the function f , in the topological space f(O) (i.e. withrespect to the relativization of the Euclidean topology to f(O)) is contained in the unionf(∂OS) ∪ f(C), that is

∂f(O)f(S) ⊆ f(∂OS) ∪ f(C),

where: (1) C is the critical set of the function f in S (that is the set of all points x of Ssuch that the Jacobian matrix Jf (x) is not invertible); (2) ∂OS is the topological boundaryof S in O (with respect to the relative topology of O).

Note. Observe for example the following trivial case. Let O be the unit open ballB(02, 1) of the plane and let f be the canonical set-immersion (injection) of O into theplane R2 (that is the function f : O → R2 : x �→ x). If S := O, then f(S) = O; theboundary of f(S) in f(O) is empty (since f(O) = O), the boundary of S in O is emptytoo, and the theorem gives the trivial inclusion ∅ ⊆ ∅.

Note. We note, however, that when S is a compact subset of the open set O (itdoesn’t matter in what topology...), then the boundaries of S and f(S) in O and f(O)coincides with the boundaries of S and f(S) in Rn.

4. Two players parametric games

In this section we shall introduce the definitions of parametric games, as it is employedin the following.

Definition 3. Let E = (Et)t∈T and F = (Ft)t∈T be two families of non empty setsand let f = (ft)t∈T be a family of functions, where ft : Et × Ft → R2, for each t ∈ T .We define parametric gain game over the strategy pair (E,F ) and with family of


payoff functions f the pair G = (f,>), where the symbol > stands for the usual strictupper order of the Euclidean plane R2. We define the payoff space of the parametricgame G as the union of all the payoff spaces of the game family g = ((ft, >))t∈T , that is,as the union of the payoff family

P = (ft(Et × Ft))t∈T .

Dynamics. We will refer to the above family g as to the dynamical path of the gameG, since we can see it as a curve of games:

g : T → g(T ) : t �→ (ft, >).

4.1. Payoff set-dynamics. We note also that the family P can be identified withthe multi-valued path in R2

p : T → R2 : t �→ ft(Et × Ft),

(multivalued means that to each value t ∈ T the mappings p associates a subset of theplane, and not one unique single point of it) and that the graph of this path p is a subsetof the Cartesian product T × R2, on the other hand, the trace of the curve p, is a subsetof the plane and it is the union of all the values of the multi-valued path p.

5. The algorithm

5.1. The game framework of the algorithm. In particular we are concentratedon the following specific kind of parametric game:

• parametric games in which the families E and F consist of only one set, respec-tively.

In the latter case, we can identify a parametric game with a pair (f,>), where f is afunction from a Cartesian product T × E × F into the plane R2, where T , E and F arethree non-empty sets.

Definition 4. When the triple (T,E,F ) is a triple of subsets of normed spaces, wedefine the parametric game (f,>) of class C1 if the function f is of class C1.

5.2. Structure of the algorithm. The algorithm for the representation of thepayoff trajectory of dynamical game generalizes the procedure presented in [1, 25], fordiscrete dynamical games of the type (fn)n∈N. In particular, it has been extended tosequences (of payoff functions) recursively defined. We define, for all (x, y) in the strategyspace S := E × F := [x1, x3]× [y1, y3] ⊂ R2,{

f0(x, y) = (f 10 (x, y), f 2

0 (x, y))

fn(x, y) = (f 1n (x, y), f 2

n (x, y)),

for all integers n ≥ 1. Note that if, for each n, the function fn is defined by means of thefunction fn−1, then it has to be evaluated.

Our aim is to represent the payoff family scenario, varying the parameter n ∈ N,performing the iteration for n varying from 1 to a fixed natural number N , with N ∈ Nfixed a priori sufficiently great.

To this order, all the points in the topological boundary T of the strategy space Sand the critical zone Cn have to be transformed by using each payoff function fn. Thus,let be

T ′n = {fn(x1, y)}y∈F ∪ {fn(x, y1)}x∈E ∪ {fn(x3, y)}y∈F ∪ {fn(x, y3)}x∈E ,

be the transformation of the topological boundary. Moreover, let us denote

Cn ={(x, y) ∈ S : ∂1f1

n (x, y) · ∂2f2

n (x, y)− ∂2f1

n (x, y) · ∂1f2

n (x, y) = 0}C ′

n ={(f 1n (x, y), f 2

n (x, y)) : (x, y) ∈ Cn}


the critical zone and its transformation, respectively. It has been proved that the topo-logical boundary of the payoff scenario of the whole dynamical game is contained in theunion

⋃Nn=0(T

′n ∪C ′

n). Taking into account the introduced notation, our algorithm can besummarized in few steps as follows. INPUT:

E = [x1, x3], F = [y1, y3],

N (Maximum number of Iteration)

f0(x, y) and fn(x, y) ∀n ≥ 1(Payoff function)

PROCESSING: FOR n=0 to N- Evaluation of function fn, for every n, if necessary.

fn(x, y) = (f 1n (x, y), f 2

n (x, y))

- Transformation of the topological boundary T by fn:representation (plot) of T ′

n, as defined above;

- Evaluation of the critical zone Cn:representation (plot) of the inverse image (det ◦Jfn)

←(0);

- Transformation of the critical zone:plot of the image C ′

n := fn(Cn), described above;

- Payoff Space Pn of the game fn: Pn is the fill in of the union T ′n ∪ C ′

n.END

Payoff space of the dynamical game G = (fn):let f : E × F × N → R2 defined by f(x, y, n) := fn(x, y), for every (x, y, n), we plot

of the image f(E × F × N) as the union⋃N

n=0 Pn.OUTPUT: plots of the payoff boundary scenario family scenario and of payoff sce-

nario family f(E × F × N).

5.3. Principal aims of the algorithm. Our algorithm gives us:

• the dynamical evolution of the payoff family P , in the sense of the dynamicalevolution (in real time) when we consider the parameter set T as the real timestraight-line (this by movies);

• the trace of this dynamical path, i.e., the very payoff space of the parametricgame G.

6. Examples

In the following subsections we shall consider the following examples:1. the parametric game G = (fa)a∈T , defined by

fa(x, y) = ||(1, a)||−2(x(1− 2x+ y), y(1− 4y + x)) + φ(a)(1, 1),

for all x, y ∈ [0, 1] and a ∈ [0,+∞[, where φ(a) := a(1 + a)−1.2. the parametric game G = (fa)a∈T , defined by

fa(x, y) = ||(1, a)||−2(x(1− 2x+ y), y(1− 4y + x)) + ie−iφ(a),

for all x, y in [0, 1] and a ∈ [0,+∞], where φ(a) := a(1 + a)−1.3. the parametric game G = (fa)a∈T , defined by

fa(x, y) = 4(1 + a3)−1(1− x− y)(x, y) + g(a),

with g(a) = (2φ(a),−(1/6)a−2φ(a)2((a − 3)2 + 9)), for all x, y in [0, 1] and a in [0,+∞],where φ(a) := a(1 + a)−1.

4. the parametric game G = (fn)n∈N, defined by

fn(x, y) = (1/3)nf0(x, y) + (3/2)(1− (1/3)n)w

where f0(x, y) = (−(1/2)(1− x)y, xy) and w = (1/3, 2/3).


5. the parametric game G = (fn)n∈N, defined by

fn+1(x, y) = an+1f0(x, y) + anw

where f0(x, y) = (−(1/2)(1 − x)y, xy), with a−1 = 0, a0 = 1, an+1 = 1 + (1/3)an andw = (1/3, 2/3).

6.1. First game. Here, we present a parametric game of Bertrand type (alreadyrepresented, at the initial state, in [21], by a long procedure, using Maxima), whosestrategy sets are E = F = [0, 1], the parameter set is T = R≥ and the a-biloss (disutility)function is defined by

fa(x, y) = ||(1, a)||−2(x(1− 2x+ y), y(1− 4y + x)) + φ(a)(1, 1),

for all x, y in [0, 1] and a in [0,+∞[, where φ(a) := a(1 + a)−1. The payoff scenario pathof the discrete subfamily-game G′ := (fn)n∈N is depicted in Figure 1 (up to n = 10, forsake of simplicity). This payoff scenario path is the union of the payoff scenario evolutionfamily (fn(E × F ))n∈N, which can be seen as a multi-valued discrete dynamical pathγ : N → R2 : n �→ fn(E × F ).

Figure 1. Game 1. Payoff trajectory of the subgame G′.

6.2. Second game. In this subsection, we present another Bertrand-type paramet-ric game, where strategy sets are E = F = [0, 1], the parameter set is T = R≥ and thea-biloss (disutility) function is defined by

fa(x, y) = ||(1, a)||−2(x(1− 2x+ y), y(1− 4y + x)) + ie−iφ(a),

for all x, y in [0, 1] and a in [0,+∞], where φ(a) := a(1 + a)−1. The payoff scenario pathof the discrete subfamily-game G′ := (fn)n∈N is depicted in Figure 2 (up to n = 10, forsake of simplicity).


Figure 2. Game 2. Payoff trajectory of the subgame G′.

6.3. Third game. In this subsection, we present a Cournot-type parametric game,where strategy sets are E = F = [0, 1], the parameter set is T = [2,+∞] and the a-payofffunction is defined by

fa(x, y) = c(x, y, a) + (2a/(1 + a),−((1/6)(a− 3)2 + 3/2)/(1 + a)2),

where c(x, y, a) = 4(1− x− y)(1 + a3)−1(x, y), for all x, y ∈ [0, 1] and a ∈ T . We shallconsider, this time, two subsequences of the above game. The payoff scenario path of thediscrete subfamily-game G′ := (fn)n∈N is depicted in Figure 3 and 4 (up to n = 20, forsake of simplicity).

Figure 3. Game 3. Payoff boundary trajectory of sub-game G′.

The payoff scenario path of the discrete subfamily-game G′′ := (f2n)n∈N is depictedin Figure 5 and in Figure 6 (up to n = 10, for sake of simplicity).


Figure 4. Game 3. Payoff trajectory of the sub-game G′.

Figure 5. Game 3. Payoff boundary trajectory of the sub-game G′′.

6.4. Fourth game. We present a discrete parametric game (proposed in financialliterature by D. Carfı and F. Musolino), with strategy sets E = [0, 1], F = [−1, 1] and then-payoff function is defined by

fn(x, y) = (1/3)nf0(x, y) + (3/2)(1− (1/3)nw)

where f0(x, y) = (−(1/2)y(1−x), xy) and w = (1/3, 2/3). The payoff scenario path of thediscrete family-game G := (fn)n∈N is depicted in Figure 7.

Otherwise, setting E = [−1, 1] and F = [−1, 1], the resulting payoff trajectorychanges, as shown in Figure 8.

6.5. Fifth game. Let us consider another Carfı-Musolino financial game, that de-fined by G = (fn)n∈N with fn : E × F → R2, with

fn+1(x, y) = an+1f0(x, y) + anw,


Figure 6. Game 3. Payoff trajectory of the sub-game G′′.

Figure 7. Game 4. Payoff trajectory of the fourth game.

for every (x, y) in E × F , where: f0(x, y) = (−(1/2)y(1 − x), xy); with a−1 = 0, a0 = 1,an+1 = 1 + (1/3)an and w = (1/3, 2/3). If E = [−1, 1] and F = [−1, 1], the payofftrajectory is illustrated in Figure 9.

Figure 10 refers to case E = [0, 1] and F = [−1, 1], with fn : [0, 1]× [−1, 1] → R2.

Setting E = [0, 1], F = [−1, 1] and an+1 = 1 + 0.9an, we obtain the trajectoryrepresented in Figure 11.


Figure 8. Game 4. Payoff trajectory of the fourth game extended to E =[−1, 1] and F = [−1, 1].

Figure 9. Game 5. Payoff trajectory of the fifth game (E = F = [−1, 1]).


Figure 10. Game 5. Payoff trajectory of the fifth game restricted to E =[0, 1] and F = [−1, 1].

Figure 11. Game 5. Payoff trajectory of the fifth game with E = [0, 1] F =[−1, 1] and an+1 = 1 + 0.9an.


Figure 12. Game 5. Payoff Space of the fifth game for E = [0, 1]F = [−1, 1]and an+1 = 1 + n+ 0.3an..

At the end, if we assume E = [−1, 1] and F = [−1, 1], with an+1 = 1 + n + 0.33an,we get the payoff trajectory illustrated in Figure 12.

7. Final Remarks

The Bertrand and Cournot games proposed above (in the example 1,2 and 3) belongto the classic economic games presented in the literature by Carfı and Perrone in [20–22],the model proposed there is quite general and the specific examples we propose here arenot particularly distinguished Bertrand/Cournot games, we fixed the constants only forsake of representability. Any other Bertrand or Cournot type game is of the same natureand the algorithm is straightforwardly good for any choice of the constants. By the way, wehave chosen those particular games to compare the results of our algorithm to the studiesperformed by the software Maxima (already published in those Carfı-Perrone papers) withvery long and intricate procedures. Note that Carfı and Perrone study only the initialstate of the economic dynamical game, but from this initial graphical representation ispossible to deduce all the sequence by contractions and translations: our algorithm heregives in one shot all the dynamics without further considerations and operations.

The Financial games studied in the last games derive from a complex and wide tenta-tive of Carfı and Musolino (see [10–19]) to give a robust stability to the financial marketsunder speculative attacks. In this cases Carfı and Musolino study (by softwares Graphand Grapher) all the evolution of the financial games, so that the confrontation with therepresentations of our paper is total, not only partial.

8. Resume

In [25], a new procedure to determine the payoff scenarios of non-parametric differ-entiable games has been presented; then this new procedure has been applied in [1] tonumerically determine a new 3-dimensional representation of the payoff spaces of contin-uous families of normal-form C1-games, with two players, families indexed by a compactinterval of the real line. Moreover, the method in [7] has been pointed out in [26] with theaim of realizing a numerical procedure providing, finally, the real geometrical represen-tation of the payoff scenarios of C1-families of normal-form C1-games, with two players,families indexed by a compact interval of the real line. In this present work, the method in[7] is applied to realize an algorithm for the representation of the payoff space trajectories


of discrete families of normal-form C1 games, that is a numerical procedure providingthe real geometrical representation of the payoff scenarios of sequences of normal-formC1-games.

9. Conclusions

In this paper, we have proposed a new algorithm able to represent in one shot thepayoff trajectory of two-player discrete-time dynamical games.

Specifically:

• we consider discrete dynamical games which can be modeled as sequences ofnormal-form games (the states of the dynamical game) with payoff functions ofclass C1.

• In this context, the payoff trajectory of such type of dynamical games is thesequence of the payoff spaces of their game-states.

• The formulation of the algorithm is motivated - especially in several applicativecontexts such as Economics, Finance, Politics, Management Sciences, Medicineand so on ... - by the need of a complete knowledge of the payoff evolution(problem which is still open in the most part of the cases), especially when thereal problem requires a Complete Analysis beyond the study of just the Nashequilibria.

• We consider, to prove the efficiency and strength of our method, the develop-ment (by the algorithm itself) of some non-linear parametric games taken fromapplications to Micro-Economics and Finance.

• The dynamical games that we shall examine are already deeply studied andrepresented, at their initial state, by the application of the topological methodpresented by Carfı in [7], in several applicative papers by Carfı, Musolino andPerrone (see [10], [11], [12], [13], [14], [16], [20]) by a long, indirect and stepby step implementations of other standard computational softwares (such asAutocad, Derive, Grapher, Graph and Maxima) or from a pure mathematicalway (see for example [8]);

• contrary to the classic softwares present in the literature, our algorithm providesthe direct and one shot graphical representation of the entire evolution of thosegames.

• Finally, the applicative games we consider in the paper (inspired by Economicsand Finance) have a natural dynamics having fractal-like trajectories.

Acknowledgement. The authors wish to thank an anonymous referee that helpedvery much to deeply improve the paper.

References

[1] S. Agreste, D. Carfı, A. Ricciardello, An algorithm for payoff space in C1 parametric games,Applied Sciences (APPS), vol. 13 (2011).

[2] Jean-Pierre Aubin, Mathematical methods of game and economic theory, Studies in Math-ematics and its Applications, vol. 7, North-Holland Publishing Co., Amsterdam, 1979.MR556865 (83a:90005)

[3] Jean-Pierre Aubin, Optima and equilibria, 2nd ed., Graduate Texts in Mathematics, vol. 140,Springer-Verlag, Berlin, 1998. An introduction to nonlinear analysis; Translated from theFrench by Stephen Wilson. MR1729758 (2001g:49002)

[4] N. Bhat, K. Leyton-Brown, Computing Nash equilibria of action-graph games, Proceedingsof the 20th conference on Uncertainty in artificial intelligence. AUAI Press, 2004.

[5] Tamer Basar and Geert Jan Olsder, Dynamic noncooperative game theory, 2nd ed., AcademicPress Ltd., London, 1995. MR1311921 (96b:90001)


[6] M. Bloem, T. Alpcan, T. Basar, A Stackelberg game for power control and channel alloca-tion in cognitive radio networks, Proceedings of the 2nd international conference on Perfor-mance evaluation methodologies and tools. ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering), 2007.

[7] David Carfi, Payoff space in C1-games, Appl. Sci. 11 (2009), 35–47. MR2534055(2010j:91025)

[8] D. Carfı, Differentiable game complete analysis for tourism firm decisions, in Proceedings of

the 2009 International Conference on Tourism and Workshop on Sustainable Tourism withinHigh Risk Areas of Environmental Crisis, Messina, Italy, 2009.

[9] D. Carfı, Optimal boundaries for decisions, AAPP, Vol. LXXXVI issue 1, 2008, pp. 1-12.[10] D. Carfı, F. Musolino (2011), Fair redistribution in financial markets: a game theory complete

analysis, Journal of Advanced Studies in Finance (JASF) 2, 74.[11] D. Carfı, F. Musolino (2012), A coopetitive approach to financial markets stabilization and

risk management, in Advances in Computational Intelligence, pp. 578-592 (Springer-Verlag,Berlin, 2012); book series: Communications in Computer and Information Science.

[12] D. Carfı, F. Musolino (2012), Game theory and speculation on government bonds, EconomicModelling (Elsevier), 29 (6), pp. 2417-2426.

[13] D. Carfı, F. Musolino (2013), Dynamical Stabilization of Currency Market with Fractal-likeTrajectories, pre-print.

[14] D. Carfı, F. Musolino (2013), Game Theory Application of Monti’s Proposal for EuropeanGovernment Bonds Stabilization, Applied Sciences, vol. 15.

[15] D. Carfı, F. Musolino (2012), Credit crunch in the Euro area: a coopetitive multi-agent so-lution, in Multicriteria and Multiagent Decision Making with Applications to Economic andSocial Sciences, vol. 310, book series: Studies in Fuzziness and Soft Computing. ISBN: 978-3-642-35634-6.

[16] D. Carfı, F. Musolino (2013), A Coopetitive-dynamical game model for currency marketsstabilization, joint proceedings of Fractal Geometry in Pure and Applied Mathematics: inMemory of Benoit Mandelbrot (Boston, January 2012), Analysis, Fractal Geometry, Dynam-ical Systems, and Economics (Messina, November 2011), Geometry and Analysis on FractalSpaces (Hawaii, March 2012), submitted.

[17] D. Carfı, F. Musolino (2013),Model of Possible Cooperation in Financial Markets in Presenceof Tax on Speculative Transactions, AAPP|Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat.Nat., vol 91, issue 1 (2013).

[18] D. Carfı, F. Musolino (2011), Game complete analysis for financial markets stabilization,Proceeding of the 1st International on-line Conference on Global Trends in Finance, ASERS,pp. 14-42.

[19] D. Carfı, F. Musolino (2013), Speculative and Hedging Interactions Model about Oil and U.S.Dollar Markets, Economic Modelling, submitted.

[20] D. Carfı, E. Perrone, 2011, Game Complete Analysis of Bertrand Duopoly, TPREF, 2 (1),5-22, June.

[21] D. Carfı, E. Perrone, 2011, Asymmetric Bertrand Duopoly: Game Complete Analysis byAlgebra System Maxima, Mathematical Models in Economics, ASERS Publishing chapter 3,pp. 44 - 66 ISBN 9786069238660

[22] D. Carfı, E. Perrone, 2012, Game complete analysis of symmetric Cournot duopoly, RePEc- Research Papers in Economics, http://econpapers.repec.org/RePEc:pra:mprapa:35930.

[23] D. Carfı, G. Patane, S. Pellegrino, A Coopetitive approach to Project financing, Book chapter,“Moving from crisis to sustainability: Emerging issues in the international context, FrancoAngeli 2012

[24] D. Carfı, A. Trunfio, A non-linear coopetitive game in global Green Economy, Book chapter,“Moving from crisis to sustainability: Emerging issues in the international context, FrancoAngeli 2012

[25] D. Carfı, A. Ricciardello, An Algorithm for Payoff Space in C1-Games, Atti Accademia

Peloritana Pericolanti (AAPP) Cl. Sci. Fis. Mat. Nat. Vol. 88, No. 1, C1A1001003 (2010)[26] D. Carfı, A. Ricciardello, Algorithms for Payoff Trajectories in C1 Parametric Games, S.

Greco et al. (Eds.): IPMU 2012, Part IV, CCIS 300, pp. 642–654, 2012. Springer-Verlag BerlinHeidelberg 2012 (2012)


[27] D. Carfı, D. Schiliro, Crisis in the Euro area: coopetitive game solutions as new policy tools,TPREF-Theoretical and Practical Research in Economic Fields, summer issue pp. 1-30. ISSN2068 - 7710, http://www.asers.eu/journals/tpref.html, 2011

[28] D. Carfı, D. Schiliro, A coopetitive model for a global green economy, Book chapter, “Movingfrom crisis to sustainability: Emerging issues in the international context, Franco Angeli 2012

[29] X. Chen, D. Xiaotie, Settling the complexity of two-player Nash equilibrium, Foundations ofComputer Science, 2006. FOCS’06. 47th Annual IEEE Symposium on. IEEE, 2006.

[30] X. Chen, D. Xiaotie, 3-Nash is PPAD-complete, Electronic Colloquium on ComputationalComplexity. Vol. 134. 2005.

[31] V. Conitzer, T. Sandholm, (2002), Complexity results about Nash equilibria, arXiv preprintcs/0205074.

[32] Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou, The complexityof computing a Nash equilibrium, STOC’06: Proceedings of the 38th Annual ACM Symposiumon Theory of Computing, ACM, New York, 2006, pp. 71–78, DOI 10.1145/1132516.1132527.MR2277132 (2007h:68078)

[33] Itzhak Gilboa and Eitan Zemel, Nash and correlated equilibria: some complexity consider-ations, Games Econom. Behav. 1 (1989), no. 1, 80–93, DOI 10.1016/0899-8256(89)90006-7.MR1000049 (90c:90247)

[34] Itzhak Gilboa, The complexity of computing the best-response automata in repeated games, J.Econom. Theory 45 (1988), no. 2, 342–352, DOI 10.1016/0022-0531(88)90274-8. MR964626(90f:90178)

[35] Itzhak Gilboa and Dov Samet, Bounded versus unbounded rationality: the tyranny of theweak, Games Econom. Behav. 1 (1989), no. 3, 213–221, DOI 10.1016/0899-8256(89)90009-2.MR1100384 (92e:90116)

[36] Paul W. Goldberg and Christos H. Papadimitriou, Reducibility among equilibrium problems,STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, ACM,New York, 2006, pp. 61–70, DOI 10.1145/1132516.1132526. MR2277131 (2007h:68083)

[37] G. Gottlob, G. Greco, F. Scarcello, Pure Nash equilibria: hard and easy games, Proceedingsof the 9th conference on Theoretical aspects of rationality and knowledge. ACM, 2003.

[38] M. Hansen, Airline competition in a hub-dominated environment: An application of nonco-

operative game theory,Transportation Research Part B: Methodological 24.1 (1990): 27-43.[39] C. Yang, et al, Power allocation based on noncooperative game theory in cognitive radio [J],

Journal of Xidian University 1 (2009): 002.[40] M. Kearns, M. L. Littman, S. Singh, Graphical models for game theory, Proceedings of the

Seventeenth conference on Uncertainty in artificial intelligence. Morgan Kaufmann PublishersInc., 2001.

[41] S. Koskie, Z. Gajic, A Nash game algorithm for SIR-based power control in 3G wirelessCDMA networks, IEEE/ACM Transactions on Networking (TON) 13.5 (2005): 1017-1026.

[42] M. L. Littman, M. Kearns, S. Singh, (2002, December), An Efficient Exact Algorithm forSolving Tree-Structured Graphical Games, In Proceedings of the 15th Conference on NeuralInformation Processing Systems-Natural and Synthetic (pp. 817-823).

[43] R. I. Lung, D. Dumitrescu,Considerations on Evolutionary Detecting Nash Equilibria versusPareto Frontier., MaC06 6th Joint Conference on Mathematics and Computer Science. 2006.

[44] Richard D. McKelvey and Andrew McLennan, Computation of equilibria in finite games,Handbook of computational economics, Vol. I, Handbooks in Econom., vol. 13, North-Holland, Amsterdam, 1996, pp. 87–142. MR1416607

[45] Meshkati, Farhad, et al, A non-cooperative power control game for multi-carrier CDMAsystems,Wireless Communications and Networking Conference, 2005 IEEE. Vol. 1. IEEE,2005.

[46] Roger B. Myerson, Game theory: Analysis of conflict, Harvard University Press, Cambridge,MA, 1991. MR1107301 (92h:90002)

[47] Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani (eds.), Algorithmic gametheory, Cambridge University Press, Cambridge, 2007. MR2391747 (2008k:91002)

[48] D. Niyato, E. Hossain, A noncooperative game-theoretic framework for radio resource man-agement in 4G heterogeneous wireless access networks, Mobile Computing, IEEE Transac-tions on 7.3 (2008): 332-345.

[49] M. J. Osborne, A. Rubinstein, A course in Game Theory, Academic Press (2001)


[50] Guillermo Owen, Game theory, 2nd ed., Academic Press Inc. [Harcourt Brace JovanovichPublishers], New York, 1982. MR697721 (84e:90109)

[51] Christos H. Papadimitriou and Tim Roughgarden, Computing correlated equilibria in multi-player games, J. ACM 55 (2008), no. 3, Art. 14, 29, DOI 10.1145/1379759.1379762.MR2444912 (2009h:91010)

[52] Ryan Porter, Eugene Nudelman, and Yoav Shoham, Simple search methods for find-ing a Nash equilibrium, Games Econom. Behav. 63 (2008), no. 2, 642–662, DOI

10.1016/j.geb.2006.03.015. MR2436075 (2009g:91013)[53] R. Rajabioun, E. Atashpaz-Gargari, C. Lucas, Colonial competitive algorithm as a tool for

Nash equilibrium point achievement, Computational Science and Its Applications ICCSA2008 (2008): 680-695.

[54] J. Rosenmuller, On a generalization of the Lemke-Howson algorithm to noncooperative N-person games, SIAM J. Appl. Math. 21 (1971), 73–79. MR0287921 (44 #5123)

[55] D. Stein, A. Noah, P. Parrilo, Asuman Ozdaglar, Exchangeable equilibria contradict exactnessof the Papadimitriou-Roughgarden algorithm. arXiv preprint arXiv:1010.2871 (2010).

[56] Vijay V. Vazirani, The notion of a rational convex program, and an algorithm forthe Arrow-Debreu Nash bargaining game, J. ACM 59 (2012), no. 2, Art. 7, 36, DOI10.1145/2160158.2160160. MR2915678

[57] D. Vickrey, D. Koller, Multi-agent algorithms for solving graphical games, Proceedings of theNational Conference on Artificial Intelligence, Menlo Park, CA; Cambridge, MA; London;AAAI Press; MIT Press; 1999, 2002.

[58] L. Wang, X. Yisheng, E. Schulz, Resource allocation in multicell OFDM systems based onnoncooperative game, Personal, Indoor and Mobile Radio Communications, 2006 IEEE 17thInternational Symposium on. IEEE, 2006.

IAMIS, Institute for the Applications of Mathematics and Integrated Science, Uni-

versity of California at Riverside, 900 Big Springs Road, Surge 231 Riverside, Califor-

nia 92521-0135

Department of Mathematics and Informatics, University of Messina, Messina, Italy.

PISRS, Permanent International Session of Research Seminars, University of Messina,

Via dei Verdi, 75, 98122 Messina, Italy

E-mail address: [email protected] address: [email protected]

Department of Mathematics, Faculty of Engineering, University of Enna, Italy



The Landscape of Anderson Localizationin a Disordered Medium

Marcel Filoche and Svitlana Mayboroda

Abstract. In quantum systems, the presence of a disordered potential mayinduce the appearance of strongly localized quantum states (a phenomenoncalled Anderson localization), i.e., eigenfunctions that essentially “live” in avery restricted subregion of the entire domain. We show here that solving asimple Dirichlet problem reveals a network of interconnected lines which arethe boundaries of the localization subregions, and allows one to evaluate thestrength of the confinement to these subregions. For each given eigenvalue,only a subset of this network effectively determines the confinement of thecorresponding eigenfunction. This subset becomes smaller as the eigenvalueincreases, leading to a weaker confinement and finally possibly delocalizedstates.

1. Introduction

Physical systems characterized by a spatial inhomogeneity of the material orby an irregular or disordered geometry exhibit specific vibrating properties, notfound in usual smooth or homogeneous systems. In particular, the stationaryvibrations, i.e., the eigenfunctions of the corresponding wave operator, can haveextremely uneven spatial distributions of their amplitude. More precisely, forsome eigenvalues (or frequencies), most of the vibration energy is concentratedonly in one very restricted subregion of the entire domain and remains very lowin the rest of the domain [HS]. Although still poorly understood, this phenom-enon, called localization, has been observed in acoustical, optical, mechanical,and quantum systems, and plays an essential role in numerous physical proper-ties [ERRPS,FAFS,RBVIDCW].

A particular case of localization introduced in 1958 by Anderson [A] is thedisorder-induced localization. It occurs in systems in which the properties of thematerial vary spatially in a random way. For a large enough amplitude of thevariation (i.e. for a sufficiently large disorder), the eigenfunctions of the waveoperator are strongly localized inside the system; they mostly “live” in a very small

2010 Mathematics Subject Classification. Primary 35P05, 47A75; Secondary 81V99.The current work was partially supported for M.F. by the ANR Program Silent Wall ANR-

06-MAPR-00-18 and PEPS-PTI grant from CNRS..Part of this work was completed during the visit of S.M. to the Ecole Normale Superieure

(ENS) de Cachan. This work was partially supported by the Alfred P. Sloan Fellowship, theNational Science Foundation CAREER Award DMS 1056004, NSF Grant DMS 0758500, and

NSF MRSEC Seed grant.


113

114 MARCEL FILOCHE AND SVITLANA MAYBORODA

subregion and their amplitudes exponentially decay away from this region. Inquantum systems it implies that the corresponding electronic states in a disorderedenough potential are non conducting, even though the system exhibits statisticaltranslational invariance.

Despite vast literature and numerous important results, many features of thelocalization of eigenfunctions remain mysterious. In particular, it seems very dif-ficult to predict where to expect localized vibrations, and for which eigenvalues,without having to solve the full eigenvalue problem. We will address in this paperthe case of Anderson localization of quantum states, and demonstrate that one canin fact predict the localization subregions by solving only one Dirichlet problem.Further related results can be found in [FM].

2. Preliminaries

2.1. The quantum states. The stationary quantum states of a particle in

a domain Ω are the eigenfunctions of the Hamiltonian H = − �2

2mΔ + V in the

domain, where m stands for the mass of the particle and V (x) is the potentialfunction describing the external forces acting on the particle. The eigenvalues ofthe Hamiltonian correspond to the energies of these states. The electronic statesinside a disordered medium can thus be modeled by introducing a random potentialV to account for the material inhomogeneities. For instance, the domain Ω can bedivided into elementary cells on which V is piecewise constant. The value of Von each cell is taken at random, uniformly between 0 and a maximum value Vmax.The goal of this paper is to study the spatial distributions of the localized states insuch a potential.

In what follows, we will first present the main inequalities and their proofs inthe context of a general second order elliptic operator with bounded measurablecoefficients, and in numerical experiments we will come back to quantum mechanicsin a disordered medium and to the Hamiltonian H.

2.2. The wave operator. Let L be a divergence form elliptic operator oran elliptic system with bounded measurable coefficients. For the sake of simplicitywe shall work here with the second order symmetric operators with real-valuedcoefficients, which already include the main examples in the focus of the presentpaper: the Laplacian, the Hamiltonian, and their non-homogeneous analogues. Itis worth mentioning, however, that an appropriate version of the key inequalitiesremains valid for much more general elliptic operators, with complex coefficientsand/or of higher order.

To this end, let Ω be a bounded open set in Rn and denote

(2.1) L = −divA(x)∇ + V (x),

where A is an elliptic real symmetric n× n matrix with bounded measurable coef-ficients, that is,(2.2)

A(x) = {aij(x)}ni,j=1, x ∈ Ω, aij ∈ L∞(Ω),

n∑i,j=1

aij(x)ξiξj ≥ c|ξ|2, ∀ ξ ∈ Rn,

for some c > 0, and aij = aji, ∀i, j = 1, ..., n, and V ∈ L∞(Ω) is a non-negativefunction. The action of the operator L in (2.1) is understood, as usually, in the

THE LANDSCAPE OF ANDERSON LOCALIZATION IN A DISORDERED MEDIUM 115

weak sense. Indeed, recall that the Lax-Milgram Lemma ascertains that for everyf ∈ (H1(Ω))∗ =: H−1(Ω) the boundary value problem

(2.3) Lu = f, u ∈ H1(Ω),

has a unique solution such that

(2.4)

∫Rn

(A∇u∇v + V uv) dx =

∫Rn

fv dx, for every v ∈ H1(Ω).

Here H1(Ω) is the Sobolev space of functions given by the completion of C∞0 (Ω)

in the norm

(2.5) ‖u‖H1(Ω) := ‖∇u‖L2(Ω).

For later reference, we also define the Green function of L, as conventionally,by

(2.6) LxG(x, y) = δy(x), for allx, y ∈ Ω, G(·, y) ∈ H1(Ω) for all y ∈ Ω,

in the sense of (2.4), so that

(2.7)

∫Rn

LxG(x, y)v(x) dx = v(y), y ∈ Ω,

for every v ∈ H1(Ω).Remark. The solution given by the Lax-Milgram Lemma can be thought of as asolution of the Dirichlet problem with zero boundary data, and for relatively nicedomains it can be shown that u is a classical solution:

(2.8) −Δu = f in Ω, u|∂Ω = 0,

where u|∂Ω denotes the pointwise limit at the boundary, i.e.,

(2.9) u(x) = limy→x, y∈Ω

u(y), x ∈ ∂Ω.

In principle, on “bad” domains the definition (2.9) might not make sense, i.e.,such a limit might not exist, and then the solution can only be interpreted in thesense of (2.4). For the Laplacian, and all homogeneous second order operators withbounded measurable coefficients as above it is known which domain are “good”and which are “bad”, due to the 1924 Wiener criterion and its generalization byLittman, Stampacchia and Weinberger [W], [LSW]. The gist of the matter is thatthe boundary should not have too sharp inward cusps, cracks or isolated points.

3. The control inequalities

3.1. Control of the eigenfunctions by the solution to the Dirichletproblem. Having at hand (2.3)–(2.4), one can consider the eigenvalue problem:

(3.1) Lϕ = λϕ, ϕ ∈ Hm(Ω),

where λ ∈ R. If for a given λ ∈ R there exists a non-trivial solution to (3.1),interpreted, as before, in the weak sense, then the corresponding λ is called aneigenvalue and ϕ ∈ H1(Ω) is an eigenvector. Under the assumptions on the operatorimposed in the previous section (which, in particular, yield self-adjointness), thestandard methods of functional analysis directly apply to show that the eigenvaluesof L form a positive sequence going to +∞, and the eigenfunctions of L define aHilbert basis of L2(Ω) (cf. [E], [H]).


Proposition 3.1. Let L be an arbitrary elliptic operator as defined by (2.1) –

(2.5), and assume that λ is an eigenvalue L and ϕ ∈ Hm(Ω) is the correspondingeigenfunction, i.e., (3.1) is satisfied. Then for every x ∈ Ω

(3.2)|ϕ(x)|

‖ϕ‖L∞(Ω)≤ λu(x), for all x ∈ Ω,

where u is the solution of the boundary problem

(3.3) Lu = 1, u ∈ Hm(Ω).

Proof. By (3.1) and (2.6) (with the roles of x and y interchanged), for everyx ∈ Ω

(3.4) ϕ(x) =

∫Ω

Lyϕ(y)G(x, y) dy =

∫Ω

λϕ(y)G(x, y) dy,

and hence,

(3.5) |ϕ(x)| ≤ λ ‖ϕ‖L∞(Ω)

∫Ω

|G(x, y)| dy, x ∈ Ω.

The Green function is positive in Ω and eigenfunctions are bounded for all secondorder elliptic operators (2.1) in all dimensions due to the strong maximum principle(see, e.g., [GT], Section 8.7). Hence,

(3.6)

∫Ω

|G(x, y)| dy =

∫Ω

G(x, y) · 1 dy, x ∈ Ω,

which is by definition a solution of (3.3). �The inequality (3.2) provides the “landscape of localization”, as the map of u

in (3.2) draws the lines separating potential subdomains. The exact meaning ofthis statement is to be clarified below.

3.2. Analysis of localized modes on the subdomains. The gist of theforthcoming discussion is that, roughly speaking, a mode of Ω localized to a sub-domain D ⊂ Ω must be fairly close to an eigenmode of this subdomain, and aneigenvalue of Ω for which localization takes place, must be close to some eigenvalueof D.

Assume that ϕ is one of the eigenmodes of Ω, which exhibits localization to D– a subdomain of Ω. This means, in particular, that the boundary values of ϕ on∂D are small. The “smallness” of ϕ on the boundary of D is to be interpreted inthe sense that an L-harmonic function, with the same data as ϕ on ∂D, is small.More precisely, let us define ε = εϕ > 0 as

ε = ‖v‖L2(D), where v ∈ H1(D) is such that

w := ϕ− v ∈ H1(D) (that is, ϕ and v on ∂D coincide),(3.7)

and Lv = 0 on D in the sense of distributions.

Proposition 3.2. Assume that Ω is an arbitrary bounded open set and that Lis an elliptic operator defined in (2.1) – (2.5). Let ϕ ∈ H1(Ω) be one of the eigen-functions of L in Ω and denote by λ the eigenvalue corresponding to ϕ. Supposefurther that D is a subset of Ω and denote by ε the norm of the boundary data ofϕ on ∂D in the sense of (3.7). Then either λ is an eigenvalue of D or

(3.8) ‖ϕ‖L2(D) ≤(

1 +λ

dD(λ)

)ε


Here, dD(λ) is the distance from λ to the spectrum of the operator L in the subre-gion D (defined as: dD(λ) = min

λk,D

{|λ− λk,D|}, the minimum being taken over all

eigenvalues (λk,D) of L in D).

Proof. If λ is the eigenvalue of L in Ω corresponding to ϕ, then we have

(3.9) (L− λ)w = λv on D,

as usually, in the sense of distributions.Whenever λ is, in addition, an eigenvalue of a subdomain D, there is nothing to

prove. Otherwise we proceed as follows. The eigenfunctions of L on D, {ψk}, forman orthogonal basis of L2(D). In particular, for every f ∈ L2(D) there are con-

stants ck(f) such that f =∑

k ck(f)ψk in L2(D) and ‖f‖L2(D) =(∑

k ck(f)2)1/2

.Therefore, for every λ not belonging to the spectrum of L on D and for everyf ∈ L2(D)

‖(L− λ)f‖L2(D) =

∥∥∥∥∥(L− λ)∑k

ck(f)ψk

∥∥∥∥∥L2(D)

=

∥∥∥∥∥∑k

(λk(D) − λ)ck(f)ψk

∥∥∥∥∥L2(D)

=

(∑k

(λk(D) − λ)2ck(f)2

)1/2

≥ mink

|λk(D) − λ|(∑

k

ck(f)2

)1/2

= mink

|λk(D) − λ| ‖f‖L2(D),(3.10)

which leads to

(3.11) ‖w‖L2(D) = ‖(L− λ)−1λv‖L2(D) ≤ maxλk(D)

{1

|λ− λk(D)|

}‖λv‖L2(D),

where the maximum is taken over all eigenvalues of L in D.Going further, (3.11) yields

(3.12) ‖w‖L2(D) ≤ maxλk(D)

{∣∣∣∣1 − λk(D)

λ

∣∣∣∣−1}

‖v‖L2(D) ≤ maxλk(D)

{∣∣∣∣1 − λk(D)

λ

∣∣∣∣−1}ε,

and therefore,

(3.13) ‖ϕ‖L2(D) ≤(

1 + maxλk(D)

{∣∣∣∣1 − λk(D)

λ

∣∣∣∣−1})

ε.

The inequality (3.13) then immediately yields (3.8). �

The presence of dD(λ) in the denominator of the right-hand side of Eq. (3.8)assures that whenever λ is far from any eigenvalue of L in D in relative value,the norm of ϕ in the entire subregion, ‖ϕ‖L2(D), has to be smaller than 2‖ε‖.Consequently, such a mode ϕ is expelled from D and must “live” in its complement,exhibiting weak localization. Conversely, ϕ can only be substantial in the subregionD when λ almost coincides with one of local eigenvalues of the operator L in D.Moreover, in that case ϕ itself almost coincides with the corresponding eigenmodeof the subregion D.


3.3. The definition of the valleys. Note that |ϕ|∣∣∣∂D

is majorized pointwise

by u∣∣∣∂D

according to the inequality (3.2). In fact, without loss of generality we

can assume that all ϕ are normalized so that ‖ϕ‖L∞(Ω) = 1, so that, in particu-lar, |ϕ(x)| ≤ λu(x) for every x ∈ ∂Ω. The normalization clearly does not affectthe statement of the Proposition 3.2, one just has to make sure to use the samenormalization in the definition (3.7). Then, according to the maximum principle,

the L2 norm of the L-harmonic extension of (|ϕ|)∣∣∣∂D

to D, that is, ‖v‖L2(D), is

majorized by λ times the L2 norm of the L-harmonic extension of u∣∣∣∂D

to D. The

latter is, in turn, controlled in the appropriate sense by u∣∣∣∂D

.

These observations, together with Proposition 3.2, suggest that the localization

will take place in the subregions delimited by curves where u∣∣∣∂D

is minimal, in the

exact sense described above. These lines will be called valleys of the landscape.One yet has to stress that the values of the control landscape u have to be

significantly small in the valleys (i.e. much smaller that 1/λ where λ is a typicaleigenvalue of the localized eigenfunction) for the inequality (3.2) to be effective. Tothis day, it still remains to be investigated under which circumstances this conditionis fulfilled, and why these circumstances occur in Anderson localization.

4. Numerical simulations

We have tested the above theory by numerically solving the Schrodinger equa-tion which is the eigenvalue problem associated to the Hamiltonian H. The domainΩ has been chosen as the unit square. This domain Ω is divided into 20×20 elemen-tary square cells, and the potential V (x) is defined as a piecewise constant functionon each of these cells. The values of the potential on the cells are independent ran-dom variables uniformly distributed between 0 and a maximum value (here 8,000,see Figure 1).

The simulations have been carried out on two different realizations of the ran-dom potential. First, the landscapes u have been computed by numerically solving(3.3) using second order rectangular Hermite elements. Figure 2 displays level setrepresentations of both landscapes. For the purposes of numerical simulations weuse as valleys the lines of steepest descent starting from the saddle points of thelandscape. The deepest valleys (hence leading to the stronger confinement) aredrawn in thick white lines while the higher valleys are plotted in thiner white lines.

One can observe that the valley lines form in each case a complicated and in-terconnected network. Both networks are very different but still exhibit similarfeatures, dividing the unit square into a partition of much smaller subregions ofvarious shapes and sizes. The eigenvalue problems associated to the two differentpotentials are then numerically solved using the same finite elements scheme. Fig-ure 3 displays 8 eigenfunctions (the corresponding eigenvalues are found above eachgraph) for both potentials. One can observe in both cases that the subregions ofthe domain Ω delimited by the corresponding networks indeed extremely accuratelypredict the localization regions of the eigenfunctions.

However, for higher eigenvalues, the control achieved by the valleys lines on theeigenfunctions through (3.2) becomes weaker. Due to the presence of the L∞(Ω)-norm in (3.2), the control partially disappears in a subregion when the eigenvalue


Figure 1. Level set representation of two realizations of a ran-dom potential V . The domain Ω is divided into 20×20 elementarysquare cells. The potential is piecewise constant, and on each cell,the value of the potential is is a uniform random variable between0 (dark blue) and Vmax = 8000 (red).

Figure 2. Level set representation of two landscapes, solutions ofLu = 1, for the two different realizations of the random potentialV (x) given in Figure 1. The thick white lines delineate the deepestvalleys of the landscapes while the thinner white lines show thehigher valleys.

λ is such that u(x) ≥ 1/λ along the valley lines surrounding the subregion. Toillustrate this, we superimpose over each representation of an eigenfunction thecorresponding valley network (deduced from Figure 2) from which we have removedthe segments where λu(x) ≥ 1. One can now observe how the progressive fading ofthe remaining valley network for higher eigenvalues coincides with the emergenceof less localized eigenfunctions. Yet, the spatial structures of these higher ordereigenfunctions is still dictated by the remaining network.


Figure 3. Left: Level set representation of 8 eigenfunctions (num-ber 1, 2, 3, 11, 31, 45, 56, and 59) of the Hamiltonian with the firstrealization of the random potential. The corresponding eigenvalueis displayed above the eigenfunction. Right: Level set representa-tion of 8 eigenfunctions (same numbers as before) of the Hamil-tonian with the second realization of the random potential. Thethicker dark lines delineate the deepest valleys of the landscapeswhile the thinnest represent the highest valleys. One can observehow accurately these lines predict the main existence regions of thelocalized eigenfunctions.


5. Conclusion

The theory and the numerical experiments presented in this paper show thatthe strong localization of eigenfunctions of the Hamiltonian H in a random potential(also called Anderson localization) is the consequence of two control inequalities.All eigenfunctions of a given elliptic operator, e.g., , the Hamiltonian, are controlledby the same function u, called here the landscape. This landscape is obtained bysolving the Dirichlet problem Hu = 1.

The valley lines of this effective landscape divide the entire domain into an invis-ible partition of disjoint subregions which correspond to the localization subregionsof the eigenfunctions. The control achieved by the function u locally disappearswhen λ is such that λu(x) ≥ 1 along a given closed curve of the network. As aconsequence, the relative number of localized modes decreases at higher eigenvalues.

The network of valleys of the landscape therefore appears as a geometricalobject that plays a major role in understanding the spatial distribution and thelocalization properties of the eigenfunctions of the Hamiltonian, hence the physicalproperties that depend on the quantum states. In the limit of a Brownian poten-tial, one may conjecture that both the landscape and its valley network becomestatistical objects with fractal properties.

References

[A] P. W.Anderson, Absence of diffusion in certain random lattices, Phys. Rev. (1958)109:1492-1505.

[E] Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studiesin Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010.MR2597943 (2011c:35002)

[ERRPS] C. Even, S. Russ, V. Repain, P. Pieranski, B. Sapoval, Localizations in fractaldrums: An experimental study, Phys. Rev. Lett. (1999) 83:726-729.

[FAFS] S. Felix, M. Asch, M. Filoche, B. Sapoval, Localization and increased damping inirregular acoustical cavities, J. Sound. Vib. (2007) 299:965-976.

[FM] M. Filoche, S. Mayboroda, Universal mechanism for Anderson and weak localiza-tion, Proc. Natl Acad. Sci. USA (2013) 109:14761-14766.

[GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of secondorder, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998edition. MR1814364 (2001k:35004)

[HS] Steven M. Heilman and Robert S. Strichartz, Localized eigenfunctions: here yousee them, there you don’t, Notices Amer. Math. Soc. 57 (2010), no. 5, 624–629.MR2664041 (2011c:35101)

[H] Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiersin Mathematics, Birkhauser Verlag, Basel, 2006. MR2251558 (2007h:35242)

[LSW] W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equa-tions with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963),43–77. MR0161019 (28 #4228)

[RBVIDCW] F. Riboli, P. Barthelemy, S. Vignolini, F. Intonti, A. De Rossi, S. Combrie, D.S. Wiersma, Anderson localization of near-visible light in two dimensions, (2011)Opt. Lett. 36:127-129.

[W] M. Wiener (1924) The Dirichlet problem, J. Math. Phys. (1924) 3:127-147.

Physique de la Matiere Condensee, Ecole Polytechnique, CNRS, 91128 Palaiseau,

France


School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church

Street SE, Minneapolis, Minnesota 55455



Zeta Functions for Infinite Graphsand Functional Equations

Daniele Guido and Tommaso Isola

Abstract. The definitions and main properties of the Ihara and Bartholdizeta functions for infinite graphs are reviewed. The general question of thevalidity of a functional equation is discussed, and various possible solutionsare proposed.

0. Introduction

In this paper, we review the main results concerning the Ihara zeta functionand the Bartholdi zeta function for infinite graphs. Moreover, we propose variouspossible solutions to the problem of the validity of a functional equation for thosezeta functions.

The zeta function associated to a finite graph by Ihara, Sunada, Hashimotoand others, combines features of Riemann’s zeta function, Artin L-functions, andSelberg’s zeta function, and may be viewed as an analogue of the Dedekind zetafunction of a number field [3,14–17,25,26]. It is defined by an Euler product overproper primitive cycles of the graph.

A main result for the Ihara zeta function ZX(z) associated with a graph X,is the so called determinant formula, which shows that the inverse of this functioncan be written, up to a polynomial, as det(I−Az+Qz2), where A is the adjacencymatrix and Q is the diagonal matrix corresponding to the degree minus 1. As aconsequence, for a finite graph, ZX(z) is indeed the inverse of a polynomial, hencecan be extended meromorphically to the whole plane.

A second main result is the fact that, for (q+1)-regular graphs, namely graphswith degree constantly equal to (q + 1), ZX , or better its so called completionξX , satisfies a functional equation, namely is invariant under the transformationz → 1

qz .

The first of the mentioned results has been proved for infinite (periodic orfractal) graphs in [10, 11], by introducing the analytic determinant for operator

2010 Mathematics Subject Classification. Primary 05C25, 05C38, 46Lxx, 11M41.Key words and phrases. Ihara zeta function, Bartholdi zeta function, functional equation,

determinant formula.The authors were partially supported by GNAMPA, MIUR, the European Network Quantum

Spaces - Noncommutative Geometry HPRN-CT-2002-00280, GDRE GREFI GENCO, and theERC Advanced Grant 227458 OACFT .


123

124 DANIELE GUIDO AND TOMMASO ISOLA

algebras. For first results and discussions about the functional equation we stillrefer to [10,11] and to [8].

The Bartholdi zeta function ZX(z, u) was introduced by Bartholdi in [2] as atwo-variable generalization of the Ihara zeta function. Such function coincides withthe Ihara zeta function for u = 0, and gives the Euler product on all primitive cyclesfor u = 1. Bartholdi also showed that some results for the Ihara zeta function extendto this new zeta function. We quote [5,18–20] for further results and generalizationsof the Bartholdi zeta function. The extension to the case of infinite periodic simplegraphs is contained in [12], where a functional equation for regular graphs and adeterminant formula are proved. Sato [24] generalised the determinant formula tothe non simple case, and also proved it for the case of fractal graphs [23].

The aim of this paper is two-fold: on the one hand we illustrate all the men-tioned results both for the periodic and the fractal case, using a unified approachin all the statements and also in some proofs, while for others we only treat thefractal case, referring the readers to [12] for the periodic case. On the other hand,we analyze the meaning and validity of the functional equation for infinite graphs.

Let us recall that the functional equation may be seen as a simple corollary ofthe determinant formula, which can be written in such a way that the argument ofthe determinant is itself invariant under the desired transformation of the complexplane. However, for infinite graphs, the determinant is no longer a polynomial,and its zeroes are no longer isolated. As a consequence, the singularities of theIhara zeta may constitute a barrier to the possibility of extending it analitically toan unbounded domain. In particular, for (q + 1)-regular graphs, the singularitiesare contained in the curve Ωq which disconnects the plane, hence may confine

11q

1

q

Figure 1. The set Ωq

ZX to the bounded component of Ωcq. In this case the functional equation loses its

meaning: one may still use the determinant formula to define ZX in the unboundedcomponent of Ωc

q, but in this case the functional equation is not a theorem but adefinition.

A first solution, due to Clair [8], consists in the observation that in some casesthe Ihara zeta naturally extends to a holomorphic function on a branched coveringof the complex plane. There, the functional equation holds if we extend the trans-formation z → 1/qz of C to a transformation of the covering which interchangesthe two branches.

Another solution, presented here, shows that a small amount of analyticityof the distribution function F given by the trace of the spectral function of the

ZETA FUNCTIONS FOR INFINITE GRAPHS AND FUNCTIONAL EQUATIONS 125

adjacency operator is sufficient to provide a suitable analytic extension of the Iharazeta function, which satisfies the functional equation.

As we shall see, the problem of the analytic extension does not arise for theBartholdi zeta function on infinite graphs. Indeed, in the case of (q + 1)-regulargraphs, namely graphs with degree constantly equal to (q + 1), the determinantformula takes the following simpler form:

ZX(z, u) = (1 − (1 − u)2z2)−(q−1)/2(detτ ((1 + (1 − u)(q + u)z2)I − zA)

)−1.

While the left-hand side is defined only in a suitable neighborhood of the origin inC2, the right-hand side is a holomorphic function on an open set whose complementis always contained in a three-dimensional real submanifold Ω of C2 containing allpossible singularities. We show that the complement of Ω is connected, hence theBartholdi zeta function holomorphically extends to Ωc. Moreover, it satisfies afunctional equation on such domain.

We shall use such result on the Bartholdi zeta to give a third solution to theanalytic extension problem for the Ihara zeta, which now works in full generality,and satisfies the functional equation. The procedure is the following: add a variableto the Ihara zeta so to get the Bartholdi zeta ZX(z, u), extend it holomorphicallyto Ωc and then set u = 0. Such function is the desired extension of the Ihara zetaZX(z) to Ωc

q, and satisfies the functional equation.

1. Zeta functions for infinite graphs

1.1. Preliminaries. In this section, we recall some terminology from graphtheory, and introduce the class of geometric operators on an infinite graph.

A simple graph X = (V X,EX) is a collection V X of objects, called vertices,and a collection EX of unordered pairs of distinct vertices, called edges. The edgee = {u, v} is said to join the vertices u, v, while u and v are said to be adjacent,which is denoted u ∼ v. A path (of length m) in X from v0 ∈ V X to vm ∈ V X,is (v0, . . . , vm), where vi ∈ V X, vi+1 ∼ vi, for i = 0, ...,m − 1 (note that m is thenumber of edges in the path). A path is closed if vm = v0.

We assume that X is countable and connected, i.e. there is a path betweenany pair of distinct vertices. Denote by deg(v) the degree of v ∈ V X, i.e. thenumber of vertices adjacent to v. We assume that X has bounded degree, i.e. d :=supv∈V X deg(v) < ∞. Denote by ρ the combinatorial distance on V X, that is, forv, w ∈ V X, ρ(v, w) is the length of the shortest path between v and w. If Ω ⊂ V X,r ∈ N, we write Br(Ω) := ∪v∈ΩBr(v), where Br(v) := {v′ ∈ V X : ρ(v′, v) ≤ r}.

Recall that the adjacency matrix of X, A =(A(v, w)

)v,w∈V X

, and the degree

matrix of X, D =(D(v, w)

)v,w∈V X

are defined by

(1.1) A(v, w) =

{1 v ∼ w

0 otherwise

and

(1.2) D(v, w) =

{deg(v) v = w

0 otherwise.

Then, considered as an operator on �2(V X), ‖A‖ ≤ d := supv∈V X deg(v) < ∞,(see [22], [21]).


1.1.1. Periodic graphs. In this section, we introduce the classes of periodicgraphs and operators, see [9,10] for more details.

Let Γ be a countable discrete subgroup of automorphisms of X, which acts freelyon X (i.e. any γ ∈ Γ, γ �= id doesn’t have fixed points), and with finite quotientB := X/Γ (observe that B needn’t be a simple graph). Denote by F ⊂ V X a setof representatives for V X/Γ, the vertices of the quotient graph B. Let us definea unitary representation of Γ on �2(V X) by (λ(γ)f)(x) := f(γ−1x), for γ ∈ Γ,f ∈ �2(V X), x ∈ V (X). Then the von Neumann algebra N(X,Γ) := {λ(γ) : γ ∈ Γ}′of bounded operators on �2(V X) commuting with the action of Γ inherits a tracegiven by TrΓ(T ) =

∑x∈F T (x, x), for T ∈ N(X,Γ).

It is easy to see that A,D ∈ N(X,Γ).1.1.2. Self-similar graphs. In this section, we introduce the class of self-similar

graphs and the geometric operators over them (see [11] for more details). This classcontains many examples of what are usually called fractal graphs, see e.g. [1,13].

If K is a subgraph of X, we call frontier of K, and denote by F(K), the familyof vertices in V K having distance 1 from the complement of V K in V X.

Definition 1.1 (Local Isomorphisms). A local isomorphism of the graph X isa triple

(1.3)(S(γ) , R(γ) , γ

),

where S(γ) , R(γ) are subgraphs of X and γ : S(γ) → R(γ) is a graph isomorphism.

Definition 1.2 (Amenable graphs). A countably infinite graph with boundeddegree X is amenable if it has an amenable exhaustion, namely, an increasing familyof finite subgraphs {Kn : n ∈ N} such that ∪n∈NKn = X and

|F(Kn)||Kn|

→ 0 as n → ∞ ,

where |Kn| stands for |V Kn| and | · | denotes the cardinality.

Definition 1.3 (Self-similar graphs). A countably infinite graph with boundeddegree X is self-similar if it has an amenable exhaustion {Kn} such that the fol-lowing conditions (i) and (ii) hold:(i) For every n ∈ N, there is a finite set of local isomorphisms G(n, n+1) such that,for all γ ∈ G(n, n + 1), one has S(γ) = Kn,

(1.4)⋃

γ∈G(n,n+1)

γ(Kn) = Kn+1,

and moreover, if γ, γ′ ∈ G(n, n + 1) with γ �= γ′,

(1.5) V (γKn) ∩ V (γ′Kn) = F(γKn) ∩ F(γ′Kn).

(ii) We then define G(n,m), for n < m, as the set of all admissible productsγm−1 · · · · · γn, γi ∈ G(i, i + 1), where “admissible” means that, for each term ofthe product, the range of γj is contained in the source of γj+1. We also let G(n, n)consist of the identity isomorphism on Kn, and G(n) := ∪m≥nG(n,m). We can nowdefine the G-invariant frontier of Kn:

FG(Kn) =⋃

γ∈G(n)

γ−1F(γKn),


and we require that

(1.6)|FG(Kn)|

|Kn|→ 0 as n → ∞ .

In the rest of the paper, we denote by G the family of all local isomorphismswhich can be written as (admissible) products γε1

1 γε22 ...γεk

k , where γi ∈ ∪n∈NG(n),εi ∈ {−1, 1}, for i = 1, ..., k and k ∈ N.

We refer to [11] for several examples of self-similar graphs.1.1.3. The C∗-algebra of geometric operators.

Definition 1.4 (Finite propagation operators). A bounded linear operatorA on �2(V X) has finite propagation r = r(A) ≥ 0 if, for all v ∈ V X, we havesupp(Av) ⊂ Br(v) and supp(A∗v) ⊂ Br(v), where we use v to mean the functionwhich is 1 on the vertex v and 0 otherwise, and A∗ is the Hilbert space adjoint ofA.

Definition 1.5 (Geometric Operators). A local isomorphism γ of the graphX defines a partial isometry U(γ) : �2(V X) → �2(V X), by setting

U(γ)(v) :=

{γ(v) v ∈ V (S(γ))

0 v �∈ V (S(γ)),

and extending by linearity. A bounded operator T acting on �2(V X) is calledgeometric if there exists r ∈ N such that T has finite propagation r and, for anylocal isomorphism γ, any v ∈ V X such that Br(v) ⊂ S(γ) and Br(γv) ⊂ R(γ), onehas

(1.7) TU(γ)v = U(γ)Tv, T ∗U(γ)v = U(γ)T ∗v .

Proposition 1.6. Geometric operators form a ∗-algebra containing the adja-cency operator A and the degree operator D.

Theorem 1.7. Let X be a self-similar graph, and let A(X) be the C∗-algebradefined as the norm closure of the ∗-algebra of geometric operators. Then, on A(X),there is a well-defined trace state TrG given by

(1.8) TrG(T ) = limn

Tr(P (Kn)T

)Tr(P (Kn)

) ,

where P (Kn) is the orthogonal projection of �2(V X) onto its closed subspace �2(V Kn).

1.2. Combinatorial results. The Bartholdi zeta function is defined by meansof equivalence classes of primitive cycles. Therefore, we need to introduce some ter-minology from graph theory, following [25] with some modifications.

Definition 1.8 (Types of closed paths).(i) A path C = (v0, . . . , vm) in X has backtracking if vi−1 = vi+1, for some i ∈{1, . . . ,m− 1}. We also say that C has a bump at vi. Then, the bump count bc(C)of C is the number of bumps in C. Moreover, if C is a closed path of length m,the cyclic bump count is cbc(C) := | {i ∈ Zm : vi−1 = vi+1} |, where the indices areconsidered in Zm, and Zm is the cyclic group on m elements.(ii) A closed path is primitive if it is not obtained by going k ≥ 2 times aroundsome other closed path.


(iii) A closed path C = (v0, . . . , vm = v0) has a tail if there is k ∈ {1, . . . , [m/2]−1}such that vj = vm−j , for j = 1, . . . , k. Denote by C the set of closed paths, by Ctail

the set of closed paths with tail, and by Cnotail the set of tail-less closed paths.Observe that C = Ctail ∪ Cnotail, Ctail ∩ Cnotail = ∅.

For any m ∈ N, u ∈ C, let us denote by Am(u)(x, y) :=∑

P ubc(P ), wherethe (finite) sum is over all paths P in X, of length m, with initial vertex x andterminal vertex y, for x, y ∈ V X. Then A1 = A. Let A0 := I and Q := D − I.Finally, let U ⊂ C be a bounded set containing {0, 1}, and denote by M(U) :=

supu∈U max {|u|, |1 − u|} ≥ 1, and α(U) :=d+

√d2+4M(U)(d−1+M(U))

2 .

Remark 1.9. In the sequel, in order to unify the notation, we will denote by(B(X), τ ) the pair (N(X,Γ),TrΓ), or (A(X),TrG), as the case may be. Moreover,

∑∗

x∈X

f(x) =

⎧⎪⎨⎪⎩∑x∈F

f(x), if X is a periodic graph

limn→∞

1|Kn|

∑x∈Kn

f(x), if X is a self-similar graph,

denotes a mean on the graph. Of course, in the self-similar case, the limit must beshown to exist.

Lemma 1.10.(i) A2(u) = A2 − (1 − u)(Q + I) ∈ B(X),(ii) for m ≥ 3, Am(u) = Am−1(u)A− (1 − u)Am−2(Q + uI) ∈ B(X),(iii) supu∈U ‖Am(u)‖ ≤ α(U)m, for m ≥ 0.

Proof. (i) If x = y, then A2(u)(x, x) = deg(x)u = (Q + I)(x, x)u becausethere are deg(x) closed paths of length 2 starting at x, whereas A2(x, x) = deg(x) =(Q+ I)(x, x), so that A2(u)(x, x) = A2(x, x) − (1 − u)(Q+ I)(x, x). If x �= y, thenA2(x, y) is the number of paths of length 2 from x to y, so A2(u)(x, y) = A2(x, y) =A2(x, y) − (1 − u)(Q + I)(x, y).

(ii) For x, y ∈ V X, consider all the paths P = (v0, . . . , vm) of length m, withv0 = x and vm = y. They can also be considered as obtained from a path P ′ oflength m−2 going from x ≡ v0 to vm−2, followed by a path of length 2 from vm−2 toy ≡ vm. There are four types of such paths: (a) those P for which y ≡ vm �= vm−2,vm−1 �= vm−3, so that bc(P ) = bc(P ′); (b) those P for which y ≡ vm �= vm−2,vm−1 = vm−3, so that bc(P ) = bc(P ′) + 1; (c) those P for which y ≡ vm = vm−2,but vm−1 �= vm−3, so that bc(P ) = bc(P ′)+1; (d) those P for which y ≡ vm = vm−2

and vm−1 = vm−3, so that bc(P ) = bc(P ′) + 2.Therefore, the terms corresponding to those four types in Am(u)(x, y) are

ubc(P ′), ubc(P ′)+1, ubc(P ′)+1, and ubc(P ′)+2, respectively.On the other hand, the sum

∑z∈V X Am−1(u)(x, z)A(z, y) assigns, to those four

types, respectively the values ubc(P ′), ubc(P ′)+1, ubc(P ′), and ubc(P ′)+1. Hence weneed to introduce corrections for paths of types (c) and (d).

Therefore Am(u)(x, y) =∑

z∈V X Am−1(u)(x, z)A(z, y)+Am−2(u)(x, y)(deg(y)−1)(u− 1) + Am−2(u)(x, y)(u2 − u), where the second summand takes into accountpaths of type (c), and the third is for paths of type (d). The statement follows.

(iii) We have ‖A1(u)‖ = ‖A‖ ≤ d ≤ α(U), ‖A2(u)‖ ≤ d2 + M(U)d ≤ α(U)2,and ‖Am(u)‖ ≤ d‖Am−1(u)‖ + M(U)(d − 1 + M(U))‖Am−2(u)‖, from which theclaim follows by induction. �


We now want to count the closed paths of length m which have a tail.

Lemma 1.11. For m ∈ N, let

tm(u) :=∑∗

x∈X

∑C=(x,...)∈Ctail

m

ubc(C)

Then(i) in the self-similar case, the above mean exists and is finite,(ii) t1(u) = 0, t2(u) = uτ (Q + I), t3(u) = 0,(iii) for m ≥ 4, tm(u) = τ

((Q− (1 − 2u)I)Am−2(u)

)+ (1 − u)2tm−2(u),

(iv) for any m ∈ N,

tm(u) = τ((Q− (1 − 2u)I)

[m−12 ]∑

j=1

(1 − u)2j−2Am−2j(u))

+ δeven(m)u(1 − u)m−2τ (Q + I),

where δeven(m) =

{1 m is even

0 m is odd.

(v) supu∈U |tm(u)| ≤ 4mα(U)m.

Proof. We consider only the case of self-similar graphs, for the periodic casesee [12]. Denote by (C, v) the closed path C with the origin in v ∈ V X.

(i) For n ∈ N, n > m, let

Ωn := V (Kn) \Bm(FG(Kn)), Ω′n := V (Kn) ∩Bm(FG(Kn)).

Then, for all p ∈ N,

V (Kn+p) =

( ⋃γ∈G(n,n+p)

γΩn

)∪( ⋃

γ∈G(n,n+p)

γΩ′n

).

Let tm(x, u) :=∑

C=(x,...)∈Ctailm

ubc(C) so that |tm(x, u)| ≤ dm−2M(U)m−1. Then∣∣∣∣∣∣ 1

|Kn+p|∑

x∈Kn+p

tm(x, u) − 1

|Kn|∑

x∈Kn

tm(x, u)

∣∣∣∣∣∣≤∣∣∣∣∣ |G(n, n + p)|

|Kn+p|∑x∈Ωn

tm(x, u) − 1

|Kn|∑

x∈Kn

tm(x, u)

∣∣∣∣∣+ 1

|Kn+p|∑

γ∈G(n,n+p)

∑x∈Ω′

n

|tm(γx, u)|

≤∣∣∣∣ |G(n, n + p)|

|Kn+p|− 1

|Kn|

∣∣∣∣ ∑x∈Kn

|tm(x, u)| + |G(n, n + p)||Kn+p|

∑x∈Bm(FG(Kn))

|tm(x, u)|

+1

|Kn+p|∑

γ∈G(n,n+p)

∑x∈Ω′

n

|tm(γx, u)|

≤∣∣∣∣1 − |Kn||G(n, n + p)|

|Kn+p|

∣∣∣∣ dm−2M(U)m−1

+ 2|Kn||G(n, n + p)|

|Kn+p||Bm(FG(Kn))|

|Kn|dm−2M(U)m−1

≤ 6dm−2(d + 1)mM(U)m−1εn → 0, as n → ∞,


where, in the last inequality, we used [11] equations (3.2), (3.8) [with r = 1], and

the fact that εn =|FG(Kn)|

|Kn|→ 0.

(ii) is easy to prove.(iii) Let us define Ω := {v ∈ V X : v �∈ Kn, ρ(v,Kn) = 1} ⊂ B1(FG(Kn)). We

have

1

|Kn|∑

x∈Kn

∑y∼x

∑C=(x,y,...)∈Ctail

m

ubc(C) =

=1

|Kn|∑y∈Kn

∑x∼y


m

ubc(C)

+1

|Kn|∑y∈Ω

∑x∈Kn,x∼y


m

ubc(C)

− 1

|Kn|∑y∈Kn

∑x∈Ω,x∼y


m

ubc(C).

Since

1

|Kn|∑y∈Ω

∑x∈Kn,x∼y


m

|ubc(C)| ≤ 1

|Kn||FG(Kn)|(d+1)dm−2M(U)m−1 → 0

and

1

|Kn|∑y∈Kn

∑x∈Ω,x∼y


m

|ubc(C)| =

=1

|Kn|∑

y∈FG(Kn)

∑x∈Ω,x∼y


m

|ubc(C)|

≤ 1

|Kn||FG(Kn)|dm−2M(U)m−1 → 0,

we obtain

tm = limn→∞

1

|Kn|∑

x∈Kn


m

ubc(C)

= limn→∞

1

|Kn|∑

x∈Kn

∑y∼x


m

ubc(C)

= limn→∞

1

|Kn|∑y∈Kn

∑x∼y


m

ubc(C).

A path C in the last set goes from x to y, then over a closed path D =(y, v1, . . . , vm−3, y) of length m − 2, and then back to x. There are two kinds ofclosed paths D at y: those with tails and those without.

Case 1 : D does not have a tail.

Then C can be of two types: (a) C1, where x �= v1 and x �= vm−3; (b) C2,where x = v1 or x = vm−3. Hence, bc(C1) = bc(D), and bc(C2) = bc(D) + 1, andthere are deg(y)− 2 possibilities for x to be adjacent to y in C1, and 2 possibilitiesin C2.


Case 2 : D has a tail.

Then C can be of two types: (c) C3, where v1 = vm−3 �= x; (d) C4, wherev1 = vm−3 = x. Hence, bc(C3) = bc(D), and bc(C4) = bc(D) + 2, and there aredeg(y) − 1 possibilities for x to be adjacent to y in C3, and 1 possibility in C4.Therefore,∑x∼y


m

ubc(C)

= (deg(y) − 2)∑

D=(y,...)∈Cnotailm−2

ubc(D) + 2u∑

D=(y,...)∈Cnotailm−2

ubc(D)

+ (deg(y) − 1)∑

D=(y,...)∈Ctailm−2

ubc(D) + u2∑


ubc(D)

= (deg(y) − 2 + 2u)∑

D=(y,...)∈Cm−2

ubc(D) + (1 − 2u + u2)∑


ubc(D),

so that

tm(u) = limn→∞

1

|Kn|∑y∈Kn

((Q(y, y) − 1 + 2u) ·Am−2(u)(y, y)

+ (1 − u)2∑


ubc(D))

= TrG((Q− (1 − 2u)I)Am−2(u)

)+ (1 − u)2tm−2(u).

(iv) Follows from (iii), and the fact that TrG((Q− (1 − 2u)I)A) = 0.(v) Let us first observe that M(U) < α(U), so that, from (iv) we obtain, with

α := α(U), M := M(U),

|tm(u)| ≤ ‖Q− (1 − 2u)I‖[m−1

2 ]∑j=1

|1 − u|2j−2‖Am−2j(u)‖ + |u||1 − u|m−2d

≤ (d− 2 + 2M)

[m−12 ]∑

j=1

M2j−2αm−2j + Mm−1d

≤ (d− 2 + 2M)[m− 1

2

]αm−2 + Mm−1d

≤([m− 1

2

]3αm−1 + αm

)≤ 4mαm. �

Lemma 1.12. Let us define

Nm(u) :=∑∗

x∈X

∑C=(x,...)∈Cm

ucbc(C).

Then, for all m ∈ N,(i) in the self-similar case, the above mean exists and is finite,(ii) Nm(u) = τ (Am(u)) − (1 − u)tm,(iii) |Nm(u)| ≤ Kmα(U)m+1, where K > 0 is independent of m.

Proof. We consider only the case of self-similar graphs, for the periodic casesee [12].


(i) the existence of limn→∞1

|Kn|∑

x∈Kn

∑(C,x)∈Cm

ucbc(C) can be proved as in

Lemma 1.11 (i).(ii) Therefore,

Nm(u) = limn→∞

1

|Kn|∑

x∈Kn

∑(C,x)∈Cm

ucbc(C)

= limn→∞

1

|Kn|∑

x∈Kn

( ∑(C,x)∈Cnotail

m

ubc(C) +∑

(C,x)∈Ctailm

ubc(C)+1

)

= limn→∞

1

|Kn|∑

x∈Kn

( ∑(C,x)∈Cm

ubc(C) + (u− 1)∑

(C,x)∈Ctailm

ubc(C)

)

= limn→∞

1

|Kn|∑

x∈Kn

Am(u)(x, x) + (u− 1) limn→∞

1

|Kn|∑

x∈Kn


m

ubc(C)

= TrG(Am(u)) + (u− 1)tm.

(iii) This follows from (ii). �

Remark 1.13. Observe that in the self-similar case we can also write

(1.9) Nm(u) = limn→∞

1

|Kn|∑

C∈CmC⊂Kn

ucbc(C).

Indeed,

0 ≤ 1

|Kn|

∣∣∣∣∣ ∑x∈Kn

∑(C,x)∈Cm

ucbc(C) −∑

C∈CmC⊂Kn

ucbc(C)

∣∣∣∣∣≤ 1

|Kn|∑

(C,x)∈Cm,C �⊂Kn

x∈Kn

|ucbc(C)|

≤ 1

|Kn||{(C, x) ∈ Cm : x ∈ Bm(FG(Kn))}|M(U)m

=M(U)m

|Kn|∑

x∈Bm(FG(Kn))

Am(1)(x, x) =M(U)m

|Kn|Tr(P (Bm(FG(Kn)))Am(1)

)≤ M(U)m‖Am(1)‖ |Bm(FG(Kn))|

|Kn|

≤ M(U)mα(U)m (d + 1)m|FG(Kn)|

|Kn|→ 0, as n → ∞.

1.3. The Zeta function. In this section, we define the Bartholdi zeta func-tion for a periodic graph and for a self-similar graph, and prove that it is a holo-morphic function in a suitable open set. In the rest of this work, U ⊂ C will denotea bounded open set containing {0, 1}.

Definition 1.14 (Cycles). We say that two closed paths C = (v0, . . . , vm = v0)and D = (w0, . . . , wm = w0) are equivalent, and write C ∼o D, if there is an integerk such that wj = vj+k, for all j, where the addition is taken modulo m, that is,the origin of D is shifted k steps with respect to the origin of C. The equivalence


class of C is denoted [C]o. An equivalence class is also called a cycle. Therefore, aclosed path is just a cycle with a specified origin.

Denote by K the set of cycles, and by P ⊂ K the subset of primitive cycles.

Definition 1.15 (Equivalence relation between cycles). Given C, D ∈ K, wesay that C and D are G-equivalent, and write C ∼G D, if there is a local isomorphismγ ∈ G such that D = γ(C). We denote by [K]G the set of G-equivalence classes ofcycles, and analogously for the subset P. The notion of Γ-equivalence is analogous(see [12] for details), and we denote by [·]G also a Γ-equivalence class.

We recall from [11] and [12] several quantities associated to a cycle.

Definition 1.16. Let C ∈ K, and call(i) effective length of C, denoted �(C) ∈ N, the length of the primitive cycle Dunderlying C, i.e. such that C = Dk, for some k ∈ N, whereas the length of C isdenoted by |C|,(ii) if C is contained in a periodic graph, stabilizer of C in Γ the subgroup ΓC ={γ ∈ Γ : γ(C) = C}, whose order divides �(C),(ii) if C is contained in a self-similar graph, size of C, denoted s(C) ∈ N, the leastm ∈ N such that C ⊂ γ(Km), for some local isomorphism γ ∈ G(m),(iii) average multiplicity of C, the number in [0,∞) given by

μ(C) :=

{1

|ΓC | , if C is contained in a periodic graph,

limn→∞

|G(s(C),n)||Kn| , if C is contained in a self-similar graph.

That the limit actually exists is the content of the following

Proposition 1.17. Let (X,G) be a self-similar graph.(i) Let C ∈ K, then the following limit exists and is finite:

limn

|G(s(C), n)||Kn|

,

(ii) s(C), �(C), and μ(C) only depend on [C]G ∈ [K]G; moreover, if C = Dk forsome D ∈ P, k ∈ N, then s(C) = s(D), �(C) = �(D), μ(C) = μ(D).

Proof. See [11] Proposition 6.4. �

Proposition 1.18. For m ∈ N, Nm(u) =∑

[C]G∈[Km]G

μ(C)�(C)ucbc(C),

where, as above, the subscript m corresponds to cycles of length m.


Proof. We prove only the self-similar case, for the periodic case see [12]. Wehave successively:

Nm(u) = limn→∞

1

|Kn|∑

C∈CmC⊂Kn

ucbc(C)

= limn→∞

∑[C]G∈[Km]G

1

|Kn|�(C)

∑D∈Km,D∼GC

D⊂Kn

ucbc(D)

= limn→∞

∑[C]G∈[Km]G

1

|Kn|�(C) |G(s(C), n)|ucbc(C)

=∑

[C]G∈[Km]G

μ(C)�(C)ucbc(C),

where, in the last equality, we used dominated convergence. �

Definition 1.19 (Zeta function).

ZX(z, u) :=∏

[C]G∈[P]G

(1 − z|C|ucbc(C))−μ(C), z, u ∈ C.

Proposition 1.20.(i) ZX(z, u) :=

∏[C]∈[P]G

(1 − z|C|ucbc(C))−μ(C) defines a holomorphic function in{(z, u) ∈ C2 : |z| < 1

α(U) , u ∈ U},

(ii) z ∂zZX (z,u)ZX(z,u) =

∑∞m=1 Nm(u)zm, where Nm(u) is defined in Lemma 1.12,

(iii) ZX(z, u) = exp(∑∞

m=1Nm(u)

m zm).

Proof. Let us observe that, for any u ∈ U, and z ∈ C such that |z| < 1α(U) ,

∞∑m=1

Nm(u)zm =∞∑

m=1

∑[C]G∈[Km]G

μ(C)�(C)ucbc(C) zm

=∑

[C]G∈[K]G

μ(C)�(C)ucbc(C) z|C|

=∑

[C]G∈[P]G

∞∑m=1

μ(C)|C|ucbc(Cm)z|Cm|

=∑

[C]G∈[P]G

μ(C)

∞∑m=1

|C|z|C|mucbc(C)m

=∑

[C]G∈[P]G

μ(C) z∂

∂z

∞∑m=1

z|C|mucbc(C)m

m

= −∑

[C]G∈[P]G

μ(C) z∂

∂zlog(1 − z|C|ucbc(C))

= z∂

∂zlogZX(z, u),


where, in the last equality we used uniform convergence on compact subsets of{(z, u) ∈ C2 : u ∈ U, |z| < 1

α(U)

}. The proof of the remaining statements is now

clear. �

1.4. The determinant formula. In this section, we prove the main resultin the theory of Bartholdi zeta functions, which says that the reciprocal of Z is, upto a factor, the determinant of a deformed Laplacian on the graph. We first needsome technical results. Let us recall that d := supv∈V X deg(v), U ⊂ C is a boundedopen set containing {0, 1}, M(U) := supu∈U max {|u|, |1 − u|}, and α ≡ α(U) :=d+

√d2+4M(U)(d−1+M(U))

2 .

Lemma 1.21. For any u ∈ U, |z| < 1α , one has

(i)(∑

m≥0 Am(u)zm) (

I −Az + (1 − u)(Q + uI)z2)

= (1 − (1 − u)2z2)I,

(ii)(∑

m≥0

(∑[m/2]k=0 (1 − u)2kAm−2k(u)

)zm) (

I −Az + (1 − u)(Q + uI)z2)

= I.

Proof. (i) From Lemma 1.10, we obtain that(∑m≥0

Am(u)zm)(

I −Az + (1 − u)(Q + uI)z2)

=∑m≥0

Am(u)zm −∑m≥0

Am(u)Azm+1 +∑m≥0

(1 − u)Am(u)(Q + uI)zm+2

= A0(u) + A1(u)z + A2(u)z2 +∑m≥3

Am(u)zm

−A0(u)Az −A1(u)Az2 −∑m≥3

Am−1(u)Azm

+ (1 − u)A0(u)(Q + uI)z2 +∑m≥3

(1 − u)Am−2(Q + uI)zm

= I + Az +(A2 − (1 − u)(Q + I)

)z2 −Az −A2z2 + (1 − u)(Q + uI)z2

= (1 − (1 − u)2z2)I.

(ii)

I = (1 − (1 − u)2z2)−1

(∑m≥0

Am(u)zm)(

I −Az + (1 − u)(Q + uI)z2)

=

(∑m≥0

Am(u)zm)( ∞∑

j=0

(1 − u)2jz2j)(

I −Az + (1 − u)(Q + uI)z2)

=

(∑k≥0

∞∑j=0

Ak(u)(1 − u)2jzk+2j

)(I −Az + (1 − u)(Q + uI)z2

)=

(∑m≥0

([m/2]∑j=0

Am−2j(u)(1 − u)2j)zm)(

I −Az + (1 − u)(Q + uI)z2).

�


Lemma 1.22. Define⎧⎪⎨⎪⎩B0(u) := I,

B1(u) := A,

Bm(u) := Am(u) − (Q− (1 − 2u)I)∑[m/2]

k=1 (1 − u)2k−1Am−2k(u), m ≥ 2.

Then(i) Bm(u) ∈ B(X),

(ii) Bm(u) = Am(u)+(1−u)−1(Q− (1−2u)I

)Am(u)−

(Q− (1−2u)I

)∑[m/2]k=0 (1−

u)2k−1Am−2k(u),(iii)

τ (Bm(u)) =

{Nm(u) − (1 − u)mτ (Q− I) m even

Nm(u) m odd,

(iv) ∑m≥1

Bm(u)zm =(Au− 2(1 − u)(Q + uI)z2

)×(I − Az + (1 − u)(Q + uI)z2

)−1, u ∈ U, |z| < 1

α.

Proof. (i) and (ii) follow from computations involving bounded operators.(iii) It follows from Lemma 1.11 (ii) that, if m is odd,

τ (Bm(u)) = τ (Am(u)) − (1 − u)tm(u) = Nm(u),

whereas, if m is even,

τ (Bm(u)) = τ (Am(u)) − (1 − u)m−1τ (Q− (1 − 2u)I)

− (1 − u)tm(u) + (1 − u)m−1uτ (Q + I)

= Nm(u) − (1 − u)mτ (Q− I).

(iv) Using (ii) we obtain(∑m≥0

Bm(u)zm)

(I −Az + (1 − u)(Q + uI)z2)

=

((I + (1 − u)−1(Q− (1 − 2u)I)

) ∑m≥0

Am(u)zm

− (1 − u)−1(Q− (1 − 2u)I)∑m≥0

[m/2]∑j=0

Am−2j(u)(1 − u)2jzm)

× (I −Az + (1 − u)(Q + uI)z2)

(by Lemma 1.21)

=(I + (1 − u)−1(Q− (1 − 2u)I)

)(1 − (1 − u)2z2)I − (1 − u)−1(Q− (1 − 2u)I)

= (1 − (1 − u)2z2)I − (1 − u)(Q− (1 − 2u)I)z2.


Since B0(u) = I, we get(∑m≥1

Bm(u)zm)

(I −Az + (1 − u)(Q + uI)z2)

= (1 − (1 − u)2z2)I − (1 − u)(Q− (1 − 2u)I)z2

−B0(u)(I −Az + (1 − u)(Q + uI)z2)

= Az − 2(1 − u)(Q + uI)z2. �

Lemma 1.23. [11] Let f : z ∈ Bε ≡ {z ∈ C : |z| < ε} �→ f(z) ∈ B(X), be a C1-function such that f(0) = 0 and ‖f(z)‖ < 1, for all z ∈ Bε. Then

τ

(− d

dzlog(I − f(z))

)= τ(f ′(z)(I − f(z))−1

).

Corollary 1.24.

τ

⎛⎝∑m≥1

Bm(u)zm

⎞⎠ = τ

(−z

∂

∂zlog(I −Az + (1 − u)(Q + uI)z2)

), u ∈ U, |z| < 1

α.

Proof. It follows from Lemma 1.22 (iv) that

τ

(∑m≥1

Bm(u)zm)

= τ((Az − 2(1 − u)(Q + uI)z2)(I −Az + (1 − u)(Q + uI)z2)−1

)and using the previous lemma with f(z) := Az − (1 − u)(Q + uI)z2

= τ(−z

∂

∂zlog(I −Az + (1 − u)(Q + uI)z2)

). �

We now recall the definition and main properties of the analytic determinanton tracial C∗-algebras studied in [11]

Definition 1.25. Let (A, τ ) be a C∗-algebra endowed with a trace state, andconsider the subset A0 := {A ∈ A : 0 �∈ convσ(A)}, where σ(A) denotes thespectrum of A and convσ(A) its convex hull. For any A ∈ A0 we set

detτ (A) = exp ◦ τ ◦(

1

2πi

∫Γ

log λ(λ−A)−1dλ

),

where Γ is the boundary of a connected, simply connected region Ω containingconvσ(A), and log is a branch of the logarithm whose domain contains Ω.

Since two Γ’s as above are homotopic in C \ conv σ(A), we have

Corollary 1.26. The determinant function defined above is well-defined andanalytic on A0.

We collect several properties of our determinant in the following result.

Proposition 1.27. Let (A, τ ) be a C∗-algebra endowed with a trace state, andlet A ∈ A0. Then

(i) detτ (zA) = zdetτ (A), for any z ∈ C \ {0},(ii) if A is normal, and A = UH is its polar decomposition,

detτ (A) = detτ (U)detτ (H),


(iii) if A is positive, then we have detτ (A) = Det(A), where the latter is the Fuglede–Kadison determinant.

We now recall from [11] the notion of average Euler–Poincare characteristicof a self-similar graph, and from [4] that of L2-Euler characteristic of a periodicgraph.

Lemma 1.28. Let X be a self-similar graph. The following limit exists and isfinite:

χav(X) := limn→∞

χ(Kn)

|Kn|= −1

2TrG(Q− I),

where χ(Kn) = |V Kn|−|EKn| is the Euler–Poincare characteristic of the subgraphKn. The number χav(X) is called the average Euler–Poincare characteristic of theself-similar graph X.

Definition 1.29. Let (X,Γ) be a periodic graph. Then χ(2)(X) :=∑v∈F0

1

|Γv|−

1

2

∑e∈F1

1

|Γe|is the L2-Euler–Poincare characteristic of (X,Γ).

Remark 1.30.(1) It was proved in [11] that χav(X) = − 1

2 TrG(Q− I).

(2) It is easy to prove that χ(2)(X) = − 12 TrΓ(Q− I) = χ(X/Γ).

In the next Theorem we denote by χ(X) the average or L2- Euler–Poincarecharacteristic of X, as the case may be.

Theorem 1.31 (Determinant formula). Let X be a periodic or self-similargraph. Then

1

ZX(z, u)= (1− (1−u)2z2)−χ(X)detτ

(I−Az+(1−u)(Q+uI)z2

), u ∈ U, |z| < 1

α.

Proof.

τ

(∑m≥1

Bm(u)zm)

=∑m≥1

τ (Bm(u))zm

(by Lemma 1.22 (iii))

=∑m≥1

Nm(u)zm −∑k≥1

(1 − u)2kτ (Q− I)z2k

=∑m≥1

Nm(u)zm − τ (Q− I)(1 − u)2z2

1 − (1 − u)2z2.

Therefore, from Proposition 1.20 and Corollary 1.24, we obtain

z∂

∂zlogZX(z, u) =

∑m≥1

Nm(u)zm

= τ

(−z

∂

∂zlog(I −Az + (1 − u)(Q + uI)z2)

)− z

2

∂

∂zlog(1 − (1 − u)2z2)τ (Q− I)


so that, dividing by z and integrating from z = 0 to z, we get

logZX(z, u) = −τ(log(I−Az+(1−u)(Q+uI)z2)

)− 1

2τ (Q−I) log(1− (1−u)2z2),

which implies that

1

ZX(z, u)= (1 − (1 − u)2z2)

12 τ(Q−I) · exp ◦τ ◦ log(I −Az + (1 − u)(Q + uI)z2),

and the thesis follows from Lemma 1.28 and Definition 1.25. �

2. Functional equations for infinite graphs

2.1. Functional equations for the Bartholdi zeta function of an infi-nite graph. In this subsection, we shall prove that a suitable completion of theBartholdi zeta functions for essentially (q+1)-regular infinite graphs satisfy a func-tional equation, where a graph is called essentially (q + 1)-regular if deg(v) = q + 1for all but a finite number of vertices, and d ≡ supv∈V X deg(v) = q + 1.

The completion considered here is the function

ξX(z, u) = (1 − (1 − u)2z2)(q−1)/2(1 − (q + 1)z + (1 − u)(q + u)z2)ZX(z, u).

We shall show that

Theorem 2.1 (Functional equation). Let X be an essentially (q + 1)-regular

infinite graph, periodic or self-similar, and set g(z, u) = 1+(1−u)(q+u)z2

z . Then(1) the function ξX analytically extends to the complement V of the set

Ω = {(z, u) ∈ C2 : g(z, u) ∈ [−d, d]},(2) the set V0 = {(z, u) ∈ Ωc : z �= 0, u �= 1, u �= −q} is invariant w.r.t. thetrasformation ψ : (z, u) �→ ( 1

(1−u)(q+u)z , u),

(3) the analytic extension of ξX satisfies the functional equation

ξX(z, u) = ξX ◦ ψ(z, u), (z, u) ∈ V0.

Lemma 2.2. Let d be a positive number, and consider the set

Ωw = {z ∈ C :1 + wz2

z∈ [−d, d]}, w ∈ C.

Then Ωw disconnects the complex plane iff w is real and 0 < w ≤ d2

4 .

Proof. If w = 0, Ωw consists of the two disjoint half lines (−∞,− 1d ], [ 1d ,∞).

If w �= 0, the set Ωw is closed and bounded. Moreover, setting z = x + iy and

w = a + ib, the equation Im 1+wz2

z = 0 becomes

(2.1) (x2 + y2)(ay + bx) − y = 0.

Let us first consider the case b = 0. If a < 0, (2.1) implies y = 0, therefore Ωw

is bounded and contained in a line, thus does not disconnect the plane.If a > 0, Ωw is determined by

(a(x2 + y2) − 1)y = 0,(2.2)

|x + ax(x2 + y2)| ≤ d(x2 + y2).(2.3)

If a > d2

4 , condition (2.3) is incompatible with y = 0, while condition (2.3) and

a(x2 + y2) − 1 = 0 give 2|x| ≤ da , namely only an upper and a lower portion of the

circle x2 + y2 = 1a remain, thus the plane is not disconnected.


A simple calculation shows that, when 0 < w ≤ d2

4 , Ωw as a shape similar toΩq in Figure 1.

Let now b �= 0. We want to show that the cubic in (2.1) is a simple curve,namely is non-degenerate and has no singular points, see Figure 2.

Figure 2. The cubic containing Ωw for Im w �= 0

Up to a rotation, the cubic can be rewritten as

(a2 + b2)(x2 + y2)y − ay + bx = 0.

The condition for critical points gives the system⎧⎪⎨⎪⎩(a2 + b2)(x2 + y2)y = ay − bx

(a2 + b2)(x2 + 3y2) = a

2(a2 + b2)xy = −b.

The first two equations give 2(a2+b2)y4 = bxy, which is incompatible with the thirdequation, namely the cubic curve is simple. Since only a finite portion of the cubichas to be considered, because Ωw is bounded, again the plane is not disconnectedby Ωw. �

Lemma 2.3. The set Ω does not disconnect C2. The function

(z, u) �→ detτ ((1 + (1 − u)(q + u)z2)I − zA)

is a non-vanishing analytic function on V = Ωc.

Proof. We first observe that the plane {0} × C is contained in V. Set now

T = {v ∈ C : (1 − v)(q + v) ∈ R, 0 < (1 − v)(q + v) ≤ d2

4 }. If u �∈ T , anyz ∈ C : (z, u) ∈ V is connected to the point (0, u) by the preceding Lemma. Since V

is open and T is 1-dimensional, for any z ∈ C : (z, u) ∈ V there exists a ball centeredin (z, u) still contained in V, and such a ball contains a (z, u′) with u′ �∈ T . Thisproves that V is connected. We now prove the second statement. For (z, u) ∈ V,the operator (1 + (1 − u)(q + u)z2)I − zA is invertible. Indeed this is clearly truefor z = 0 and, for z �= 0, it may be written as −z (A− g(z, u)I), which is invertiblesince the spectrum of A is contained in the interval [−d, d] of the real line (see[22], [21]), and the condition (z, u) ∈ V means g(z, u) �∈ [−d, d]. Such invertibilityimplies that the determinant is defined and invertible. Analyticity follows fromCorollary 1.26. �


Proof of Theorem 2.1. The determinant formula gives

ZX(z, u) =(1 − (1 − u)2z2)−(q−1)/2

detτ ((1 + (1 − u)(q + u)z2)I − zA),

ξX(z, u) =(1 − (q + 1)z + (1 − u)(q + u)z2)

detτ ((1 + (1 − u)(q + u)z2)I − zA).

Now the first statement is a consequence of Lemma 2.3. As for the other twostatements, we have

ξX(z, u) =(g(z, u) − (q + 1)

)(detτ (g(z, u)I −A)

)−1,

Ω = {(z, u) ∈ C2 : g(z, u) ∈ [−d, d]},

hence the the results follow by the equality g ◦ ψ = g. �

2.2. Functional equations for the Ihara zeta function on infinite graphs.As discussed in the introduction, the possibility of proving functional equations forthe Ihara zeta function on regular infinite graphs, relies on the possibility of ex-tending ZX to a domain U which is invariant under the transformation z → 1

qz ,

and then to check the invariance properties under the mentioned transformation.Functional equation may fail for two reasons. The first is that the spectrum of

the adjacency operator A may consist of the whole interval [−d, d], and that ZX

may be singular in all points of the curve Ωq in Figure 1, so that no analyticalextension is possible outside Ωq. The second is more subtle, and was noticed byB. Clair in [8]. In one of his examples, which we describe below in Example 2.8,the completion ξX analytically extends to the whole complex plane, but the points{1,−1} are ramification points for ZX , which lives naturally on a double cover of C.Then the functional equation makes sense only on the double cover, interchangingthe two copies, so that it is false on C.

2.2.1. Criteria for analytic extension. We first make a simple observation.

Proposition 2.4. Let X be an infinite (q + 1)-regular graph as above. Denoteby E(λ) the spectral family of the adjacency operator A, and set F (λ) = τ (E(λ)).If F (λ) is constant in a neighborhood of a point x ∈ (−2

√q, 2

√q), then ZX extends

analytically to a domain U which is invariant under the transformation z → 1qz ,

and a suitable completion of such extension satisfies the functional equation.

Proof. Let z± be the two solutions of the equation 1 + qz2 − xz = 0. SincedF (λ) vanishes in a neighborhood of x, the function

detτ ((1 + qz2)I − zA) = exp

∫σ(A)

log(1 + qz2 − λz) dF (λ)

is analytic in a neighborhood of the points z±, hence the singularity region {z ∈C : 1−λz+ qz2 = 0, λ ∈ σ(A)} does not disconnect the plane. Its complement U istherefore connected and invariant under the transformation z → 1

qz . Consider now

the completion

ξX(z) = (1 − z2)(q−1)/2(1 − (q + 1)z + qz2)ZX(z) .


The determinant formula in Theorem 1.31 gives, for |z| < 1q , z �= 0,

ξX(z) =(1 − (q + 1)z + qz2

)(detτ ((1 + qz2)I − zA)

)−1

=(1

z+ qz − (q + 1)

)(detτ

((1

z+ qz)I −A

))−1

=(1

z+ qz − (q + 1)

)exp

(−∫σ(A)

log((1

z+ qz) − λ

)dF (λ)

).

As explained above, ξX analytically extends to the region U, where the functionalequation follows by the invariance of the expression ( 1z + qz) under the transforma-

tion z → 1qz . �

The following criterion is valid also when there are no holes in the spectrum.A regularity assumption on the spectral measure of A will guarantee that the be-haviour of ZX on the critical curve Ωq is not too singular, allowing analytic contin-uation outside Ωq and the validity of a functional equation for suitable completions.With the notation above,

detτ ((1 + qz2)I − zA) = exp

∫σ(A)

log(1 + qz2 − λz) dF (λ)

= (1 − (q + 1)z + qz2) · exp

∫ d

−d

z

1 + qz2 − λzF (λ) dλ ,

where we used integration by parts and the fact that σ(A) ⊆ [−d, d], hence

(2.4) ξX(z) = exp

(−∫ d

−d

z

1 + qz2 − λzF (λ) dλ

).

Theorem 2.5 (Functional equation). Assume there exist ε > 0, σ, τ ∈ {−1, 1},and a function ϕ such that ϕ is analytic in {σ%z > 0, |z − 2τ

√q| < ε}, and F is

the boundary value of ϕ on [2τ√q − ε, 2τ

√q + ε], Then, there exists a connected

domain U containing {|z| < 1/q} such that

(i) ξX extends analytically to U,(ii) U \ {0} is invariant under the transformation z → 1

qz ,

(iii) the function ξX , extended as above, verifies

ξX(1

qz) = ξX(z), z ∈ U \ {0}.

Proof. We give the proof for σ = τ = 1, the other cases being analogous. LetΓ be the oriented curve in C made of the segment [−d, 2

√q−ε], the upper semicircle

{%λ ≥ 0, |λ−2√q| = ε}, and the segment [2

√q+ε, d]. It is not restrictive to assume

that ϕ has a continuous extension to the upper semicircle. We have∫ d

−d

z

1 + qz2 − λzF (λ) dλ−

∫Γ

z

1 + qz2 − λzϕ(λ) dλ =

∫S

z

1 + qz2 − λzϕ(λ) dλ,

if S is the contour of the semi-disc D = {%λ > 0, |λ− 2√q| < ε}. If |z| < 1/q and

ε is small enough, 1 + qz2 − λz does not vanish for λ ∈ D, hence the last integrandis analytic, and the contour integral vanishes. As a consequence, for |z| < 1/q,

(2.5) ξX(z) = exp

(−∫Γ

z

1 + qz2 − λzϕ(λ) dλ

).


We now observe that the singularities of the integral are contained in the set Ωq :={z ∈ C : 1

z + qz ∈ Γ} shown in Fig. 3, which does not disconnect the plane, andproperty (i) follows.

Figure 3. The set Ωq for q = 4, ε = 0.2

The definition of Ωq guarantees its invariance under the transformation z → 1qz ,

i.e. (ii) is proved. Property (iii) follows directly from equation (2.5). �

Remark 2.6. Let us notice that, when q = 1, the criterion above is useless.Indeed, since F (λ) is constant before −2 and after 2, analyticity implies it shouldbe constant in a neighborhood of either −2 or 2, namely already Proposition 2.4applies. In particular, the results above do not apply to the Example 2.8.

2.2.2. An extension via Bartholdi zeta function. The extension we discuss hereis again based on analytic extension, but in the sense of two-variable functions.Moreover, it does not require either holes in the spectrum or regularity assumptionson the function F (λ).

As shown above, the Ihara zeta function coincides with the Bartholdi zetafunction for u = 0, |z| < 1/q, and the latter has a unique analytic extension to theset Ωc. We may therefore extend the Ihara zeta function via

(2.6) ZX(z) := ZX(z, 0), (z, 0) ∈ Ωc,

where the Bartholdi zeta function has been extended to Ωc. Let us remark that{z ∈ C : (z, 0) ∈ Ωc} = Ωc

q, cf. Fig. 1.The following result follows directly by Theorem 2.1.

Corollary 2.7. Assume X is an infinite graph (either periodic or self-similar),which is essentially (q + 1)-regular. Then, the domain Ωc

q contains {|z| < 1/q} and

is invariant under the transformation z → 1qz . Moreover, setting

ξX(z) = (1 − z2)(q−1)/2(1 − (q + 1)z + qz2)ZX(z).

where ZX is extended to Ωcq as above, we have ξ(z) = ξ( 1

qz ), for any z ∈ Ωcq.


Example 2.8. Let us consider the graph X = Z, and the group Γ = Z, whichacts on X by translations. Using results from [8], we compute the Bartholdi zetafunction of (X,Γ). We obtain

1

ZX,Γ(z, u)= detΓ(I −Az + (1 − u2)z2) = exp

∫T

Log(1 − 2 cosϑ z + (1 − u2)z2) dϑ

= 2z exp

∫T

Log(z−1 + (1 − u2)z

2− cosϑ

)dϑ

= 2z exp(

arcosh(z−1 + (1 − u2)z

2

)− log 2

)= z(z−1 + (1 − u2)z

2

)(1 +

√1 − 4

(z−1 + (1 − u2)z)2

)

=1 + (1 − u2)z2

2

(1 +

√1 − 4z2

(1 + (1 − u2)z2)2

),

which extends to an analytic function on the complement of the set Ω = {(z, u) ∈C2 : 1+(1−u2)z2

z ∈ [−d, d]}. Therefore the Ihara zeta function, extended via theBartholdi zeta, is defined on the complement of {z ∈ C : |z| = 1}, where it is givenby

ZX,Γ(z) = ZX,Γ(z, 0) =2

1 + z2

(1 +

√(1 − z2

1 + z2

)2)−1

=

⎧⎪⎨⎪⎩2

1 + z2

(1 +

1 − z2

1 + z2

)−1

= 1, |z| < 1,

2

1 + z2

(1 − 1 − z2

1 + z2

)−1

= z−2, |z| > 1.

and the completion ξX,Γ is given by

ξX,Γ(z) = (z − 1)2ZX,Γ(z) =

{(z − 1)2, |z| < 1,

( 1z − 1)2, |z| > 1.

Defined in this way, ξX,Γ satisfies the functional equation, but its behaviour outsidethe disc is not given by the analytic extension on C.

References

[1] Martin T. Barlow, Heat kernels and sets with fractal structure, (Paris, 2002), Con-temp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 11–40, DOI10.1090/conm/338/06069. MR2039950 (2005e:60166)

[2] Laurent Bartholdi, Counting paths in graphs, Enseign. Math. (2) 45 (1999), no. 1-2, 83–131.MR1703364 (2000f:05047)

[3] Hyman Bass, The Ihara-Selberg zeta function of a tree lattice, Internat. J. Math. 3 (1992),no. 6, 717–797, DOI 10.1142/S0129167X92000357. MR1194071 (94a:11072)

[4] Jeff Cheeger and Mikhael Gromov, L2-cohomology and group cohomology, Topology 25(1986), no. 2, 189–215, DOI 10.1016/0040-9383(86)90039-X. MR837621 (87i:58161)

[5] Young-Bin Choe, Jin Ho Kwak, Yong Sung Park, and Iwao Sato, Bartholdi zeta and L-functions of weighted digraphs, their coverings and products, Adv. Math. 213 (2007), no. 2,865–886, DOI 10.1016/j.aim.2007.01.013. MR2332613 (2008f:05111)

[6] Bryan Clair and Shahriar Mokhtari-Sharghi, Zeta functions of discrete groups acting ontrees, J. Algebra 237 (2001), no. 2, 591–620, DOI 10.1006/jabr.2000.8600. MR1816705(2002f:11119)


[7] Bryan Clair and Shahriar Mokhtari-Sharghi, Convergence of zeta functions of graphs, Proc.Amer. Math. Soc. 130 (2002), no. 7, 1881–1886 (electronic), DOI 10.1090/S0002-9939-02-06532-2. MR1896018 (2003f:11130)

[8] Bryan Clair, Zeta functions of graphs with Z actions, J. Combin. Theory Ser. B 99 (2009),no. 1, 48–61, DOI 10.1016/j.jctb.2008.04.002. MR2467818 (2010e:05135)

[9] Daniele Guido, Tommaso Isola, and Michel L. Lapidus, Ihara zeta functions for periodicsimple graphs, C∗-algebras and elliptic theory II, Trends Math., Birkhauser, Basel, 2008,

pp. 103–121, DOI 10.1007/978-3-7643-8604-7 5. MR2408138 (2010a:05096)[10] Daniele Guido, Tommaso Isola, and Michel L. Lapidus, Ihara’s zeta function for periodic

graphs and its approximation in the amenable case, J. Funct. Anal. 255 (2008), no. 6, 1339–1361, DOI 10.1016/j.jfa.2008.07.011. MR2565711 (2011g:05126)

[11] Daniele Guido, Tommaso Isola, and Michel L. Lapidus, A trace on fractal graphs and the Iharazeta function, Trans. Amer. Math. Soc. 361 (2009), no. 6, 3041–3070, DOI 10.1090/S0002-9947-08-04702-8. MR2485417 (2010g:11148)

[12] Daniele Guido, Tommaso Isola, and Michel L. Lapidus, Bartholdi zeta functions for periodicsimple graphs, Analysis on graphs and its applications, Proc. Sympos. Pure Math., vol. 77,Amer. Math. Soc., Providence, RI, 2008, pp. 109–122. MR2459866 (2010a:11173)

[13] Ben M. Hambly and Takashi Kumagai, Heat kernel estimates for symmetric random walkson a class of fractal graphs and stability under rough isometries, Fractal geometry and ap-plications: a jubilee of Benoıt Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, Amer.Math. Soc., Providence, RI, 2004, pp. 233–259. MR2112125 (2005k:60141)

[14] Ki-ichiro Hashimoto and Akira Hori, Selberg-Ihara’s zeta function for p-adic discrete groups,Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math., vol. 15,Academic Press, Boston, MA, 1989, pp. 171–210. MR1040608 (91g:11053)

[15] Ki-ichiro Hashimoto, Zeta functions of finite graphs and representations of p-adic groups,Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math., vol. 15,Academic Press, Boston, MA, 1989, pp. 211–280. MR1040609 (91i:11057)

[16] Yasutaka Ihara, On discrete subgroups of the two by two projective linear group over p-adicfields, J. Math. Soc. Japan 18 (1966), 219–235. MR0223463 (36 #6511)

[17] Motoko Kotani and Toshikazu Sunada, Zeta functions of finite graphs, J. Math. Sci. Univ.

Tokyo 7 (2000), no. 1, 7–25. MR1749978 (2001f:68110)[18] Jin Ho Kwak, Jaeun Lee, and Moo Young Sohn, Bartholdi zeta functions of graph

bundles having regular fibers, European J. Combin. 26 (2005), no. 5, 593–605, DOI10.1016/j.ejc.2004.05.002. MR2127683 (2006a:05090)

[19] Hirobumi Mizuno and Iwao Sato, Bartholdi zeta functions of graph coverings, J. Combin.Theory Ser. B 89 (2003), no. 1, 27–41, DOI 10.1016/S0095-8956(03)00043-1. MR1999735(2004j:05101)

[20] Hirobumi Mizuno and Iwao Sato, A new Bartholdi zeta function of a digraph, Linear Al-gebra Appl. 423 (2007), no. 2-3, 498–511, DOI 10.1016/j.laa.2007.02.009. MR2312421(2008c:05112)

[21] Bojan Mohar and Wolfgang Woess, A survey on spectra of infinite graphs, Bull. LondonMath. Soc. 21 (1989), no. 3, 209–234, DOI 10.1112/blms/21.3.209. MR986363 (90d:05162)

[22] Bojan Mohar, The spectrum of an infinite graph, Linear Algebra Appl. 48 (1982), 245–256,DOI 10.1016/0024-3795(82)90111-2. MR683222 (84d:05123)

[23] Iwao Sato, Bartholdi zeta functions of fractal graphs, Electron. J. Combin. 16 (2009), no. 1,Research Paper 30, 21 pp. MR2482098 (2010a:05120)

[24] Iwao Sato, Bartholdi zeta functions of periodic graphs, Linear Multilinear Algebra 59 (2011),no. 1, 11–24, DOI 10.1080/03081080903171914. MR2769347 (2012c:05153)

[25] H. M. Stark and A. A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121(1996), no. 1, 124–165, DOI 10.1006/aima.1996.0050. MR1399606 (98b:11094)


[26] Toshikazu Sunada, L-functions in geometry and some applications, Curvature and topologyof Riemannian manifolds (Katata, 1985), Lecture Notes in Math., vol. 1201, Springer, Berlin,1986, pp. 266–284, DOI 10.1007/BFb0075662. MR859591 (88g:58152)

Dipartimento di Matematica, Universita di Roma “Tor Vergata”, I–00133 Roma,

Italy.


Dipartimento di Matematica, Universita di Roma “Tor Vergata”, I–00133 Roma,

Italy.



Vector Analysis on Fractals and Applications

Michael Hinz and Alexander Teplyaev

Abstract. The paper surveys some recent results concerning vector analysison fractals. We start with a strongly local regular Dirichlet form and use theframework of 1-forms and derivations introduced by Cipriani and Sauvageotto set up some elements of a related vector analysis in weak and non-localformulation. This allows to study various scalar and vector valued linear andnon-linear partial differential equations on fractals that had not been accessible

before. Subsequently a stronger (localized, pointwise or fiberwise) version ofthis vector analysis can be developed, which is related to previous work ofKusuoka, Kigami, Eberle, Strichartz, Hino, Ionescu, Rogers, Rockner, and theauthors.

Contents

1. Introduction2. Dirichlet forms and energy measures3. 1-forms and vector fields4. Scalar PDE involving first order terms5. Navier-Stokes equations6. Magnetic Schrodinger equationsReferences

1. Introduction

In the present article we survey some recent results concerning vector analy-sis based on symmetric strongly local regular Dirichlet forms on locally compactseparable metric spaces, cf. [17, 32, 73, 79]. The notions and results we discusshave been introduced in the papers [44–46]. They are based on the approach todifferential 1-forms as proposed by Cipriani and Sauvageot in [22] in much greatergenerality, and later investigated by several authors, [21,23,43,50]. The construc-tions are sufficiently robust to apply to symmetric diffusions on fractals such asp.c.f. self-similar sets [55, 56, 85], nested fractals [72], finitely ramified fractals[81,90], generalized Sierpinski carpets [8,10,68], spaces of Barlow-Evans-Laakso

2010 Mathematics Subject Classification. Primary 28A80, 31E05, 53C23, 60J25, 60J35,81Q35, 35A01, 35Q30.

The first author’s research was supported in part by NSF grant DMS-0505622 and by theAlexander von Humboldt Foundation Feodor (Lynen Research Fellowship Program).

The second author’s research was supported in part by NSF grant DMS-0505622.


147

148 M. HINZ AND A. TEPLYAEV

type [82,83], and some random fractals [36,37]. As they are based on Dirichletforms they also apply to classical situations such as Euclidean spaces, domains withsufficiently regular boundary and smooth compact Riemannian manifolds. In thesecases we recover well-known results.

A general theme motivating our studies consists of the questions for whichelements of differential geometry and vector analysis one can find analogs builtsolely upon the notion of energy and how these analogs can be used to formulateand study physical models on non-smooth spaces. The space H of 1-forms asconstructed in [22,23] is a Hilbert space. Therefore one can identify 1-forms andvector fields, and furthermore introduce other notions of vector analysis, as recentlydone in [44] (which generalizes earlier approaches to vector analysis on fractals, see[55, 58, 67, 75, 84, 90]). This is a part of a comprehensive program to introduceand study vector equations on general non-smooth spaces which carry a diffusionprocess (or, equivalently, a local regular Dirichlet form).

Much of the existing literature on analysis on fractals has been concerned withthe primary problems of construction diffusions on fractals ([6,10,13,34,40,55,64–66, 68, 72, 81] and references therein), studying their heat kernel decay ([7,12, 35, 52, 53, 59, 63] and references therein), their potential theory ([9, 11, 14,15, 49, 74] and references therein), their spectral properties ([1–4, 24–26, 33, 38,41,51,61,62,69,71,87,89] and references therein), and some related elliptic andparabolic partial differential equations ([28,29,47,48] and references therein). Forsome recent physics applications of analysis on fractals see [1,2,18,30,54,78,86]and references therein, and for analysis on fractals in general see [5,56,85].

Once a diffusion is known to exist, we may regard its infinitesimal generator asthe Laplacian Δ, and employ general functional analytic tools (such as semigrouptheory or variational methods, [27]) to solve equations of type Δu = f and ∂u

∂t =

Δu + F (u), and even such of form ΔΦ(u) = f or ∂u∂t = ΔΦ(u), with possibly

nonlinear transformations F and Φ. Note that these equations do not includeanalogs of first order operators (gradients). However, we would like to investigatescalar equations of type

(1) div(a(∇u)) = f

or

(2) Δu + b(∇u) = f

with possibly nonlinear a and b, or vector equations like for instance the Navier-Stokes system

(3)

{∂u∂t + (u · ∇)u− Δu + ∇p = 0,

div u = 0,

or the magnetic Schrodinger equation

(4) i∂u

∂t= (−i∇ −A)2u + V u.

Previous constructions [55, 66, 84, 88] of first order operators related to dif-fusions on fractals were rather based on probabilistic and point-wise approaches,and perhaps for this reason not quite flexible enough to fit into a setup that allowsto investigate partial differential equations containing first order terms. The ma-chinery of [22,23], together with further developments in [44–46,50], provides a

VECTOR ANALYSIS ON FRACTALS AND APPLICATIONS 149

functional analytic definition of a first order derivation (respectively gradient) anda framework suitable for a comfortable analysis of problems like (1)-(4) on fractals.

It is our aim in this paper to highlight elements of this toolkit and to announcesome related results. We proceed as follows. In the next section we state our mainhypotheses and collect some useful facts on Dirichlet forms and energy measures. InSection 3 we review the basic setup of [22,23] and discuss related notions of vectoranalysis proposed in [44]. First applications to scalar valued partial differentialequations of types (1) and (2) are then presented in Section 4, and some results onanalogs of (3) in Section 5. In Section 6 we discuss an approach to (4). We alsopresent the definition of related Dirac operators proposed in abstract form in [22]and in pointwise form in [46].

2. Dirichlet forms and energy measures

Let X be a locally compact separable metric space and m a Radon measureon X such that each nonempty open set is charged positively. We assume that(E ,F) is a symmetric strongly local regular Dirichlet form on L2(X,m) with coreC := F ∩ C0(X). Endowed with the norm ‖f‖C := E(f)1/2 + supX |f | the space Cbecomes an algebra and in particular,

(5) E(fg)1/2 ≤ ‖f‖C ‖g‖C , f, g ∈ C,see [17]. For any g, h ∈ C we can define a finite signed Radon measure Γ(g, h) onX such that

2

∫X

f dΓ(g, h) = E(fg, h) + E(fh, g) − E(gh, f) , f ∈ C,

the mutual energy measure of g and h. By approximation we can also define themutual energy measure Γ(g, h) for general g, h ∈ F . Note that Γ is symmetric andbilinear, and Γ(g) ≥ 0, g ∈ F . For details we refer the reader to [32]. We providesome examples.

Examples 2.1.

(i) Dirichlet forms on Euclidean domains. Let X = Ω be a bounded domainin Rn with smooth boundary ∂Ω and

E(f, g) =

∫Ω

∇f∇g dx, f, g ∈ C∞0 (Ω).

If H10 (Ω) denotes the closure of C∞

0 (Ω) with respect to the scalar productE1(f, g) := E(f, g)+〈f, g〉L2(Ω), then (E , H1

0 (Ω)) is a local regular Dirichlet

form on L2(Ω). The mutual energy measure of f, g ∈ H10 (Ω) is given by

∇f∇g dx.(ii) Dirichlet forms on Riemannian manifolds. Let X = M be a smooth

compact Riemannian manifold and

E(f, g) =

∫M

〈df, dg〉T∗M dvol, f, g ∈ C∞(M).

Here dvol denotes the Riemannian volume measure. Similarly as in (i)the closure of E in L2(M,dvol) yields a local regular Dirichlet form. Themutual energy measure of two energy finite functions f, g is given by〈df, dg〉T∗M dvol.


(iii) Dirichlet forms induced by resistance forms on fractals. Let X be a set and(E ,F) a local resistance form on it such that X, endowed with the corre-sponding resistance metric R, is complete, separable and locally compact.For any Borel regular measure m on (X,R) such that 0 < m(B(x, r)) < ∞,the space (F ∩ L2(X,m), E1) is Hilbert, and denoting by F the closureof C0(X) ∩ F in it, we obtain a local regular Dirichlet form (E ,F) onL2(X,m) (see for instance [60, Section 9]). Here we have again used thestandard notation E1(f, g) = E(f, g) + 〈f, g〉L2(X,m).

Remark 2.1. In Examples 2.1 (i) and (ii) the energy measures have beenabsolutely continuous with respect to the given reference measure. For diffusionson self-similar fractals this is typically not true if we choose the corresponding self-similar Hausdorff type measure as reference measure, see for instance [16] or [42].We may, however, use Kusuoka type measures as reference measures to produceabsolute continuity, see for instance [44,52,58,66,90].

3. 1-forms and vector fields

Following [22, 23] we consider C ⊗ Bb(X), where Bb(X) denotes the space ofbounded Borel functions on X. We endow this tensor product with the symmetricbilinear form

(6) 〈a⊗ b, c⊗ d〉H :=

∫X

bd dΓ(a, c),

a ⊗ b, c ⊗ d ∈ C ⊗ Bb(X), let ‖·‖H denote the associated seminorm on C ⊗ Bb(X)and write

ker ‖·‖H :=

{∑i

ai ⊗ bi ∈ C ⊗ Bb(X) :

∥∥∥∥∥∑i

ai ⊗ bi

∥∥∥∥∥H

= 0

}(with finite linear combinations). To the Hilbert space H obtained as the completionof C ⊗ Bb(X)/ker ‖·‖H with respect to ‖·‖H we refer as the space of differential1-forms on X, cf. [22,23,43,50].

The space H becomes a bimodule if we declare the algebras C and Bb(X) toact on it as follows: For a⊗ b ∈ C ⊗ Bb(X), c ∈ C and d ∈ Bb(X) set

(7) c(a⊗ b) := (ca) ⊗ b− c⊗ (ab)

and

(8) (a⊗ b)d := a⊗ (bd).

In [22] and [50] it has been shown that (7) and (8) extend to well defined left andright actions of the algebras C and Bb(X) on H. From (6) and the Leibniz rulefor energy measures, see [32, Theorem 3.2.2], it can be seen that left and rightmultiplication agree for any c ∈ C, and as

max {‖(a⊗ b)c‖H , ‖c(a⊗ b)‖H} ≤ supX

|c| ‖a⊗ b‖H ,

it follows by approximation that they agree for all c ∈ Bb(X), see [50].A derivation operator ∂ : C → H can be defined by setting

∂f := f ⊗ 1.


It obeys the Leibniz rule,

(9) ∂(fg) = f∂g + g∂f, f, g ∈ C,and is a bounded linear operator satisfying

(10) ‖∂f‖2H = E(f), f ∈ C.On Euclidean domains and on smooth manifolds the operator ∂ coincides with theclassical exterior derivative (in the sense of L2-differential forms). Details can befound in [22,23,43,44,50].

Being Hilbert, H is self-dual. We therefore regard 1-forms also as vector fieldsand ∂ as the gradient operator. Let C∗ denote the dual space of C, normed by

‖w‖C∗ = sup {|w(f)| : f ∈ C, ‖f‖C ≤ 1} .Given f, g ∈ C, consider the functional

u �→ ∂∗(g∂f)(u) := −〈∂u, g∂f〉H = −∫X

g dΓ(u, f)

on C. It defines an element ∂∗(g∂f) of C∗, to which we refer as the divergence ofthe vector field g∂f .

Lemma 3.1. The divergence operator ∂∗ extends continuously to a boundedlinear operator from H into C∗ with ‖∂∗v‖C∗ ≤ ‖v‖H, v ∈ H. We have

∂∗v(u) = −〈∂u, v〉Hfor any u ∈ C and any v ∈ H.

See [44] for a proof. The Euclidean identity

div (g grad f) = gΔf + ∇f∇g

has a counterpart in terms of ∂ and ∂∗. Let (A, dom A) denote the infinitesimalL2(X,m)-generator of (E ,F), that is the non-positive definite self-adjoint operatorA on L2(X,m) such that E(f, g) =

⟨√−Af,

√−Ag

⟩L2(X,m)

for all f, g ∈ F . Given

f, g ∈ C we set (gAf)(u) := −E(gu, f), u ∈ C.

Lemma 3.2. We have

∂∗(g∂f) = gAf + Γ(f, g)

for any simple vector field g∂f with f, g ∈ C, seen as an equality in C∗. In particular,Af = ∂∗∂f for f ∈ C.

A proof is given in [44, Section 3]. This distributional perspective can becomplemented by the following point of view. The operator ∂, equipped withthe domain C, may be seen as densely defined unbounded operator

∂ : L2(X,m) → H.

Since (E ,F) is a Dirichlet form, ∂ extends uniquely to a closed linear operator ∂with domain dom ∂ = F . The divergence ∂∗, seen as an operator

∂∗ : H → L2(X,m),

will be unbounded, note that in general the inclusions C ⊂ L2(X,m) ⊂ C∗ areproper. As usual v ∈ H is said to be a member of dom ∂∗ if there exists some


v∗ ∈ L2(X,m) such that 〈u, v∗〉L2(X,m) = −〈∂u, v〉H for all u ∈ C. In this case

∂∗v := v∗ and

〈u, ∂∗v〉L2(X,m) = −〈∂u, v〉H , u ∈ C,i.e. −∂∗ is the adjoint operator of ∂. It is immediate that {∂f : f ∈ dom A} ⊂dom∂∗. As ∂ is densely defined and closed, the domain dom∂∗ of ∂∗ is automaticallydense in H.

We say that (E ,F) admits a spectral gap if there exists some c > 0 such that

(11)

∫X

(f − fX)2dm ≤ c E(f)

for any f ∈ F , where fX = 1m(X)

∫Xf dm. If (E ,F) has a spectral gap, then the

image Im ∂ of ∂ is a closed subspace of H. In this case the space H decomposesorthogonally into Im∂ and its complement (Im∂)⊥, what implies (Im∂)⊥ = ker∂∗,and as a consequence we observe the following explicit description of dom ∂∗.

Corollary 3.1. Assume that (E ,F) admits a spectral gap, ( 11). Then thedomain dom ∂∗ agrees with

{v ∈ H : v = ∂f + w : f ∈ dom A , w ∈ ker ∂∗} .

For any v = ∂f + w with f ∈ dom A and w ∈ ker ∂∗ we have ∂∗v = Af .

The proof is short and straightforward, see [46, Corollary 2.2].

4. Scalar PDE involving first order terms

The results of the preceding section may be used to obtain some results onequations of type (1) and (2). We quote from [44, Section 4]. First consider thequasilinear equation

(12) ∂∗a(∂u) = f.

on L2(X,m). In the situation of Example 2.1 (i) it agrees with (1). Assume thata : H → H satisfies the following monotonicity, growth and coercivity conditions:

(13) 〈a(v) − a(w), v − w〉H ≥ 0 for all v, w ∈ Im ∂,

(14) ‖a(v)‖H ≤ c0(1 + ‖v‖H) for all v ∈ Im ∂

with some constant c0 > 0, and

(15) 〈a(v), v〉H ≥ c1 ‖v‖2H − c2 for all v ∈ Im ∂

with constants c1 > 0, c2 ≥ 0. For simplicity we assume the validity of a Poincareinequality,

(16) ‖f‖2L2(X,m) ≤ cP E(f)

with some constant cP > 0 for all f ∈ L2(X,m) with∫Xfdm = 0. A function

u ∈ F is called a weak solution to ( 12) if

〈a(∂u), ∂v〉H = −〈f, v〉L2(X,m) for all v ∈ F .

By classical methods, [27, Section 9.1], we obtain the following result.


Theorem 4.1. Assume a satisfies ( 13), ( 14) and ( 15) and suppose ( 16) holds.Then ( 12) has a weak solution. Moreover, if a is strictly monotone, i.e.

(17) 〈a(v) − a(w), v − w〉H ≥ c3 ‖v − w‖2H for all v, w ∈ Im ∂

with some constant c3 > 0, then ( 12) has a unique weak solution.

An analog of (2) can be treated in a similar manner. Consider

(18) −Au + b(∂u) + �u = 0,

where � > 0 and b is a generally non-linear function-valued mapping on H. Assumethat b : H → L2(X,m) is such that

(19) ‖b(v)‖L2(X,m) ≤ c4(1 + ‖v‖H), v ∈ Im ∂,

with some c4 > 0. A function u ∈ F is called a weak solution to (18) if

E(u, v) + 〈b(∂u), v〉L2(X,m) + � 〈u, v〉L2(X,m) = 0 for all v ∈ F .

From [27, Section 9.2.2, Example 2], we then obtain the following.

Theorem 4.2. Assume that the embedding F ⊂ L2(X,m) is compact and that( 19) holds. Then for any sufficiently large � > 0 there exists a weak solution to( 18).

5. Navier-Stokes equations

In this section we comment on equations of type (3) which provide some moreinteresting applications for the notions discussed in Section 3.

Together with suitable boundary conditions the Navier-Stokes system (3) de-scribes the flow of an incompressible and homogeneous fluid in a Euclidean domainwith velocity field u and subject to the pressure p. In a one-dimensional situation itreduces to an Euler equation ∂u/∂t+∂p/∂x = 0 that has only stationary solutions.In [45] we have proposed to investigate an analog of (3) on compact connected topo-logically one-dimensional fractals X. We collect some items necessary to formulateit.

Assume that the metric space X is compact, connected and topologically one-dimensional and that (E ,F) admits a spectral gap, (11). Recall that a compactmetric space X is topologically zero dimensional if every open cover has a refinementconsisting of disjoint open sets such that any point of the space is contained inexactly one open set of this refinement. It is topologically one-dimensional if it is nottopologically zero dimensional but any open cover of X has a refinement such thatno point of X is contained in more than two open sets of this refinement. Combinedwith several results on Hodge decompositions and topology, cf. [45, Sections 4, 5and 6], the assumption of topological one-dimensionality had motivated to define aLaplacian Δ1 on 1-forms by

(20) Δ1 := ∂∂∗,

seen as an unbounded operator on H with domain

dom Δ1 = {ω ∈ dom ∂∗ : ∂∗ω ∈ F} .

Theorem 5.1. The operator (Δ1, dom Δ1) is a self-adjoint operator on H.


A proof can be found in [46, Section 6]. Theorem 5.1 allows to talk aboutharmonic forms: A 1-form ω ∈ H is called harmonic if ω ∈ dom Δ1 and Δ1ω = 0.From compactness and topological one-dimensionality can deduce the following, cf.[46, Theorem 6.2].

Theorem 5.2. A 1-form ω ∈ H is harmonic if and only if it is in (Im ∂)⊥.

The proof of of Theorem 5.2 is rather subtle, it involves a description of (Im∂)⊥

in terms of locally harmonic forms. We refer the reader to [45]. Note that Theorem5.2 indicates that in this situation the definition (20) is appropriate.

Also for the convection term (u · ∇)u in (3) we propose a substitute which byone-dimensionality seems reasonable. Our choice is motivated by the Euclideansituation: Given a vector field u, the quantity

(21) −∫

|u|2 div v dx,

seen as a functional on a space of test vector fields v, provides a formulation of∇|u|2 in the weak sense. In our situation we set

domc∂∗ := {v ∈ dom ∂∗ : ∂∗v ∈ C(X)}

and given u ∈ H, define

(22) ∂ΓH(u)(v) := −〈(∂∗v)u, u〉H , v ∈ domc∂∗.

This seems reasonable by a fiberwise (respectively m-a.e. pointwise) representationfor H proved in [44, Section 2] and [46, Theorem 2.2]:

Theorem 5.3. Let ν be a Radon measure such that all energy measures are ab-solutely continuous with respect to ν. There are a family of Hilbert spaces {Hx}x∈X

and surjective linear maps ω �→ ωx from H onto Hx such that the direct integral∫ ⊕K

Hxν(dx) is isometrically isomorphic to H and in particular,

‖ω‖2H =

∫K

‖ωx‖2H,x ν(dx), ω ∈ H.

Theorem 5.3 itself is more general, it does neither require X to be compact ortopologically one-dimensional nor (E ,F) to admit a spectral gap.

If we replace the Euclidean norm | · | in (21) by the norms ‖·‖H,x of the fibers

Hx, we arrive at (22). On the other hand the classical identity

1

2∇|u|2 = (u · ∇)u + u× curlu.

holds in the Euclidean case. In a one-dimensional situation there should be nononzero 2-forms, hence curlu should be trivial, so that (22) may be a good substi-tute for (u · ∇)u. See [45] for a more detailed discussion. Altogether this gives astrong heuristic motivation to regard

(23)

{∂u∂t + 1

2∂ΓH(u) − Δ1u + ∂p = 0

∂∗u = 0.

as a suitable analog of a (3) on a compact topologically one-dimensional space.Note that this is a boundary free formulation. We say that a square integrable


dom ∂∗-valued function u on [0,∞) provides a weak solution to ( 23) with initialcondition u0 ∈ ker ∂∗ if(24){

〈u(t), v〉H − 〈u0, v〉H +∫ t

0∂ΓH(u(s))(v)ds+

∫ t

0〈∂∗u(s), ∂∗v〉L2(X,m) ds = 0

∂∗u(t) = 0

for a.e. t ∈ [0,∞) and all v ∈ ker ∂∗. By some immediate simplifications we thenobserve stationarity and uniqueness of solutions. In other words, the behavior ofthe system (23) on a compact topologically one-dimensional space resembles thebehavior of (3) on a one-dimensional Euclidean space.

Theorem 5.4. Any weak solution u of ( 23) is harmonic and stationary, i.e. uis independent of t ∈ [0,∞). Given an initial condition u0 the corresponding weaksolution is uniquely determined.

Note that we have not made any restriction on the Hausdorff dimension dH ofX. Indeed there are examples of spaces of any Hausdorff dimension 1 ≤ dH < ∞such that the previous Theorem holds. It is the topological dimension that governsthe behavior of (23).

Remark 5.1. Logically Theorem 5.3 is not needed to set up the model (23), wehave just included it here to support the intuition behind our choice of substituteterms. We would also like to remark that even though the energy measures mightnot be absolutely continuous with respect to the initial reference measure m, onecan always construct a finite Radon measure ν that has this property.

For the rest of this section we specialize further to the situation of Examples2.1 (iii), that is, we assume X to be a set and (E ,F) to be a local resistanceform on it such that X, together with the resistance metric R, is a compact andconnected metric space. We further assume that m is a Borel regular measure onX as in Examples 2.1 (iii) so that consequently a local regular Dirichlet form (E ,F)is obtained by taking the closure. Then all our previous results may be applied for(E ,F). We finally assume that (X,R) is topologically one-dimensional.

Remark 5.2. We conjecture that any set that carries a regular resistance formis a topologically one-dimensional space when equipped with the associated resis-tance metric.

Note that in the resistance form case points have positive capacity. This prop-erty allows to prove the following equivalence, [45, Section 5].

Theorem 5.5. Assume that (E ,F) is a local resistance form on X with resis-tance metric R and (X,R) is compact and topologically one-dimensional. Then anontrivial solution to ( 23) exists if and only if the first Cech cohomology H1(X) ofX is nontrivial.

In the resistance form context Neumann derivatives are well-defined, and itis not difficult to see that if the Navier-Stokes system (23) is considered with anonempty boundary, it may have additional nontrivial solutions arising from solu-tions of a related Neumann problem.

Let B ⊂ X be a finite set, which is interpreted as the boundary of X. ByGB we denote the Green operator associated with the boundary B with respect to(E ,F), [57, Definition 5.6], and DL

B,0 its image in F . Let HB denote the B-harmonic


functions with respect to (E ,F), [57, Definition 2.16], and note that F = FB⊕HB ,where

FB :={u ∈ F : u|B = 0

}.

A B-harmonic function h is harmonic on Bc in the Dirichlet form sense, moreprecisely, it satisfies E(h, ψ) = 0 for all ψ ∈ FB. The space DL := DL

B,0 + HB is

seen to be independent of the choice of B, [57, Theorem 5.10]. For any u ∈ DL andany p ∈ X the Neumann derivative (du)p of u at p can be defined, [57, Theorems

6.6 and 6.8]. If ϕ is a function on B, then a function hϕ ∈ F is called a solutionto the Neumann problem on Bc with boundary values ϕ if it is harmonic on Bc

and satisfies (dh)p = ϕ(p) for all p ∈ B. Such a Neumann solution hϕ exists and isunique if and only if ϕ is such that∑

p∈B

ϕ(p) = 0.

We use the notation H(Bc) = clos span {v1Bc : v ∈ H}. A square integrabledom ∂∗-valued function u on [0,∞) provides a weak solution to ( 23) on Bc if(25){

〈u(t), v〉H − 〈u(0), v〉H +∫ t

0∂ΓH(u(s))(v)ds +

∫ t

0〈∂∗u(s), ∂∗v〉L2(X,m) ds = 0

〈u(t), ∂ψ〉H = 0

for a.e. t ∈ [0,∞), all v ∈ dom ∂∗ ∩H(Bc) and all ψ ∈ FB.

Theorem 5.6. Assume that (E ,F) is a local resistance form on X with re-sistance metric R and (X,R) is compact and topologically one-dimensional. If his the unique, up to an additive constant, harmonic function on Bc with normalderivatives ϕ on B, then

u(t) = ∂h, t ∈ [0,∞).

is the unique weak solution to ( 23) on Bc with the Neumann boundary values ϕon B.

Remark 5.3. In (25) we have considered weak solutions to (23). For weaksolutions the pressure p does not occur explicitely. However, any definition of astrong solution to (3) should lead to the relation

p(t) = −1

2Γ(h), t ∈ [0,∞),

seen as an equality of measures. A similar statement could be written for theboundary free case (24).

For more details and the Hodge theory leading to the statements of this sectionwe refer the reader to [46].

6. Magnetic Schrodinger equations

We turn to results concerning the magnetic Schrodinger equation (4). To dis-cuss this equation we do not need to assume that X is compact or topologicallyone-dimensional. As in Section 2 it may just be an arbitrary locally compact sepa-rable metric space equipped with a Radon measure m that charges any nonemptyopen set positively and carrying a symmetric strongly local regular Dirichlet form(E ,F) on L2(X,m) with core C := F ∩ C0(X). However, to investigate (4) wewill now assume that (E ,F) possesses energy densities with respect to the reference


measure m, i.e. for any g, h ∈ F the measure Γ(g, h) is absolutely continuous withrespect to m.

Remark 6.1. As previously mentioned in Remark 5.1 we can always constructa measure ν with respect to which all energy measures are absolutely continuous.For the cases that the given Dirichlet form (E ,F) on L2(X,m) is transient orinduced by a resistance form we have shown in [44] that (E , C) is a closable form onthe space L2(X, ν) of functions that are square integrable with respect to this newmeasure ν. Then this change of measure merely amounts to a change of domains.It is not difficult to show that (E , C) is always closable with respect to this measureν. We will discuss this matter in a later paper.

In [46] we have studied analogs of the magnetic Schrodinger Hamiltonian(−i∇ − A)2 + V and in particular, have verified their essential self-adjointness.To sketch this result, let L2,C(X,m), FC, CC and HC denote the natural complex-ifications of L2(X,m), F , C and H, respectively. The natural extensions to thecomplex case of E and the corresponding energy measures Γ(f, g) are again de-noted by the same symbols. Note that they are conjugate symmetric and linear inthe first argument. If both arguments agree, they yield a real nonnegative numberand a real nonnegative measure, respectively. The first result concerns related qua-dratic forms. Here we use the notion of quadratic form in the sense of [76, SectionVIII.6].

Proposition 6.1. Let a ∈ H and V ∈ L∞(X,m). The form Ea,V , given by

Ea,V (f, g) = 〈(−i∂ − a)f, (−i∂ − a)g〉H + 〈fV, g〉L2(X,m) , f, g ∈ CC,

defines a quadratic form on L2,C(X,m).

Proposition 6.1 is a slight variation of Proposition 4.1 in [46], and up to inessen-tial details it has the same proof. Here a is seen as the magnetic vector potentialreplacing A in (4) and V is the electric scalar potential.

Now recall the fiberwise representation of H from Theorem 5.3, here we use itwith m in place of ν. We define the space of real vector fields of bounded length by

H∞ :={v = (vx)x∈X ∈ H : ‖v·‖H,· ∈ L∞(X,m)

}.

If the potential a is recruited from H∞ then we can obtain the closedness of Ea,V

and the essential self-adjointness of the associated operator from straightforwardperturbation arguments, cf. [46, Theorem 4.1]. Recall that A denotes the generatorof (E ,F). We denote its complexification by the same symbol.

Theorem 6.1. Let a ∈ H∞ and V ∈ L∞(X,m).

(i) The quadratic form (Ea,V ,FC) is closed.(ii) The self-adjoint non-negative definite operator on L2,C(X,m) uniquely as-

sociated with (Ea,V ,FC) is given by

Ha,V = (−i∂ − a)∗(−i∂ − a) + V,

and the domain of the operator A is a domain of essential self-adjointnessfor Ha,V .


The operator Ha,V is a natural generalization of the quantum mechanicalSchrodinger Hamiltonian (−i∇−A)+V from (4). By Theorem 6.1 we have estab-lished a suitable framework to study a fractal counterpart

i∂u

∂t= Ha,V u

of the evolution equation (4).As they are closely related to magnetic Hamiltonians, we conclude this section

by a brief look at Dirac operators. We have introduced a local Dirac operator in[46]. Up to sign and complexity conventions it is defined as a matrix operator

(26) D =

(0 ∂∗

∂ 0

),

acting on H0 ⊕ H1, where H0 := L2(X,m) and H1 := H. We consider D asan unbounded linear operator with domain dom D := F ⊕ dom ∂∗ and have thefollowing result, obtained in abstract form in [22], and in pointwise form in [46].

Theorem 6.2. The operator (D, domD) is self-adjoint operator on H0 ⊕H1.

Note that as a consequence we also obtain a local matrix Laplacian D2 actingon H0 ⊕H1.

According to [45,46], this Dirac operator is naturally related to the topologicalstructure of the fractals space and, in a certain natural a sense, to the differentialgeometry of the fractal (see [19,20] and references therein for a discussion of thenotion of a Dirac operator in the context of non-commutative analysis). In partic-ular, for compact topologically one dimensional fractals (of arbitrary Hausdorff andspectral dimensions) our Dirac operator gives rise to a natural Hodge Laplacian∂∂∗ + ∂∗∂ on the appropriate differential complex. It will be the subject of futurework to study the Hodge Laplacian for higher order differential forms defined in aprobabilistic or Dirichlet form sense.

As a side remark we note that sometimes there may be a different conventionfor the Dirac operator in a complex setup. For instance

D =

(0 −i∂∗

−i∂ 0

)has signs and imaginary factors which are somewhat more suitable in relation tothe magnetic magnetic Schrodinger operator Ha,V .

Finally, we would like to point out related perturbation results. Assume thatb ∈ H∞ and set

Q(f, g) := E(f, g) −∫X

g(x) 〈bx, ∂xf〉Hxm(dx),

f, g ∈ F . Here bx and ∂xf denote the images of b and ∂f under the projection fromH onto Hx as in Theorem 5.3. For α ≥ 0 write

Q(f, g) := Q(f, g) + α 〈f, g〉L2(X,m) .

We may then conclude the following.

Theorem 6.3.

(i) For any α ≥ 0 the form (Qα,F) is closed on L2(X,m). It generates astrongly continuous semigroup of bounded operators on L2(X,m).


(ii) If α > 0 is sufficiently large then the associated semigroup is positivitypreserving.

(iii) The generator LQ of Q is given by

LQu(x) = Au(x) + 〈bx, ∂xf〉Hx, u ∈ dom A.

See [44, Section 10] and the references therein, in particular [31].

References

[1] E. Akkermans, G. Dunne, A. Teplyaev Physical Consequences of Complex Dimensions ofFractals. Europhys. Lett. 88, 40007 (2009).

[2] E. Akkermans, G. Dunne, A. Teplyaev Thermodynamics of photons on fractals. Phys. Rev.Lett. 105(23):230407, 2010.

[3] N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst,and A. Teplyaev, Vibration modes of 3n-gaskets and other fractals, J. Phys. A 41 (2008),no. 1, 015101, 21, DOI 10.1088/1751-8113/41/1/015101. MR2450694 (2010a:28008)

[4] N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst,and A. Teplyaev, Vibration spectra of finitely ramified, symmetric fractals, Fractals 16 (2008),no. 3, 243–258, DOI 10.1142/S0218348X08004010. MR2451619 (2009k:47098)

[5] Martin T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995), Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 1–121, DOI10.1007/BFb0092537. MR1668115 (2000a:60148)

[6] Martin T. Barlow and Richard F. Bass, The construction of Brownian motion on theSierpinski carpet, Ann. Inst. H. Poincare Probab. Statist. 25 (1989), no. 3, 225–257 (English,with French summary). MR1023950 (91d:60183)

[7] Martin T. Barlow and Richard F. Bass, Transition densities for Brownian motion onthe Sierpinski carpet, Probab. Theory Related Fields 91 (1992), no. 3-4, 307–330, DOI10.1007/BF01192060. MR1151799 (93k:60203)

[8] Martin T. Barlow and Richard F. Bass, Brownian motion and harmonic analysis on Sier-pinski carpets, Canad. J. Math. 51 (1999), no. 4, 673–744, DOI 10.4153/CJM-1999-031-4.MR1701339 (2000i:60083)

[9] Martin T. Barlow, Richard F. Bass, and Takashi Kumagai, Stability of parabolic Harnackinequalities on metric measure spaces, J. Math. Soc. Japan 58 (2006), no. 2, 485–519.MR2228569 (2007f:60064)

[10] Martin T. Barlow, Richard F. Bass, Takashi Kumagai, and Alexander Teplyaev, Uniquenessof Brownian motion on Sierpinski carpets, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3,655–701. MR2639315 (2011i:60146)

[11] M.T. Barlow, A. Grigor’yan, T. Kumagai, On the equivalence of parabolic Harnack inequal-ities and heat kernel estimates, Journal of the Mathematical Society of Japan. Volume 64,Number 4 (2012), 1091-1146.

[12] Martin T. Barlow and Takashi Kumagai, Transition density asymptotics for some diffusionprocesses with multi-fractal structures, Electron. J. Probab. 6 (2001), no. 9, 23 pp. (elec-

tronic), DOI 10.1214/EJP.v6-82. MR1831804 (2002c:60127)[13] Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpinski gasket, Probab.

Theory Related Fields 79 (1988), no. 4, 543–623, DOI 10.1007/BF00318785. MR966175(89g:60241)

[14] R.F. Bass, A stability theorem for elliptic Harnack inequalities, to appear in J. Europ. Math.Soc.

[15] Richard F. Bass and Maria Gordina, Harnack inequalities in infinite dimensions, J. Funct.Anal. 263 (2012), no. 11, 3707–3740, DOI 10.1016/j.jfa.2012.09.009. MR2984082

[16] Oren Ben-Bassat, Robert S. Strichartz, and Alexander Teplyaev, What is not in the domainof the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal. 166 (1999), no. 2, 197–217,DOI 10.1006/jfan.1999.3431. MR1707752 (2001e:31016)

[17] Nicolas Bouleau and Francis Hirsch, Dirichlet forms and analysis on Wiener space, deGruyter Studies in Mathematics, vol. 14, Walter de Gruyter & Co., Berlin, 1991. MR1133391(93e:60107)

[18] J. Chen, Statistical mechanics of Bose gas in Sierpinski carpets. Submitted. arXiv:1202.1274


[19] Erik Christensen and Cristina Ivan, Extensions and degenerations of spectral triples, Comm.Math. Phys. 285 (2009), no. 3, 925–955, DOI 10.1007/s00220-008-0657-4. MR2470910(2009i:58035)

[20] Erik Christensen, Cristina Ivan, and Michel L. Lapidus, Dirac operators and spectraltriples for some fractal sets built on curves, Adv. Math. 217 (2008), no. 1, 42–78, DOI10.1016/j.aim.2007.06.009. MR2357322 (2009a:58031)

[21] F. Cipriani, D. Guido, T. Isola, J.-L. Sauvageot, Differential 1-forms, their integrals and

potential theory on the Sierpinski gasket, preprint ArXiv (2011).[22] Fabio Cipriani and Jean-Luc Sauvageot, Derivations as square roots of Dirichlet forms, J.

Funct. Anal. 201 (2003), no. 1, 78–120, DOI 10.1016/S0022-1236(03)00085-5. MR1986156(2004e:46080)

[23] Fabio Cipriani and Jean-Luc Sauvageot, Fredholm modules on P.C.F. self-similar frac-tals and their conformal geometry, Comm. Math. Phys. 286 (2009), no. 2, 541–558, DOI10.1007/s00220-008-0673-4. MR2472035 (2010d:58029)


[25] Gregory Derfel, Peter J. Grabner, and Fritz Vogl, Complex asymptotics of Poincare functionsand properties of Julia sets, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 3, 699–718,DOI 10.1017/S0305004108001564. MR2464785 (2010a:37087)

[26] G. Derfel, P. J. Grabner, and F. Vogl, Laplace Operators on Fractals and Related FunctionalEquations, 2012 J. Phys. A: Math. Theor. 45 463001 doi:10.1088/1751-8113/45/46/463001

[27] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19,American Mathematical Society, Providence, RI, 1998. MR1625845 (99e:35001)

[28] K. J. Falconer, Semilinear PDEs on self-similar fractals, Comm. Math. Phys. 206 (1999),no. 1, 235–245, DOI 10.1007/s002200050703. MR1736985 (2001a:28008)

[29] Kenneth J. Falconer and Jiaxin Hu, Nonlinear diffusion equations on unbounded fractaldomains, J. Math. Anal. Appl. 256 (2001), no. 2, 606–624, DOI 10.1006/jmaa.2000.7331.MR1821760 (2002d:35101)

[30] Edward Fan, Zuhair Khandker, and Robert S. Strichartz, Harmonic oscillators on infinite

Sierpinski gaskets, Comm. Math. Phys. 287 (2009), no. 1, 351–382, DOI 10.1007/s00220-008-0633-z. MR2480752 (2011f:35059)

[31] Patrick J. Fitzsimmons and Kazuhiro Kuwae, Non-symmetric perturbations of symmetricDirichlet forms, J. Funct. Anal. 208 (2004), no. 1, 140–162, DOI 10.1016/j.jfa.2003.10.005.MR2034295 (2004k:31020)

[32] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetricMarkov processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co.,Berlin, 1994. MR1303354 (96f:60126)

[33] M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Anal.1 (1992), no. 1, 1–35, DOI 10.1007/BF00249784. MR1245223 (95b:31009)

[34] Sheldon Goldstein, Random walks and diffusions on fractals, (Minneapolis, Minn., 1984),IMA Vol. Math. Appl., vol. 8, Springer, New York, 1987, pp. 121–129, DOI 10.1007/978-1-4613-8734-3 8. MR894545 (88g:60245)

[35] Alexander Grigor’yan and Andras Telcs, Two-sided estimates of heat kernels on metric mea-sure spaces, Ann. Probab. 40 (2012), no. 3, 1212–1284, DOI 10.1214/11-AOP645. MR2962091

[36] B. M. Hambly, Brownian motion on a homogeneous random fractal, Probab. Theory RelatedFields 94 (1992), no. 1, 1–38, DOI 10.1007/BF01222507. MR1189083 (93m:60163)

[37] B. M. Hambly, Brownian motion on a random recursive Sierpinski gasket, Ann. Probab. 25(1997), no. 3, 1059–1102, DOI 10.1214/aop/1024404506. MR1457612 (98i:60072)

[38] B. M. Hambly, Asymptotics for functions associated with heat flow on the Sierpinski carpet,Canad. J. Math. 63 (2011), no. 1, 153–180, DOI 10.4153/CJM-2010-079-7. MR2779136(2012f:35548)

[39] Ben M. Hambly, Takashi Kumagai, Shigeo Kusuoka, and Xian Yin Zhou, Transition densityestimates for diffusion processes on homogeneous random Sierpinski carpets, J. Math. Soc.Japan 52 (2000), no. 2, 373–408, DOI 10.2969/jmsj/05220373. MR1742797 (2001e:60158)

[40] B. M. Hambly, V. Metz, and A. Teplyaev, Self-similar energies on post-critically fi-nite self-similar fractals, J. London Math. Soc. (2) 74 (2006), no. 1, 93–112, DOI10.1112/S002461070602312X. MR2254554 (2007i:31011)


[41] K.E. Hare, B.A. Steinhurst, A. Teplyaev, D. Zhou, Disconnected Julia sets and gaps in thespectrum of Laplacians on symmetric finitely ramified fractals. Math. Res. Lett. 19 (2012),537–553.

[42] Masanori Hino, On singularity of energy measures on self-similar sets, Probab. TheoryRelated Fields 132 (2005), no. 2, 265–290, DOI 10.1007/s00440-004-0396-1. MR2199293(2007g:28010)

[43] Michael Hinz, 1-Forms and Polar Decomposition on Harmonic Spaces, Potential Anal. 38

(2013), no. 1, 261–279, DOI 10.1007/s11118-012-9272-2. MR3010780[44] M. Hinz, M. Rockner, A. Teplyaev, Vector analysis for Dirichlet forms and quasilinear PDE

and SPDE on metric measure spaces, preprint arXiv:1202.0743 (2012), to appear in Stochas-tic Process. Appl.

[45] M. Hinz, A. Teplyaev, Local Dirichlet forms, Hodge theory, and the Navier-Stokes equa-tions on topologically one-dimensional fractals, preprint arXiv:1206.6644 (2012), to appearin Trans. Amer. Math. Soc.

[46] M. Hinz, A. Teplyaev, Dirac and magnetic Schrodinger operators on fractals, preprintarXiv:1207.3077 (2012), to appear in J. Funct. Anal.

[47] Michael Hinz and Martina Zahle, Semigroups, potential spaces and applications to (S)PDE,Potential Anal. 36 (2012), no. 3, 483–515, DOI 10.1007/s11118-011-9238-9. MR2892585

[48] Jiaxin Hu and Martina Zahle, Schrodinger equations and heat kernel upper bounds onmetric spaces, Forum Math. 22 (2010), no. 6, 1213–1234, DOI 10.1515/FORUM.2010.064.MR2735894 (2012c:47118)

[49] M. Ionescu, L. Rogers, R.S. Strichartz, Pseudodifferential operators on fractals, to appear inRev. Math. Iberoam.

[50] Marius Ionescu, Luke G. Rogers, and Alexander Teplyaev, Derivations and Dirichlet formson fractals, J. Funct. Anal. 263 (2012), no. 8, 2141–2169, DOI 10.1016/j.jfa.2012.05.021.MR2964679


[52] Naotaka Kajino, Heat kernel asymptotics for the measurable Riemannian structure on theSierpinski gasket, Potential Anal. 36 (2012), no. 1, 67–115, DOI 10.1007/s11118-011-9221-5.

MR2886454[53] N. Kajino, On-diagonal oscillation of the heat kernels on post-critically self-similar fractals,

Probab. Th. Relat. Fields, in press.[54] Christopher J. Kauffman, Robert M. Kesler, Amanda G. Parshall, Evelyn A. Stamey, and

Benjamin A. Steinhurst, Quantum mechanics on Laakso spaces, J. Math. Phys. 53 (2012),no. 4, 042102, 18, DOI 10.1063/1.3702099. MR2953261

[55] J. Kigami, Harmonic metric and Dirichlet form on the Sierpinski gasket, Asymptotic prob-lems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990),Pitman Res. Notes Math. Ser., vol. 283, Longman Sci. Tech., Harlow, 1993, pp. 201–218.MR1354156 (96m:31014)


[57] Jun Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), no. 2,399–444, DOI 10.1016/S0022-1236(02)00149-0. MR2017320 (2004m:31010)

[58] Jun Kigami, Measurable Riemannian geometry on the Sierpinski gasket: the Kusuoka mea-sure and the Gaussian heat kernel estimate, Math. Ann. 340 (2008), no. 4, 781–804, DOI10.1007/s00208-007-0169-0. MR2372738 (2009g:60105)

[59] Jun Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem.Amer. Math. Soc. 199 (2009), no. 932, viii+94. MR2512802 (2010e:28007)

[60] Jun Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem.Amer. Math. Soc. 216 (2012), no. 1015, vi+132, DOI 10.1090/S0065-9266-2011-00632-5.MR2919892

[61] Jun Kigami and Michel L. Lapidus, Weyl’s problem for the spectral distribution of Laplacianson p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), no. 1, 93–125. MR1243717(94m:58225)

[62] Jun Kigami and Michel L. Lapidus, Self-similarity of volume measures for Laplacianson p.c.f. self-similar fractals, Comm. Math. Phys. 217 (2001), no. 1, 165–180, DOI10.1007/s002200000326. MR1815029 (2002j:35237)


[63] Takashi Kumagai, Estimates of transition densities for Brownian motion on nested frac-tals, Probab. Theory Related Fields 96 (1993), no. 2, 205–224, DOI 10.1007/BF01192133.MR1227032 (94e:60068)

[64] Takashi Kumagai and Karl-Theodor Sturm, Construction of diffusion processes on fractals,d-sets, and general metric measure spaces, J. Math. Kyoto Univ. 45 (2005), no. 2, 307–327.MR2161694 (2006i:60113)

[65] Shigeo Kusuoka, A diffusion process on a fractal, Probabilistic methods in mathematical

physics (Katata/Kyoto, 1985), Academic Press, Boston, MA, 1987, pp. 251–274. MR933827(89e:60149)

[66] Shigeo Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res.Inst. Math. Sci. 25 (1989), no. 4, 659–680, DOI 10.2977/prims/1195173187. MR1025071(91m:60142)

[67] S. Kusuoka, Lecture on diffusion process on nested fractals. Lecture Notes in Math. 156739–98, Springer-Verlag, Berlin, 1993.

[68] Shigeo Kusuoka and Zhou Xian Yin, Dirichlet forms on fractals: Poincare constant and resis-tance, Probab. Theory Related Fields 93 (1992), no. 2, 169–196, DOI 10.1007/BF01195228.MR1176724 (94e:60069)

[69] Nishu Lal and Michel L. Lapidus, Hyperfunctions and spectral zeta functions of Lapla-cians on self-similar fractals, J. Phys. A 45 (2012), no. 36, 365205, 14, DOI 10.1088/1751-8113/45/36/365205. MR2967908

[70] Michel L. Lapidus and Machiel van Frankenhuijsen, Fractal geometry, complex dimensionsand zeta functions. Geometry and spectra of fractal strings, Springer Monographs in Math-ematics, Springer, New York, 2006. MR2245559 (2007j:11001)

[71] Michel L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partialresolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), no. 2, 465–529,DOI 10.2307/2001638. MR994168 (91j:58163)

[72] Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 (1990),no. 420, iv+128. MR988082 (90k:60157)

[73] Z. Ma and M. Rockner, Introduction to the theory of (nonsymmetric) Dirichlet forms. Uni-versitext, Springer-Verlag, Berlin, 1992.

[74] Volker Metz, Potentialtheorie auf dem Sierpinski gasket, Math. Ann. 289 (1991), no. 2, 207–237, DOI 10.1007/BF01446569 (German). MR1092171 (92b:31008)

[75] Anders Pelander and Alexander Teplyaev, Infinite dimensional i.f.s. and smooth func-tions on the Sierpinski gasket, Indiana Univ. Math. J. 56 (2007), no. 3, 1377–1404, DOI10.1512/iumj.2007.56.2991. MR2333477 (2008j:28011)

[76] Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functionalanalysis, 2nd ed., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York,1980. MR751959 (85e:46002)

[77] Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier anal-ysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York,1975. MR0493420 (58 #12429b)

[78] Martin Reuter and Frank Saueressig, Fractal space-times under the microscope: a renormal-ization group view on Monte Carlo data, Journal of High Energy Physics 2011, 2011:12.

[79] L. C. G. Rogers and David Williams, Diffusions, Markov processes, and martingales. Vol.1: Foundations, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Proba-bility and Mathematical Statistics, John Wiley & Sons Ltd., Chichester, 1994. MR1331599(96h:60116)

[80] Luke G. Rogers and Robert S. Strichartz, Distribution theory on P.C.F. fractals, J. Anal.Math. 112 (2010), 137–191, DOI 10.1007/s11854-010-0027-y. MR2762999 (2012c:46082)

[81] C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar fractals,

Ann. Sci. Ecole Norm. Sup. (4) 30 (1997), no. 5, 605–673, DOI 10.1016/S0012-9593(97)89934-X (English, with English and French summaries). MR1474807 (98h:60118)

[82] Benjamin Steinhurst, Uniqueness of Locally Symmetric Brownian Motion on Laakso Spaces,Potential Anal. 38 (2013), no. 1, 281–298, DOI 10.1007/s11118-012-9273-1. MR3010781

[83] B. Steinhurst and A. Teplyaev, Symmetric Dirichlet forms and spectral analysis on Barlow-Evans fractals, preprint, 2011.

[84] Robert S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct.Anal. 174 (2000), no. 1, 76–127, DOI 10.1006/jfan.2000.3580. MR1761364 (2001i:31018)


[85] Robert S. Strichartz,Differential equations on fractals: A tutorial, Princeton University Press,Princeton, NJ, 2006. MR2246975 (2007f:35003)

[86] Robert S. Strichartz, A fractal quantum mechanical model with Coulomb potential, Com-mun. Pure Appl. Anal. 8 (2009), no. 2, 743–755, DOI 10.3934/cpaa.2009.8.743. MR2461574(2010c:81086)

[87] Alexander Teplyaev, Spectral analysis on infinite Sierpinski gaskets, J. Funct. Anal. 159(1998), no. 2, 537–567, DOI 10.1006/jfan.1998.3297. MR1658094 (99j:35153)

[88] Alexander Teplyaev, Gradients on fractals, J. Funct. Anal. 174 (2000), no. 1, 128–154, DOI10.1006/jfan.2000.3581. MR1761365 (2001h:31012)


[90] Alexander Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure,Canad. J. Math. 60 (2008), no. 2, 457–480, DOI 10.4153/CJM-2008-022-3. MR2398758(2008m:60066)

Mathematisches Institut, Friedrich-Schiller-Universitat Jena, Ernst-Abbe-Platz

2, 07737, Germany and Department of Mathematics, University of Connecticut, Storrs,

Connecticut 06269-3009

E-mail address: [email protected] and [email protected]

Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-

3009



Non-Regularly Varying and Non-Periodic Oscillation of theOn-Diagonal Heat Kernels on Self-Similar Fractals

Naotaka Kajino

Dedicated to my mother on the occasion of her 65th birthday

Abstract. Let pt(x, y) be the canonical heat kernel associated with a self-similar Dirichlet form on a self-similar fractal and let ds denote the spectraldimension of the Dirichlet space, so that tds/2pt(x, x) is uniformly boundedfrom above and below by positive constants for t ∈ (0, 1]. In this article it isproved that, under certain mild assumptions on pt(x, y), for a “generic” (inparticular, almost every) point x of the fractal, p(·)(x, x) neither varies regu-

larly at 0 (and hence the limit limt↓0 tds/2pt(x, x) does not exist) nor admits a

periodic function G : R → R such that pt(x, x) = t−ds/2G(− log t) + o(t−ds/2)as t ↓ 0. This result is applicable to most typical nested fractals (but not tothe d-dimensional standard Sierpinski gasket with d ≥ 2 at this moment) andall generalized Sierpinski carpets, and the assertion of non-regular variation isestablished also for post-critically finite self-similar fractals (possibly withoutgood symmetry) possessing a certain simple topological property.

Contents

1. Introduction2. Framework and main results3. Proof of Theorems 2.17 and 2.184. Post-critically finite self-similar fractals4.1. Harmonic structures and resulting self-similar Dirichlet spaces4.2. Cases with good symmetry and affine nested fractals4.3. Cases possibly without good symmetry5. Sierpinski carpetsReferences

1. Introduction

Heat kernels on fractals are believed to exhibit highly oscillatory behavior asopposed to the classical case of Riemannian manifolds. For example, as a gener-alization of the results of [7,12,30] for the standard Sierpinski gasket, Lindstrøm

2010 Mathematics Subject Classification. Primary 28A80, 60J35; Secondary 31C25, 37B10.Key words and phrases. Self-similar fractals, Dirichlet form, heat kernel, oscillation, short

time asymptotics, post-critically finite self-similar fractals, generalized Sierpinski carpets.The author was supported in part by SFB 701 of the German Research Council (DFG).


165

166 NAOTAKA KAJINO

Figure 1. Examples of self-similar fractals within the reach of themain results of this paper. From the left, two-dimensional level-3Sierpinski gasket, pentagasket (5-polygasket), Hata’s tree-like setand Sierpinski carpet

[32] constructed canonical Brownian motion on a certain large class of self-similarfractals called nested fractals, and Kumagai [29] proved that its transition density(heat kernel) p = pt(x, y) satisfies the two-sided sub-Gaussian estimate(1.1)

c1.1tds/2

exp

(−(ρ(x, y)dw

c1.1t

) 1dw−1

)≤ pt(x, y) ≤ c1.2

tds/2exp

(−(ρ(x, y)dw

c1.2t

) 1dw−1

).

Here c1.1, c1.2 ∈ (0,∞) are constants, ds ∈ [1,∞) and dw ∈ [2,∞) are also constantscalled the spectral dimension and the walk dimension of the fractal, respectively,and ρ is a suitably constructed geodesic metric on the fractal which is comparableto some power of the Euclidean metric1. Later Fitzsimmons, Hambly and Kumagai[10] extended these results to a larger class of self-similar fractals called affine nestedfractals. In particular, given an affine nested fractal K, for any x ∈ K we have

(1.2) c1.1 ≤ tds/2pt(x, x) ≤ c1.2, t ∈ (0, 1],

and then it is natural to ask how tds/2pt(x, x) behaves as t ↓ 0 and especiallywhether the limit

(1.3) limt↓0

tds/2pt(x, x)

exists or not. As Barlow and Perkins conjectured in [7, Problem 10.5] in the caseof the Sierpinski gasket, this limit was believed not to exist for most self-similarfractals, but this problem had remained open until the author’s recent paper [21].

It was proved in [21] that, under very weak assumptions on the affine nestedfractal K, the limit (1.3) does not exist for “generic” (hence almost every) x ∈ K,and that the same is true for any x ∈ K when K is either the d-dimensionalstandard (level-2) Sierpinski gasket with d ≥ 2 or the N -polygasket with N ≥ 3odd (see Figure 2 below). The proofs of these facts, however, heavily relied on thetwo important features of affine nested fractals — they are finitely ramified (i.e.can be made disconnected by removing finitely many points) and highly symmetric.In particular, the results of [21] were not applicable to self-similar fractals withoutthese properties like Hata’s tree-like set, which admits no isometric symmetry asshown in Proposition 4.17 below, and the Sierpinski carpet, which is infinitelyramified (see Figure 1).

1To be precise, the heat kernel estimate in [29] had been presented in terms of the Euclideanmetric, and the geodesic metric ρ was constructed later in [10].

OSCILLATION OF ON-DIAGONAL HEAT KERNELS ON SELF-SIMILAR FRACTALS 167

The purpose of this paper is twofold. First, we replace the assumptions offinite ramification and symmetry of the fractal with certain properties of the heatkernel which are expected to be much robuster in many cases. In particular, ourmain results imply the non-existence of the limit (1.3) for “generic” points x in thecases of Hata’s tree-like set and of the Sierpinski carpet. Secondly, we establish notonly the non-existence of the limit (1.3) but also more detailed descriptions of theoscillation of pt(x, x) as t ↓ 0 for “generic” points x of the self-similar fractal.

More specifically, let K be the self-similar set determined by a finite family{Fi}i∈S of injective contraction maps on a complete metric space, so that K is acompact metrizable topological space satisfying K =

⋃i∈S Fi(K), and let V0 be

the set of boundary points of K (see Definition 2.3 for the precise definition of V0).Assume K �= V0, let μ be a Borel measure on K satisfying μ(Fw1

◦ · · · ◦Fwm(K)) =

μw1· · ·μwm

for any w1 . . . wm ∈⋃

n∈NSn for some (μi)i∈S ∈ (0, 1)S with

∑i∈S μi =

1, and assume that (E ,F) is a self-similar symmetric regular Dirichlet form on

L2(K,μ) with resistance scaling factor r given by r =(μ2/ds−1i

)i∈S

for some ds ∈(0,∞) (see Definition 2.7 for details). Further assuming that (K,μ, E ,F) admits acontinuous heat kernel p = pt(x, y) and that the upper inequality of (1.1) holds fort ∈ (0, 1] for some dw ∈ (1,∞) and a suitable metric ρ on K satisfying μ(Bs(x, ρ)) ≤c1.3s

dsdw/2, (s, x) ∈ (0, 1] × K, Bs(x, ρ) := {y ∈ K | ρ(x, y) < s}, we establish thefollowing assertions as the main results of this paper:

(NRV) p(·)(x, x) does not vary regularly at 0 for “generic” x ∈ K, if

(1.4) lim supt↓0

pt(y, y)

pt(z, z)> 1 for some y, z ∈ K \ V0.

(NP) “Generic” x ∈ K does not admit a periodic function G : R → R such that

pt(x, x) = t−ds/2G(− log t) + o(t−ds/2) as t ↓ 0, if(1.5)

lim inft↓0

pt(y, y)

pt(z, z)> 1 for some y, z ∈ K \ V0.(1.6)

Note that we still have the on-diagonal estimate (1.2) in this situation as shownin Proposition 2.16 below, and recall (see e.g. [9, Section VIII.8]) that a Borelmeasurable function f : (0,∞) → (0,∞) is said to vary regularly at 0 if and onlyif the limit limt↓0 f(αt)/f(t) exists in (0,∞) for any α ∈ (0,∞). In particular, ifx ∈ K and p(·)(x, x) does not vary regularly at 0, then it also follows that the limit(1.3) does not exist. Note also that a log-periodic behavior of the form (1.5) isvalid when x is the fixed point of Fw1

◦ · · · ◦ Fwmfor some w1 . . . wm ∈

⋃n∈N

Sn

by Proposition 3.7 below, which is a slight generalization of [16, Theorems 4.6 and5.3]. Such a log-periodic behavior has been observed in various contexts of analysison fractals such as Laplacian eigenvalue asymptotics on self-similar sets discussedin [16,19,27] and long time asymptotics of the transition probability of the simplerandom walk on self-similar graphs treated in [13,28]. Contrary to these existentresults, the combination of (NRV) and (NP) asserts that pt(x, x) oscillates as t ↓ 0in a non-log-periodic but still non-regularly varying way for “generic” x ∈ K aslong as the assumption (1.6) is satisfied.

In fact, for (NP) we will actually prove the following stronger result: if (1.6)is satisfied, then for “generic” x ∈ K and any periodic function G : R → R,

(1.7) lim supt↓0

∣∣∣tds/2pt(x, x) −G(− log t)∣∣∣ ≥ My,z

2,

168 NAOTAKA KAJINO

where My,z := lim inft↓0 tds/2(pt(y, y) − pt(z, z)

)∈ (0,∞) with y, z as in (1.6).

The proof of (NRV) and (NP) relies only on the self-similarity of the Dirichletspace, the joint continuity of the heat kernel and its sub-Gaussian upper bound,which are all known to hold quite in general, and is free of extra a priori assumptions.Instead, however, we still need certain topological properties of the fractal K toverify (1.4) or (1.6). Roughly speaking, (1.4) can be verified if the local geometry ofK around Fw1

◦· · ·◦Fwm(x) is not the same for all x ∈ V0 and w1 . . . wm ∈

⋃n∈N

Sn

with Fw1◦ · · · ◦Fwm

(x) �∈ V0, and so can (1.6) if in addition the fractal K (or moreprecisely, the Dirichlet space (K,μ, E ,F)) has good symmetry.

For example, when K is the two-dimensional level-3 Sierpinski gasket in Figure1, the barycenter is contained in three of the cells {Fi(K) | i ∈ S} but each of theother points of

(⋃i∈S Fi(V0)

)\V0 is contained only in two of them, which together

with the dihedral symmetry of K implies (1.6). The pentagasket also satisfies (1.6)for exactly the same reason, whereas only (1.4) can be verified for Hata’s tree-likeset due to the lack of symmetry although (1.6) could actually be the case. For theSierpinski carpet, and its generalizations called generalized Sierpinski carpets, (1.6)is proved by using their symmetry under the isometries of the unit cube and thefact that some faces of the cells {Fi(K) | i ∈ S} are contained only in one cell butthe others in two cells.

Unfortunately, actually the author does not have any idea whether (1.4) and(1.6) are valid for the d-dimensional standard (level-2) Sierpinski gasket with d ≥ 2;the argument in the previous paragraph does not work in this case since any x ∈(⋃

m∈N

⋃w1...wm∈Sm Fw1

◦ · · · ◦Fwm(V0)

)\V0 has exactly two neighboring cells (see

Figure 2 below). In fact, it will be proved in a forthcoming paper [22] that p(·)(x, x)does not vary regularly at 0 for any x ∈ K for certain specific post-critically finiteself-similar fractals K where very detailed information on the eigenvalues of theLaplacian is known, including the d-dimensional standard Sierpinski gasket. Thisresult alone, however, does not exclude the possibility that (1.4) is not valid.

This article is organized as follows. In Section 2, we introduce our framework ofself-similar Dirichlet forms on self-similar sets and give the precise statements of ourmain results (NRV) and (NP) in Theorems 2.17 and 2.18, respectively. Section 3 isdevoted to the proof of Theorems 2.17 and 2.18, and then they are applied to post-critically finite self-similar fractals and generalized Sierpinski carpets in Sections4 and 5, respectively. In Section 4, after recalling basics of self-similar Dirichletforms on post-critically finite self-similar fractals in Subsection 4.1, we verify (1.6)for those with good symmetry such as affine nested fractals in Subsection 4.2,and (1.4) for those possibly without good symmetry such as Hata’s tree-like setin Subsection 4.3. Finally in Section 5, we first collect important facts concerninggeneralized Sierpinski carpets and their canonical self-similar Dirichlet form andthen verify (1.6) for them.

Notation. In this paper, we adopt the following notation and conventions.(1) N = {1, 2, 3, . . . }, i.e. 0 �∈ N.(2) The cardinality (the number of elements) of a set A is denoted by #A.(3) We set sup ∅ := 0, inf ∅ := ∞ and set a ∨ b := max{a, b} and a ∧ b := min{a, b}for a, b ∈ [−∞,∞]. All functions in this paper are assumed to be [−∞,∞]-valued.(4) For d ∈ N, Rd is always equipped with the Euclidean norm | · |.


(5) Let E be a topological space. The Borel σ-field of E is denoted by B(E). We

set C(E) := {u | u : E → R, u is continuous}, suppE [u] := {x ∈ E | u(x) �= 0} and‖u‖∞ := supx∈E |u(x)| for u ∈ C(E). For A ⊂ E, intE A denotes its interior in E.(6) Let E be a set, ρ : E × E → [0,∞) and x ∈ E. We set distρ(x,A) :=infy∈A ρ(x, y) for A ⊂ E and Br(x, ρ) := {y ∈ E | ρ(x, y) < r} for r ∈ (0,∞).

2. Framework and main results

In this section, we first introduce our framework of a self-similar set and aself-similar Dirichlet form on it, and then state the main theorems of this paper.

Let us start with standard notions concerning self-similar sets. We refer to[23, Chapter 1], [25, Section 1.2] and [19, Subsection 2.2] for details. Throughoutthis and the next sections, we fix a compact metrizable topological space K with#K ≥ 2, a non-empty finite set S and a continuous injective map Fi : K → K foreach i ∈ S. We set L := (K,S, {Fi}i∈S).

Definition 2.1. (1) Let W0 := {∅}, where ∅ is an element called the emptyword, let Wm := Sm = {w1 . . . wm | wi ∈ S for i ∈ {1, . . . ,m}} for m ∈ N and letW∗ :=

⋃m∈N∪{0} Wm. For w ∈ W∗, the unique m ∈ N ∪ {0} satisfying w ∈ Wm

is denoted by |w| and called the length of w. For i ∈ S and n ∈ N ∪ {0} we writein := i . . . i ∈ Wn.(2) We set Σ := SN = {ω1ω2ω3 . . . | ωi ∈ S for i ∈ N}, which is always equippedwith the product topology, and define the shift map σ : Σ → Σ by σ(ω1ω2ω3 . . . ) :=ω2ω3ω4 . . . . For i ∈ S we define σi : Σ → Σ by σi(ω1ω2ω3 . . . ) := iω1ω2ω3 . . . . Forω = ω1ω2ω3 . . . ∈ Σ and m ∈ N ∪ {0}, we write [ω]m := ω1 . . . ωm ∈ Wm.(3) For w = w1 . . . wm ∈ W∗, we set Fw := Fw1

◦ · · · ◦ Fwm(F∅ := idK), Kw :=

Fw(K), σw := σw1◦ · · · ◦ σwm

(σ∅ := idΣ) and Σw := σw(Σ), and if w �= ∅ thenw∞ ∈ Σ is defined by w∞ := www . . . in the natural manner.

Definition 2.2. L is called a self-similar structure if and only if there exists acontinuous surjective map π : Σ → K such that Fi ◦ π = π ◦ σi for any i ∈ S. Notethat such π, if exists, is unique and satisfies {π(ω)} =

⋂m∈N

K[ω]m for any ω ∈ Σ.

In what follows we always assume that L is a self-similar structure, so that#S ≥ 2 by #K ≥ 2 and π(Σ) = K. For A ⊂ K, the closure of A in K is denotedby A.

Definition 2.3. (1) We define the critical set C and the post-critical set P ofL by

(2.1) C := π−1(⋃

i,j∈S, i �=j Ki ∩Kj

)and P :=

⋃n∈N

σn(C).

L is called post-critically finite, or p.c.f. for short, if and only if P is a finite set.(2) We set V0 := π(P), Vm :=

⋃w∈Wm

Fw(V0) for m ∈ N and V∗ :=⋃

m∈NVm.

(3) We set KI := K \ V0, KIw := Fw(KI) for w ∈ W∗ and V∗∗ :=

⋃w∈W∗

Fw(V0).

V0 should be considered as the “boundary” of the self-similar set K; recall thatKw∩Kv = Fw(V0)∩Fv(V0) for any w, v ∈ W∗ with Σw∩Σv = ∅ by [23, Proposition1.3.5-(2)]. Note that Fw(V0) =

⋃n∈N

π(σw ◦ σn(C)) ∈ B(K) for any w ∈ W∗ by thecompactness of Σ. According to [23, Lemma 1.3.11], Vm−1 ⊂ Vm for any m ∈ N,and if V0 �= ∅ then V∗ is dense in K. Furthermore by [19, Lemma 2.11], KI

w is openin K and KI

w ⊂ KI for any w ∈ W∗.

170 NAOTAKA KAJINO

Definition 2.4. Let (μi)i∈S ∈ (0, 1)S satisfy∑

i∈S μi = 1. A Borel probabilitymeasure μ on K is called a self-similar measure on L with weight (μi)i∈S if andonly if the following equality (of Borel measures on K) holds:

(2.2) μ =∑i∈S

μiμ ◦ F−1i .

Let (μi)i∈S ∈ (0, 1)S satisfy∑

i∈S μi = 1. Then there exists a self-similarmeasure on L with weight (μi)i∈S . Indeed, if ν is the Bernoulli measure on Σ withweight (μi)i∈S , then ν ◦ π−1 is such a self-similar measure on L; see [23, Section1.4] for details. Moreover by [25, Theorem 1.2.7 and its proof], if K �= V0 and μ is aself-similar measure on L with weight (μi)i∈S , then μ(Kw) = μw and μ(Fw(V0)) = 0for any w ∈ W∗, where μw := μw1

· · ·μwmfor w = w1 . . . wm ∈ W∗ (μ∅ := 1). In

particular, a self-similar measure on L with given weight is unique if K �= V0.The following lemmas are immediate from the above-mentioned facts.

Lemma 2.5. Assume K �= V0, let μ be a self-similar measure on L with weight(μi)i∈S and let w ∈ W∗. Then

∫K

|u◦Fw|dμ = μ−1w

∫Kw

|u|dμ for any Borel measur-

able u : K → [−∞,∞]. In particular, if we set F ∗wu := u◦Fw for u : K → [−∞,∞],

then F ∗w defines a bounded linear operator F ∗

w : L2(K,μ) → L2(K,μ).

Lemma 2.6. Let w ∈ W∗. For u : K → [−∞,∞], define (Fw)∗u : K →[−∞,∞] by

(2.3) (Fw)∗u :=

{u ◦ F−1

w on Kw,

0 on K \Kw.

If u is Borel measurable then so is (Fw)∗u, and if K �= V0 in addition then∫K

|(Fw)∗u|dμ = μw

∫K

|u|dμ. In particular, if K �= V0, then (Fw)∗ defines a

bounded linear operator (Fw)∗ : L2(K,μ) → L2(K,μ).

Next we define the notion of a homogeneously scaled self-similar Dirichlet spaceand state its basic properties. The following definition is a special case of [19,Definition 3.3]. See [11, Section 1.1] for basic notions concerning regular Dirichletforms.

Definition 2.7 (Homogeneously scaled self-similar Dirichlet space). AssumeK �= V0. Let μ be a self-similar measure on L with weight (μi)i∈S , let ds ∈ (0,∞)

and set ri := μ2/ds−1i for i ∈ S. (E ,F) is called a homogeneously scaled self-similar

Dirichlet form on L2(K,μ) with spectral dimension ds if and only if it is a non-zerosymmetric regular Dirichlet form on L2(K,μ) satisfying the following conditions:

(SSDF1) u ◦ Fi ∈ F for any i ∈ S and any u ∈ F ∩ C(K).(SSDF2) For any u ∈ F ∩ C(K),

(2.4) E(u, u) =∑i∈S

1

riE(u ◦ Fi, u ◦ Fi).

(SSDF3) (Fi)∗u ∈ F for any i ∈ S and any u ∈ F ∩ C(K) with suppK [u] ⊂ KI .

If (E ,F) is a homogeneously scaled self-similar Dirichlet form on L2(K,μ) withspectral dimension ds, then (L, μ, E ,F) is called a homogeneously scaled self-similarDirichlet space with spectral dimension ds, and we call (μi)i∈S its weight.


In the rest of this section, we assume that (L, μ, E ,F) is a homogeneously scaledself-similar Dirichlet space with weight (μi)i∈S and spectral dimension ds. Then by[19, Lemma 5.5], (SSDF1) and (SSDF2) still hold if F ∩C(K) is replaced with F .

Lemma 2.8. (E ,F) is conservative (i.e. 1 ∈ F and E(1,1) = 0) and stronglylocal. Moreover, V0 �= ∅.

Proof. Since K is compact and (E ,F) is regular, F ∩ C(K) is dense in(C(K), ‖ · ‖∞), so that there exists u ∈ F ∩C(K) such that ‖1−u‖∞ ≤ 1/2. Thus1 = min{2u,1} ∈ F , and then it easily follows from (SSDF2) and

∑i∈S r−1

i =∑i∈S μ

1−2/ds

i > 1 that E(1,1) = 0. Moreover, (E ,F) is local by [19, Lemma 3.4],and it is also easily seen to be strongly local by virtue of its conservativeness.

Suppose V0 = ∅, so that π : Σ → K is a homeomorphism by [23, Proposition1.3.5-(3)]. Then since Kw is compact and open, we easily see from the conserva-tiveness of (E ,F) and [11, Theorem 1.4.2-(ii) and Exercise 1.4.1] that 1Kw

∈ F andE(1Kw

,1Kw) = 0 for any w ∈ W∗. This fact together with the denseness of the lin-

ear span of {1Kw}w∈W∗ in L2(K,μ) yields F = L2(K,μ) and E = 0, contradicting

the assumption that (E ,F) is non-zero. �

We need to introduce several geometric notions to formulate the assumption ofa sub-Gaussian heat kernel upper bound which is required for our main results. Werefer the reader to [25, Sections 1.1 and 1.3] and [19, Section 2] for further details.

Definition 2.9. (1) Let w, v ∈ W∗, w = w1 . . . wm, v = v1 . . . vn. We definewv ∈ W∗ by wv := w1 . . . wmv1 . . . vn (w∅ := w, ∅v := v). We write w ≤ v if andonly if w = vτ for some τ ∈ W∗. Note that Σw ∩ Σv = ∅ if and only if neitherw ≤ v nor v ≤ w.(2) A finite subset Λ of W∗ is called a partition of Σ if and only if Σw ∩ Σv = ∅ forany w, v ∈ Λ with w �= v and Σ =

⋃w∈Λ Σw.

(3) Let Λ1,Λ2 be partitions of Σ. We say that Λ1 is a refinement of Λ2, and writeΛ1 ≤ Λ2, if and only if for each w1 ∈ Λ1 there exists w2 ∈ Λ2 such that w1 ≤ w2.

Definition 2.10. (1) Set γw := μ1/dsw for w ∈ W∗. We define Λ1 := {∅},

(2.5) Λs := {w | w = w1 . . . wm ∈ W∗ \ {∅}, γw1...wm−1> s ≥ γw}

for each s ∈ (0, 1), and S := {Λs}s∈(0,1]. We call S the scale on Σ associated with(L, μ, E ,F).

(2) For each (s, x) ∈ (0, 1] × K, we define Λ0s,x := {w ∈ Λs | x ∈ Kw}, U (0)

s (x) :=⋃w∈Λ0

s,xKw, and inductively for n ∈ N,

(2.6) Λns,x := {w ∈ Λs | Kw ∩ U (n−1)

s (x) �= ∅} and U (n)s (x) :=

⋃w∈Λn

s,x

Kw.

Clearly lims↓0 min{|w| | w ∈ Λs} = ∞, and it is easy to see that Λs is a partitionof Σ for any s ∈ (0, 1] and that Λs1 ≤ Λs2 for any s1, s2 ∈ (0, 1] with s1 ≤ s2. Thesefacts together with [23, Proposition 1.3.6] imply that for any n ∈ N ∪ {0} and any

x ∈ K, {U (n)s (x)}s∈(0,1] is non-decreasing in s and forms a fundamental system of

neighborhoods of x in K. Note also that Λns,x and U

(n)s (x) are non-decreasing in

n ∈ N ∪ {0} for any (s, x) ∈ (0, 1] ×K.

172 NAOTAKA KAJINO

We would like to consider U(n)s (x) to be a “ball of radius s centered at x”. The

following definition formulates the situation where U(n)s (x) may be thought of as

actual balls with respect to a distance function on K.

Definition 2.11. (1) Let ρ : K ×K → [0,∞). For α ∈ (0,∞), ρ is called anα-qdistance on K if and only if ρα := ρ(·, ·)α is a distance on K. Moreover, ρ iscalled a qdistance on K if and only if it is an α-qdistance on K for some α ∈ (0,∞).(2) A qdistance ρ on K is called adapted to S if and only if there exist β1, β2 ∈ (0,∞)and n ∈ N such that for any (s, x) ∈ (0, 1] ×K,

(2.7) Bβ1s(x, ρ) ⊂ U (n)s (x) ⊂ Bβ2s(x, ρ).

If ρ is an α-qdistance on K adapted to S, then ρα is compatible with the original

topology of K, since {U (n)s (x)}s∈(0,1] is a fundamental system of neighborhoods of

x in the original topology of K.

Definition 2.12. We say that S is locally finite with respect to L, or simply(L, S) is locally finite, if and only if sup{#Λ1

s,x | (s, x) ∈ (0, 1] ×K} < ∞.

Note that by [23, Lemma 1.3.6], (L, S) is locally finite if and only if sup{#Λns,x |

(s, x) ∈ (0, 1] × K} < ∞ for any n ∈ N. The local finiteness of (L, S) is closelyrelated with local behavior of μ. In fact, we have the following proposition.

Proposition 2.13. Set γ := mini∈S γi and let n ∈ N ∪ {0}. Then for any(s, x) ∈ (0, 1] ×K,

(2.8) γdssds#Λns,x ≤ μ

(U (n)s (x)

)≤ sds#Λn

s,x.

In particular, for fixed n ∈ N, (L, S) is locally finite if and only if there exist

cV,n ∈ (0,∞) such that μ(U

(n)s (x)

)≤ cV,ns

ds for any (s, x) ∈ (0, 1] ×K,

Proof. We easily see from the definition (2.5) of Λs that

(2.9) γdssds < μw ≤ sds , s ∈ (0, 1], w ∈ Λs.

Since μ(Kw) = μw and μ(Fw(V0)) = 0 for any w ∈ W∗ by the assumption thatK �= V0, (2.9) implies that for any (s, x) ∈ (0, 1] ×K,

γdssds#Λns,x ≤

∑w∈Λn

s,x

μw =∑

w∈Λns,x

μ(Kw) = μ(U (n)s (x)

)≤ sds#Λn

s,x,

proving (2.8). The latter assertion is immediate from (2.8). �Next we prepare fundamental conditions for our main results concerning the

heat kernel of (K,μ, E ,F).

Definition 2.14 (CHK). We say that (K,μ, E ,F) satisfies (CHK), or sim-ply (CHK) holds, if and only if the Markovian semigroup {Tt}t∈(0,∞) on L2(K,μ)associated with (E ,F) admits a continuous integral kernel p, i.e. a continuous func-tion p = pt(x, y) : (0,∞) × K × K → R such that for any u ∈ L2(K,μ) and anyt ∈ (0,∞),

(2.10) Ttu =

∫K

pt(·, y)u(y)dμ(y) μ-a.e.

Such p, if exists, is unique and satisfies pt(x, y) = pt(y, x) ≥ 0 for any (t, x, y) ∈(0,∞)×K×K by a standard monotone class argument. p is called the (continuous)heat kernel of (K,μ, E ,F).


Definition 2.15 (CUHK). We say that (L, μ, E ,F) satisfies (CUHK), or sim-ply (CUHK) holds, if and only if (L, S) is locally finite, (K,μ, E ,F) satisfies (CHK)and there exist dw ∈ (1,∞), a (2/dw)-qdistance ρ on K adapted to S and c2.1, c2.2 ∈(0,∞) such that for any (t, x, y) ∈ (0, 1] ×K ×K,

(2.11) pt(x, y) ≤ c2.1t−ds/2 exp

(−c2.2

(ρ(x, y)2t

) 1dw−1

).

Note that (CUHK) remains the same if we replace t−ds/2 with 1/μ(B√

t(x, ρ))

in (2.11) and omit the condition that (L, S) is locally finite; indeed, this equivalenceeasily follows from Definition 2.11-(2), Proposition 2.13 and [19, Proposition 5.8].

Proposition 2.16. Suppose that (CUHK) holds. Then there exist c2.3, c2.4 ∈(0,∞) such that for any x ∈ K,

(2.12) c2.3 ≤ tds/2pt(x, x) ≤ c2.4, t ∈ (0, 1].

Proof. tds/2pt(x, x) ≤ c2.1 for any (t, x) ∈ (0, 1] ×K by (2.11). For the lowerbound we follow [24, Proof of Theorem 2.13]. Let ρ be the qdistance on K as inDefinition 2.15. Since (L, S) is assumed to be locally finite, Definition 2.11-(2) andProposition 2.13 easily imply that μ(Br(x, ρ)) ≤ c2.5r

ds for any (r, x) ∈ (0,∞)×Kfor some c2.5 ∈ (0,∞), and the same calculation as [24, Proof of Lemma 4.6-(1)]shows that

∫K\Bδ

√t(x,ρ)

pt(x, y)dμ(y) ≤ 1/2 for any (t, x) ∈ (0, 1] × K for some

δ ∈ (0,∞). Now for (t, x) ∈ (0, 1] × K, the conservativeness of (E ,F) yields∫Kpt(x, y)dμ(y) = 1, and hence

1

2≤ 1 −

∫K\Bδ

√t(x,ρ)

pt(x, y)dμ(y) =

∫Bδ

√t(x,ρ)

pt(x, y)dμ(y)

≤√μ(Bδ

√t(x, ρ)

) ∫K

pt(x, y)2dμ(y) ≤√

c2.5δdstds/2p2t(x, x)

by the symmetry and the semigroup property of the heat kernel p, proving thelower inequality in (2.12). �

Now we are in the stage of stating the main theorems of this paper. Note thatany Borel measure on K vanishing on V∗(∈ B(K)) is of the form ν ◦ π−1 with ν aBorel measure on Σ, since π|Σ\π−1(V∗) : Σ\π−1(V∗) → K \V∗ is a homeomorphism.Recall the following notions: a Borel measure ν on Σ is called σ-ergodic if and onlyif ν ◦ σ−1 = ν and ν(A)ν(Σ \A) = 0 for any A ∈ B(Σ) with σ−1(A) = A, and it issaid to have full support if and only if ν(U) > 0 for any non-empty open subset Uof Σ. Recall also that we set KI := K \ V0 and V∗∗ :=

⋃w∈W∗

Fw(V0).

Theorem 2.17. Suppose that (CUHK) holds and that

(2.13) lim supt↓0

pt(y, y)

pt(z, z)> 1 for some y, z ∈ KI .

Then there exists NRV ∈ B(K) satisfying V∗∗ ⊂ NRV and ν ◦π−1(NRV) = 0 for anyσ-ergodic finite Borel measure ν on Σ with full support, such that p(·)(x, x) does not

vary regularly at 0 for any x ∈ K \NRV. In particular, the limit limt↓0 tds/2pt(x, x)

does not exist for any x ∈ K \NRV.

Note that (2.13) does not hold if and only if limt↓0 pt(y, y)/pt(z, z) = 1 for anyy, z ∈ KI .

174 NAOTAKA KAJINO

Theorem 2.18. Suppose that (CUHK) holds and that

(2.14) lim inft↓0

pt(y, y)

pt(z, z)> 1 for some y, z ∈ KI .

Then there exists NP ∈ B(K) satisfying V∗∗ ⊂ NP and ν ◦ π−1(NP) = 0 for any σ-ergodic finite Borel measure ν on Σ with full support, such that for any x ∈ K \NP

and any periodic function G : R → R,

(2.15) lim supt↓0

∣∣∣tds/2pt(x, x) −G(− log t)∣∣∣ ≥ My,z

2,

where My,z := lim inft↓0 tds/2(pt(y, y) − pt(z, z)

)∈ (0,∞) with y, z as in (2.14).

Note that by (2.12), for each y, z ∈ K, lim inft↓0 pt(y, y)/pt(z, z) > 1 if and

only if lim inft↓0 tds/2(pt(y, y) − pt(z, z)

)∈ (0,∞).

Remark 2.19. Let y, z ∈ KI be as in (2.13) or (2.14). Then the sets NRV

in Theorem 2.17 and NP in Theorem 2.18 can be given explicitly in terms of (andhence can be determined solely by) y, z and π; see (3.8), Lemmas 3.10 and 3.12below.

The proof of Theorems 2.17 and 2.18 is given in the next section. As we will seein Sections 4 and 5, the conditions (2.13) and (2.14) are satisfied for many typicalexamples such as most nested fractals and all generalized Sierpinski carpets.

3. Proof of Theorems 2.17 and 2.18

Throughout this section, we fix a homogeneously scaled self-similar Dirichletspace (L = (K,S, {Fi}i∈S), μ, E ,F) with weight (μi)i∈S and spectral dimension dsand assume that (CUHK) holds with dw and ρ as in Definition 2.15.

Definition 3.1. Let U be a non-empty open subset of K. We define μ|U :=μ|B(U),

(3.1) FU := {u ∈ F ∩ C(K) | suppK [u] ⊂ U} and EU := E|FU×FU,

where the closure is taken in the Hilbert space F with inner product E1(u, v) :=E(u, v) +

∫Kuvdμ. (EU ,FU ) is called the part of the Dirichlet form (E ,F) on U .

Since u = 0 μ-a.e. on K \ U for any u ∈ FU , we can regard FU as a linearsubspace of L2(U, μ|U ) in the natural manner. Under this identification, we havethe following lemma.

Lemma 3.2. Let U be a non-empty open subset of K. Then (EU ,FU ) is astrongly local regular Dirichlet form on L2(U, μ|U ) whose associated Markoviansemigroup {TU

t }t∈(0,∞) admits a unique continuous integral kernel pU = pUt (x, y) :(0,∞) × U × U → R, called the Dirichlet heat kernel on U , similarly to (2.10).Moreover, 0 ≤ pUt (x, y) = pUt (y, x) ≤ pt(x, y) for any (t, x, y) ∈ (0,∞) × U × U .

Proof. Recall Lemma 2.8. The regularity of (E ,F) yields that of (EU ,FU )by (3.1) and [11, Lemma 1.4.2-(ii)], and the strong locality of (E ,F) impliesthat of (EU ,FU ). Since (E ,F) is conservative, a continuous integral kernel pU

of {TUt }t∈(0,∞) exists by [19, Lemma 7.11-(2)] and (CUHK), and a monotone class

argument immediately shows the uniqueness of such pU . Finally, the last assertioneasily follows from [25, (C.2)] and a monotone class argument again. �


Lemma 3.3. Let U be a non-empty open subset of K. Then for any (t, x, y) ∈(0,∞) × U × U ,

(3.2) pt(x, y) − pUt (x, y) ≤ sups∈[t/2, t]

supz∈U\U

ps(x, z) + sups∈[t/2, t]

supz∈U\U

ps(z, y).

Proof. This is immediate from [15, Theorem 5.1] (or [14, Theorem 10.4]), thecontinuity of the heat kernels pt(x, y) and pUt (x, y) and the compactness of U . �

Lemma 3.4. Let w ∈ W∗. Then for any (t, x, y) ∈ (0,∞) ×KI ×KI ,

(3.3) (γ2wt)

ds/2pKI

w

γ2wt

(Fw(x), Fw(y)

)= tds/2pK

I

t (x, y).

Proof. Fw|KI : KI → KIw is clearly a homeomorphism, and F ∗

w defines abijection F ∗

w : L2(KIw, μ|KI

w) → L2(KI , μ|KI ) such that F ∗

w(FKIw) = FKI by [19,

Lemma 5.5]. Moreover, γdsw

∫KI (F

∗wu)2dμ =

∫KI

wu2dμ for any u ∈ L2(KI

w, μ|KIw)

by Lemma 2.5 and γds−2w E(F ∗

wu, F∗wu) = E(u, u) for any u ∈ FKI

wby (SSDF2). It

easily follows from these facts and [11, Lemma 1.3.4-(i)] that F ∗wT

KIw

γ2wt = TKI

t F ∗w for

any t ∈ (0,∞), which together with the uniqueness of the continuous heat kernels

pKIw and pK

I

implies (3.3). �Lemma 3.5. There exists c3.1 ∈ (0,∞) such that for any x ∈ K and any

w ∈ W∗,

(3.4) distρ(Fw(x), Fw(V0)

)≥ c3.1γw distρ(x, V0).

Proof. Let β1, β2 ∈ (0,∞) and n ∈ N be as in Definition 2.11-(2) for theqdistance ρ, let x ∈ K, w ∈ W∗ and set δ := distρ(x, V0). The assertion is obvious

for x ∈ V0. Assuming x ∈ K \ V0 = KI , by K = U(n)1 (x) ⊂ Bβ2

(x, ρ) we haveρ(x, y) < β2 for any y ∈ K and hence δ ∈ (0, β2) (recall that V0 �= ∅ by Lemma

2.8), and U(n)δ/β2

(x) ∩ V0 = ∅ since U(n)δ/β2

(x) ⊂ Bδ(x, ρ) ⊂ KI . Then an induction in

k easily shows that Λkγwδ/β2,Fw(x) =

{wv∣∣ v ∈ Λk

δ/β2,x

}for any k ∈ {0, . . . , n}, and

hence

Bγwδβ1/β2(Fw(x), ρ) ⊂ U

(n)γwδ/β2

(Fw(x)) = Fw

(U

(n)δ/β2

(x))⊂ KI

w = Kw \ Fw(V0).

Thus ρ(Fw(x), y) ≥ γwδβ1/β2 = (β1/β2)γw distρ(x, V0) for any y ∈ Fw(V0) and(3.4) follows with c3.1 := β1/β2. �

Lemma 3.6. There exist c3.2, c3.3 ∈ (0,∞) such that for any (t, x) ∈ (0, 1]×KI

and any w ∈ W∗,∣∣∣(γ2wt)

ds/2pγ2wt

(Fw(x), Fw(x)

)− tds/2pt(x, x)

∣∣∣≤ c3.2 exp

(−c3.3 distρ(x, V0)

2dw−1 t−

1dw−1

).

(3.5)

Proof. We easily see from (2.11), Lemmas 3.3 and 3.5 that, with c3.2 :=

21+ds/2c2.1 and c3.3 := c2.2c2

dw−1

3.1 , for any w ∈ W∗ and any (t, x) ∈ (0, γ−2w ] ×KI ,

0 ≤ (γ2wt)

ds/2(pγ2

wt

(Fw(x), Fw(x)

)− p

KIw

γ2wt

(Fw(x), Fw(x)

))≤ c3.2 exp

(−c3.3 distρ(x, V0)

2dw−1 t−

1dw−1

),

(3.6)

which together with Lemma 3.4 immediately shows (3.5). �

176 NAOTAKA KAJINO

Proposition 3.7. Let w ∈ W∗ \ {∅}, set xw := π(w∞) and suppose xw ∈ KI .Then there exist constants c3.4, c3.5 ∈ (0,∞) independent of w and a continu-ous log(γ−2

w )-periodic function Gw : R → (0,∞) such that for any t ∈(0, 1 ∧

distρ(xw, V0)2],

(3.7)∣∣∣tds/2pt(xw, xw) −Gw(− log t)

∣∣∣ ≤ c3.4 exp(−c3.5 distρ(xw, V0)

2dw−1 t−

1dw−1

).

Proof. Note that Fw(xw) = xw ∈ KIw. Since pK

Iw ≤ pK

I ≤ p on (0,∞)×KIw×

KIw by KI

w ⊂ KI , [25, (C.2)] and a monotone class argument, we see from (3.6) and

Lemma 3.4 that for any t ∈ (0, γ−2w ], (γ2

wt)ds/2pK

I

γ2wt(xw, xw) − tds/2pK

I

t (xw, xw) is

subject to the same upper and lower bounds as those in (3.6) with x = xw. On the

other hand, by [8, Theorem 2.1.4], the generator ΔKI of {TKI

t }t∈(0,∞) has compact

resolvent, pKI

admits the eigenfunction expansion [8, (2.1.4)], and it easily follows

from these facts that pKI

t (x, y) ≤(2dsc22.4e

λ01

)e−λ0

1t for any t ∈ [1,∞), where λ01

denotes the smallest eigenvalue of −ΔKI . Moreover, we have λ01 > 0; indeed, if

E(u, u) = 0 for some u ∈ FKI \ {0}, then for any w ∈ W∗, (Fw)∗u ∈ FKIw

⊂FKI by [19, Lemma 5.5] and E

((Fw)∗u, (Fw)∗u

)= 0 by (SSDF2), contradicting

dim ker ΔKI = dim{v ∈ FKI | E(v, v) = 0} < ∞. Now exactly the same argumentas [16, Proof of Theorem 4.6] easily shows the existence of a continuous log(γ−2

w )-

periodic function Gw : R → R satisfying (3.7) with pKI

t in place of pt (see also[23, Proof of Theorem B.4.3] for the remainder estimate). Then (3.7) follows byusing (3.6) with xw and ∅ in place of x and w, respectively, and Proposition 2.16implies that Gw is (0,∞)-valued. �

In the rest of this section, we fix the following setting:

(3.8)

Let y, z ∈ KI , ωy ∈ π−1(y) and ωz ∈ π−1(z). Define wn := [ωy]n,vn := [ωz]n, xn := π((wnvn)∞) and xn := π((vnwn)∞) for n ∈ N.Also set n0 := 1 + sup{n ∈ N | {xn, xn} �⊂ KI} and for n ≥ n0 letGn denote the periodic function Gwnvn given in Proposition 3.7 forw = wnvn. Finally, set D := infn≥n0

distρ(xn, V0) ∧ distρ(xn, V0).

Note that n0 < ∞ and D > 0 by limn→∞ xn = y ∈ KI and limn→∞ xn = z ∈ KI .

Lemma 3.8. Let ε ∈ (0,∞), δ ∈ (0, 1 ∧ D2] and α ∈ (0, 1], and suppose that

c3.4 exp(−c3.5(D

2/δ)1

dw−1)

≤ ε/2. Then there exists n1 ≥ n0 such that for anyn ≥ n1 and any t ∈ [αδ, δ],

(3.9)∣∣∣tds/2pt(y, y) −Gn(− log t)

∣∣∣ ≤ ε,∣∣∣tds/2pt(z, z) −Gn

(− log(γ2

wnt))∣∣∣ ≤ ε.

Proof. Let n ≥ n0 and let Gn be the periodic function Gvnwngiven in Propo-

sition 3.7 for w = vnwn. Then limt↓0∣∣Gn(− log t)−Gn

(− log(γ2

wnt))∣∣ = 0 by Lemma

3.6 and Proposition 3.7, and hence Gn = Gn

(· − log(γ2

wn))

in view of the fact that

Gn and Gn are both log(γ−2wnvn)-periodic.

Since limn→∞ xn = y, limn→∞ xn = z and the heat kernel p is uniformlycontinuous on [αδ, δ] ×K ×K, we can choose n1 ≥ n0 so that

(3.10) tds/2∣∣pt(xn, xn) − pt(y, y)

∣∣ ≤ ε

2and tds/2

∣∣pt(xn, xn) − pt(z, z)∣∣ ≤ ε

2


for any n ≥ n1 and any t ∈ [αδ, δ]. Now for such n and t, Proposition 3.7, (3.10),

c3.4 exp(−c3.5(D

2/δ)1

dw−1)≤ ε/2 and Gn = Gn

(·− log(γ2

wn))

together immediatelyyield (3.9). �

Lemma 3.9. Assume lim inft↓0 pt(y, y)/pt(z, z) > 1. Let ε ∈ (0,∞), δ0 ∈(0, 1∧D2], α ∈ (0, 1] and set My,z := lim inft↓0 t

ds/2(pt(y, y)−pt(z, z)

). Then there

exist δ ∈ (0, δ0] and n2 ≥ n0 such that for any n ≥ n2 and any t ∈ [αδ, δ],

(3.11) Gn(− log t) ≥ minR

Gn + My,z − ε.

Proof. Choose δ ∈ (0, δ0] so that inft∈(0,δ] tds/2(pt(y, y) − pt(z, z)

)≥ My,z −

ε/3 and c3.4 exp(−c3.5(D

2/δ)1

dw−1)≤ ε/6. Also let n1 ≥ n0 be as in Lemma 3.8

for ε/3, δ, α and set n2 := n1. Then by (3.9), for any n ≥ n2 and any t ∈ [αδ, δ],

Gn(− log t) ≥ tds/2pt(y, y) − ε/3 ≥ tds/2pt(z, z) + My,z − 2ε/3

≥ Gn

(− log(γ2

wnt))

+ My,z − ε ≥ minR

Gn + My,z − ε,

completing the proof. �

Lemma 3.10. Let q ∈ KI and define Nq ⊂ K by

(3.12) Nq :=

{x ∈ K

∣∣∣∣ π(σmk(ω)) does not converge to q as k → ∞ for anyω ∈ π−1(x) and any strictly increasing {mk}k∈N ⊂ N

}.

Then Nq ∈ B(K), V∗∗ ⊂ Nq and ν ◦ π−1(Nq) = 0 for any σ-ergodic finite Borelmeasure ν on Σ with full support.

Proof. V∗ ⊂ Nq since σm(π−1(Vm)) = P for m ∈ N∪{0} by [23, Proposition1.3.5-(1)]. Noting that π|Σ\π−1(V∗) : Σ \ π−1(V∗) → K \ V∗ is a homeomorphism,

we get Nq = V∗ ∪ π({ω ∈ Σ \ π−1(V∗) | lim infm→∞ ρ(π(σm(ω)), q) > 0}) ∈ B(K).

Let x ∈ V∗∗ \ V∗, so that x = Fw(π(ω)) for some w ∈ W∗ and ω ∈ π−1(V0 \ V∗).By π(ω) ∈ V0, π−1(V0) = P and the compactness of Σ, there exist τ ∈ Σ and{ωn}n∈N ⊂ P such that π(ωn) → π(ω) and ωn → τ as n → ∞, but then π(ω) =

π(τ ) ∈ K \V∗ and hence ω = τ ∈ P. Thus σm(ω) ∈ σm(P) ⊂ P and π(σm(ω)) ∈ V0

for any m ∈ N and therefore x ∈ Nq on account of π−1(x) = {σw(ω)}, provingV∗∗ ⊂ Nq. Finally, since Nq can be written as⋃

n∈N

{x ∈ K

∣∣∣ limm→∞

distρ(π(σm(ω)),K \B1/n(q, ρ)

)= 0 for any ω ∈ π−1(x)

},

the last assertion follows in exactly the same way as [21, Proposition 3.2]. �

Lemma 3.11. Let x ∈ K, ω ∈ π−1(x), n ≥ n0 and let {mk}k∈N ⊂ N be strictlyincreasing and satisfy limk→∞ π(σmk(ω)) = xn. Let ε ∈ (0,∞), let δ ∈ (0, 1 ∧D2]

satisfy c3.6 exp(−c3.7(D

2/δ)1

dw−1)< ε/3 with c3.6 := c3.2∨c3.4 and c3.7 := c3.3∧c3.5,

and let α ∈ (0, 1]. Then there exists k1 ∈ N such that for any k ≥ k1 and anyt ∈ [αδ, δ],

(3.13)∣∣∣(γ2

[ω]mkt)ds/2pγ2

[ω]mkt(x, x) −Gn(− log t)

∣∣∣ ≤ ε.

178 NAOTAKA KAJINO

Proof. Since limk→∞ π(σmk(ω)) = xn and the heat kernel p is uniformlycontinuous on [αδ, δ] ×K ×K, we can choose k1 ∈ N so that for any k ≥ k1,

tds/2∣∣pt(π(σmk(ω)), π(σmk(ω))

)− pt(xn, xn)

∣∣ ≤ ε

3, t ∈ [αδ, δ],(3.14)

c3.6 exp(−c3.7 distρ

(π(σmk(ω)), V0

) 2dw−1 δ−

1dw−1

)≤ ε

3.(3.15)

Now for such k and t, noting that π(σmk(ω)) ∈ K \Nxn⊂ KI by Lemma 3.10, we

obtain (3.13) from Lemma 3.6 with π(σmk(ω)) and [ω]mkin place of x and w, (3.15),

(3.14), Proposition 3.7 with w = wnvn, and c3.6 exp(−c3.7(D

2/δ)1

dw−1)< ε/3. �

Lemma 3.12. Recalling (3.12), set N :=⋃

k≥n0

⋂n≥k Nxn

and let x ∈ K \N .

(1) If lim supt↓0 pt(y, y)/pt(z, z) > 1, then p(·)(x, x) does not vary regularly at 0.(2) If lim inft↓0 pt(y, y)/pt(z, z) > 1, then (2.15) holds for any periodic function

G : R → R, where My,z := lim inft↓0 tds/2(pt(y, y) − pt(z, z)

).

Proof. Let ω ∈ π−1(x) and let c2.3, c2.4 ∈ (0,∞) be as in Proposition 2.16.(1) Set M := lim supt↓0 pt(y, y)/pt(z, z) − 1 and ε := c2.3M/10. Choose δ ∈ (0, 1 ∧D2] so that c3.6 exp

(−c3.7(D

2/δ)1

dw−1)< ε/3 and pδ(y, y)/pδ(z, z) ≥ 1 +M/2, and

let n1 ≥ n0 be as in Lemma 3.8 for these ε, δ and α = 1. By x ∈ K \ N wecan take n ≥ n1 such that x ∈ K \ Nxn

, and then limk→∞ π(σmk(ω)) = xn forsome strictly increasing {mk}k∈N ⊂ N. Let k1 ∈ N be as in Lemma 3.11 for thesex, ω, n, {mk}k∈N, ε, δ and α = γ2

wn. Then by (3.13), (3.9) and (2.12), for any k ≥ k1,(

γ2[ω]mk

δ)ds/2

pγ2[ω]mk

δ(x, x) −(γ2[ω]mk

γ2wn

δ)ds/2

pγ2[ω]mk

γ2wn

δ(x, x)

≥ Gn(− log δ) −Gn

(− log(γ2

wnδ))− 2ε ≥ δds/2pδ(y, y) − δds/2pδ(z, z) − 4ε

≥ (1 + M/2 − 1)δds/2pδ(z, z) − 4ε ≥ c2.3M/2 − 4ε = ε,

which together with (2.12) yields, by letting k → ∞,

(3.16) lim supt↓0

tds/2pt(x, x)

(γ2wn

t)ds/2pγ2wn

t(x, x)≥ 1 +

ε

c2.4> 1.

Now suppose that p(·)(x, x) varies regularly at 0, so that by [9, Section VIII.8,Lemma 1], pt(x, x) = tβL(t) for any t ∈ (0,∞) for some β ∈ R and L : (0,∞) →(0,∞) varying slowly at 0 (i.e. such that limt↓0 L(αt)/L(t) = 1 for any α ∈ (0,∞)).

Then (2.12) yields c2.3 ≤ tβ+ds/2L(t) ≤ c2.4 for t ∈ (0, 1], which together with[9, Section VIII.8, Lemma 2] implies β = −ds/2. It follows that tds/2pt(x, x) = L(t)varies slowly at 0, which contradicts (3.16). Thus pt(x, x) does not vary regularlyat 0.(2) Let G : R → R be T -periodic with T ∈ (0,∞). Let ε ∈ (0,∞), let δ0 ∈ (0, 1∧D2]

be such that c3.6 exp(−c3.7(D

2/δ0)1

dw−1)< ε/3, and let δ ∈ (0, δ0] and n2 ≥ n0

be as in Lemma 3.9 for ε, δ0 and α = e−T . By x ∈ K \ N we can choose n ≥ n2

so that γ2wnvn ≤ e−T and x ∈ K \ Nxn

, and then limk→∞ π(σmk(ω)) = xn forsome strictly increasing {mk}k∈N ⊂ N. Let k1 ∈ N be as in Lemma 3.11 for thesex, ω, n, {mk}k∈N, ε, δ and α = γ2

wnvn . Since Gn is log(γ−2wnvn)-periodic, minRGn =

Gn(− log tn,1) for some tn,1 ∈ [γ2wnvnδ, δ], and then there exist tn,0 ∈ [e−T δ, δ]

and ln ∈ N ∪ {0} such that − log tn,1 = − log tn,0 + lnT . Now it follows from


G = G(· + lnT ), (3.13) with α = γ2wnvn , (3.11) with α = e−T and Gn(− log tn,1) =

minR Gn that for any k ≥ k1,∑j∈{0,1}

(−1)j((

γ2[ω]mk

tn,j)ds/2pγ2

[ω]mktn,j

(x, x) −G(− log(γ2

[ω]mktn,j)

))=(γ2[ω]mk

tn,0)ds/2pγ2

[ω]mktn,0

(x, x) −(γ2[ω]mk

tn,1)ds/2pγ2

[ω]mktn,1

(x, x)

≥ Gn(− log tn,0) −Gn(− log tn,1) − 2ε ≥ My,z − 3ε,

which implies

(3.17) maxj∈{0,1}

∣∣∣(γ2[ω]mk

tn,j)ds/2pγ2

[ω]mktn,j

(x, x)−G(− log(γ2

[ω]mktn,j)

)∣∣∣ ≥ My,z

2−3

2ε.

Letting k → ∞ and then ε ↓ 0 in (3.17) shows (2.15). �

Proof of Theorems 2.17 and 2.18. Setting NRV := NP := N with N ∈B(K) as in Lemma 3.12, we conclude Theorems 2.17 and 2.18 from Lemmas 3.10and 3.12. �

4. Post-critically finite self-similar fractals

In this and the next sections, we apply Theorems 2.17 and 2.18 to concrete ex-amples. First in this section, we consider the case of post-critically finite self-similarfractals, and the next section treats the case of generalized Sierpinski carpets.

Throughout this section, we assume that L = (K,S, {Fi}i∈S) is a post-criticallyfinite self-similar structure with K connected and #K ≥ 2; see [23, Theorem 1.6.2]for a simple equivalent condition for K to be connected. In particular, 2 ≤ #V0 < ∞and V∗ is countably infinite and dense in K, so that K �= V0 = V0 and V∗∗ = V∗.

4.1. Harmonic structures and resulting self-similar Dirichlet spaces.First in this subsection, we briefly describe the construction of a homogeneouslyscaled self-similar Dirichlet space over K; see [23, Chapter 3] for details. LetD = (Dxy)x,y∈V0

be a real symmetric matrix of size #V0 (which we also regard asa linear operator on RV0) such that

(D1) {u ∈ RV0 | Du = 0} = R1V0,

(D2) Dxy ≥ 0 for any x, y ∈ V0 with x �= y.

We define E(0)(u, v) := −∑

x,y∈V0Dxyu(y)v(x) for u, v ∈ RV0 , so that (E(0),RV0)

is a Dirichlet form on L2(V0,#). Furthermore let r = (ri)i∈S ∈ (0,∞)S and define

(4.1) E(m)(u, v) :=∑

w∈Wm

1

rwE(0)(u ◦ Fw|V0

, v ◦ Fw|V0), u, v ∈ RVm

for each m ∈ N, where rw := rw1rw2

· · · rwmfor w = w1w2 . . . wm ∈ Wm (r∅ := 1).

Definition 4.1. The pair (D, r) with D and r as above is called a harmonicstructure on L if and only if E(0)(u, u) = infv∈RV1 , v|V0

=u E(1)(v, v) for any u ∈ RV0 ;

note that then E(m)(u, u) = minv∈RVm+1 , v|Vm=u E(m+1)(v, v) for any m ∈ N ∪ {0}

and any u ∈ RVm . If r ∈ (0, 1)S in addition, then (D, r) is called regular.

In the rest of this section, we assume that (D, r) is a regular harmonic structure

on L. Let dH ∈ (0,∞) be such that∑

i∈S rdHi = 1, set μi := rdH

i for i ∈ S and letμ be the self-similar measure on L with weight (μi)i∈S . We set ds := 2dH/(dH +1),

180 NAOTAKA KAJINO

so that ri = μ2/ds−1i for each i ∈ S. In this case, {E(m)(u|Vm

, u|Vm)}m∈N∪{0} is

non-decreasing and hence has the limit in [0,∞] for any u ∈ C(K). Then we define

F := {u ∈ C(K) | limm→∞ E(m)(u|Vm, u|Vm

) < ∞},E(u, v) := limm→∞ E(m)(u|Vm

, v|Vm) ∈ R, u, v ∈ F ,

(4.2)

so that (E ,F) is easily seen to satisfy the conditions (SSDF1), (SSDF2) and (SSDF3)of Definition 2.7. By [23, Theorem 3.3.4], (E ,F) is a resistance form on K whoseresistance metric R : K × K → [0,∞) is compatible with the original topologyof K, and then [26, Corollary 6.4 and Theorem 9.4] imply that (E ,F) is a non-zero symmetric regular Dirichlet form on L2(K,μ); see [23, Definition 2.3.1] or[26, Definition 3.1] for the definition of resistance forms and their resistance met-rics. Thus (L, μ, E ,F) is a homogeneously scaled self-similar Dirichlet space withweight (μi)i∈S and spectral dimension ds. Note that ds ∈ (0, 2) in this case.

Remark 4.2. As described in [23, Sections 3.1–3.3], even for a non-regularharmonic structure (D, r = (ri)i∈S) on L, in a similar way as above we can stillconstruct a resistance form (E ,F) on (a certain proper Borel subset of) K whichsatisfies (suitable modifications of) (SSDF1), (SSDF2) and (SSDF3). Such (D, r),however, does not give rise to a homogeneously scaled self-similar Dirichlet spacesince ri < 1 for some i ∈ S by [23, Proposition 3.1.8] and rj ≥ 1 for some j ∈ S bythe non-regularity of (D, r). This is why we have assumed from the beginning thatour harmonic structure (D, r) on L is regular.

Let S = {Λs}s∈(0,1] be the scale on Σ associated with (L, μ, E ,F) and set dw :=dH+1. Then (L, S) is locally finite by [23, Lemma 4.2.3] and [25, Lemma 1.3.6], andby [23, Proof of Lemma 4.2.4] there exists cR ∈ (0,∞) such that R(x, y) ≥ cRs

2/dw

for any s ∈ (0, 1], any w, v ∈ Λs with Kw ∩ Kv = ∅ and any (x, y) ∈ Kw × Kv,which together with [23, Lemma 3.3.5] easily implies that Rdw/2 is adapted to S.Finally, (CHK) holds by [26, Theorem 10.4] (or by [23, Section 5.1]) and so does(CUHK) with ρ := Rdw/2 by [26, Theorem 15.10] (see also [21, Lemma 2.5]).

Thus in order to apply Theorems 2.17 and 2.18 to the present case, it sufficesto verify (2.13) and (2.14). In the rest of this section, we give a few criteria for(2.13) and (2.14) and apply them to concrete examples. In Subsection 4.2 we treatthe case where (L, (D, r), μ) possesses certain good symmetry, including the case ofaffine nested fractals, and Subsection 4.3 presents alternative criteria for (2.13) and(2.14) which are applicable for some cases with weaker (or even without) symmetry.

The following definitions play central roles in the rest of this section.

Definition 4.3. (1) We define the symmetry group G of (L, (D, r), μ) by

(4.3) G :=

{g

∣∣∣∣ g is a homeomorphism from K to itself, g(V0) = V0, μ◦g = μ,u ◦ g, u ◦ g−1 ∈ F and E(u ◦ g, u ◦ g) = E(u, u) for any u ∈ F

},

which clearly forms a subgroup of the group of homeomorphisms of K.(2) For each x ∈ V∗, we define

(4.4) mx := min{m ∈ N ∪ {0} | x ∈ Vm} and nx := #{w ∈ Wmx| x ∈ Kw}.

4.2. Cases with good symmetry and affine nested fractals. Assumingcertain good symmetry of (L, (D, r), μ), we have the following criterion for (2.13)and (2.14), which is an immediate consequence of [21, Remark 6.4].


Proposition 4.4. Let q ∈ V0 and suppose that {g(q) | g ∈ G} = V0 and thatri = r for any i ∈ S for some r ∈ (0, 1). Then for each x ∈ V∗, nx = #{w ∈ Wm |x ∈ Kw} for any m ≥ mx and limt↓0 pt(x, x)/pt(q, q) = n−1

x . In particular, if

(4.5) ny �= nz for some y, z ∈ V∗ \ V0

in addition, then the conditions (2.13) and (2.14) are satisfied.

Next we recall the definition of affine nested fractals and apply Proposition 4.4to them. Throughout the rest of this subsection, we assume the following:

(4.6)d ∈ N, K is a compact subset of Rd, and for each i ∈ S, Fi = fi|Kfor some contractive similitude fi on Rd with contraction ratio αi.

Recall that f : Rd → Rd is called a contractive similitude on Rd if and only if thereexist α ∈ (0, 1), U ∈ O(d) and b ∈ Rd such that f(x) = αUx + b for any x ∈ Rd.Then such α is called the contraction ratio of f . According to [23, Theorem 1.2.3],any finite family of contractive similitudes on Rd actually gives rise to a self-similarstructure satisfying (4.6) by taking the associated self-similar set.

Notation. For x, y ∈ Rd with x �= y, let gxy : Rd → Rd denote the reflectionin the hyperplane Hxy := {z ∈ Rd | |z − x| = |z − y|}.

Definition 4.5. (1) A homeomorphism g : K → K is called a symmetry of Lif and only if, for any m ∈ N∪ {0}, there exists an injective map g(m) : Wm → Wm

such that g(Fw(V0)) = Fg(m)(w)(V0) for any w ∈ Wm.

(2) We set Gs := {g | g is a symmetry of L, g = f |K for some isometry f of Rd}.(3) L is called an affine nested fractal if and only if it is post-critically finite, K isconnected and gxy|K ∈ Gs for any x, y ∈ V0 with x �= y. An affine nested fractal Lis called a nested fractal if and only if αi = α for any i ∈ S for some α ∈ (0, 1).(4) A real matrix A = (Axy)x,y∈V0

is called Gs-invariant if and only if Axy =Ag(x)g(y) for any x, y ∈ V0 and any g ∈ Gs. Also a = (ai)i∈S ∈ (0,∞)S is calledGs-invariant if and only if ai = aj for any i, j ∈ S satisfying g(Fi(V0)) = Fj(V0) forsome g ∈ Gs.

By [23, Proof of Proposition 3.8.9], if L is an affine nested fractal, then A =(Axy)x,y∈V0

is Gs-invariant if and only if Axy = Ax′y′ whenever |x− y| = |x′ − y′|.Now we can conclude the following theorem for affine nested fractals.

Theorem 4.6. Assume that L = (K,S, {Fi}i∈S) is an affine nested fractal,that D is Gs-invariant and that ri = r for any i ∈ S for some r ∈ (0, 1). Furtherassume that

(4.7) #(Fi(V0) ∩ Fj(V0)) ≤ 1 for any i, j ∈ S with i �= j.

If L satisfies (4.5), then the conclusions of Theorems 2.17 and 2.18 hold true.

Proof. We have Gs ⊂ G by [21, Proof of Theorem 4.5] and [23, Corollary3.8.21]. Since gxy|K ∈ Gs and gxy(x) = y for any x, y ∈ V0 with x �= y, Proposition4.4 is applicable and hence so are Theorems 2.17 and 2.18. �

Remark 4.7. (1) If L = (K,S, {Fi}i∈S) is an affine nested fractal satisfying(4.7), then a harmonic structure (D, r) on L as in Theorem 4.6 exists and is unique(up to constant multiples of D). Here the existence part is essentially due to Lind-strøm [32]; see [23, Section 3.8] and references therein for further details. Also see[17,33–35] for more recent results on existence of harmonic structures.

182 NAOTAKA KAJINO

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Figure 2. Examples of nested fractals. From the upper left, two-dimensional level-l Sierpinski gasket (l = 2, 3, 4), three-dimensionalstandard (level-2) Sierpinski gasket, pentagasket (5-polygasket),heptagasket (7-polygasket), snowflake and the Vicsek set. In eachfractal, the set V0 of its boundary points is marked by solid circles.

(2) For the same reason as [21, Theorem 4.5] (see [21, Remark 4.6-(2)]), it is unclearwhether the (technical) assumption (4.7) can be removed from Theorem 4.6.

At the last of this subsection, we provide some examples of nested fractals.

Example 4.8 (Sierpinski gaskets). Let d, l ∈ N \ {1}, let L = (K,S, {Fi}i∈S)be the d-dimensional level-l Sierpinski gasket as in [21, Example 5.1] and let (D, r)be the harmonic structure on L described there. Then clearly L is a nested fractaland the conditions of Theorem 4.6 except (4.5) are satisfied. Moreover, it is easyto see that (4.5) is satisfied if and only if l ≥ 3 (see Figure 2). Thus by Theorem4.6, if l ≥ 3 then the conclusions of Theorems 2.17 and 2.18 are valid.

On the other hand, it is unclear whether (2.13) and (2.14) hold when l = 2, asalready remarked at the end of the introduction.

Example 4.9 (Polygaskets). Let L = (K,S, {Fi}i∈S) be the (N, l)-polygasketwith N, l ∈ N, N ≥ 4, l < N/2 in [21, Example 5.5] and let (D, r) be the harmonicstructure on L described there. Then we easily see that the conditions of Propo-sition 4.4 including (4.5) are satisfied and hence the conclusions of Theorems 2.17and 2.18 hold in this case.

Note that this example includes the case of the N -polygasket with N ∈ N,N ≥ 5, N/4 �∈ N in [21, Example 5.3] (see Figure 2), which is the (N, �N/4 )-polygasket with �N/4 := min{n ∈ N | n ≥ N/4} and is realized in R2 as a nestedfractal.

4.3. Cases possibly without good symmetry. We follow the frameworkof Subsection 4.1 throughout this subsection. Recall that Proposition 4.4 above isbased on the assumption of good symmetry of (L, (D, r), μ). On the other hand,even under weaker assumptions on symmetry of (L, (D, r), μ), we can still verify(2.13) or (2.14) in some cases, as follows. Recall that KI = K \ V0 = K \ V0.


Proposition 4.10. Let y, z ∈ KI and let Λy,Λz be partitions of Σ. DefineΓy := {w ∈ Λy | y ∈ Kw} and Γz := {w ∈ Λz | z ∈ Kw}, let wy ∈ Γy and assumey ∈ Fwy

(V0) if Γy = {wy}. Let ϕ : Γy → Γz and suppose that for each w ∈ Γy,

rϕ(w)/rw = rϕ(wy)/rwyand there exists gw ∈ G such that gw

(F−1w (y)

)= F−1

ϕ(w)(z).

Set y := Fϕ(wy)(y) and z := Fwy(z) (note that y, z ∈ KI).

(1) If ϕ is injective and not surjective, then lim supt↓0 pt(y, y)/pt(z, z) > 1.

(2) If n ∈ N and #ϕ−1(v) = n for any v ∈ Γz, then limt↓0 pt(y, y)/pt(z, z) = n−1.

We need the following definition and lemma for the proof of Proposition 4.10.

Definition 4.11. Let U be a non-empty open subset of K.(1) Let λ ∈ (0,∞) and set Eλ(u, v) := E(u, v) +

∫Kuvdμ for u, v ∈ F . We define

capUλ (B) := inf{Eλ(u, u) | u ∈ FU , u ≥ 1 μ-a.e. on B}, B ⊂ U open in U ,(4.8)

CapUλ (A) := inf

{capU

λ (B)∣∣ B ⊂ U open in U , A ⊂ B

}, A ⊂ U,(4.9)

so that CapUλ is an extension of capU

λ . We call CapUλ the λ-order capacity on U .

(2) We define the Dirichlet resolvent kernel uU = uUλ (x, y) on U by

(4.10) uUλ (x, y) :=

∫ ∞

0

e−λtpUt (x, y)dt, (λ, x, y) ∈ (0,∞) × U × U,

where pU = pUt (x, y) is the Dirichlet heat kernel on U introduced in Lemma 3.2.

By (2.11), pU ≤ p and ds ∈ (0, 2), uUλ (x, y) ≤ uK

λ (x, y) ≤ c4.1λds/2−1 for any

(λ, x, y) ∈ (0,∞)×U×U for some c4.1 ∈ (0,∞), and uU : (0,∞)×U×U → [0,∞) is

continuous by the continuity of pU . Note that CapUλ ({x}) ∈ (0,∞) for any (λ, x) ∈

(0,∞) × U by [26, (3.1)] and hence FU = {u ∈ F | u|K\U = 0} by [11, Corollary

2.3.1]. Then since pU = pUt (x, y) is the transition density of a μ|U -symmetricdiffusion on U whose Dirichlet form on L2(U, μ|U ) is (EU ,FU ) by [26, Theorem10.4] (or by [19, Lemma 7.11-(2)]), we easily see from [11, (2.2.11), (2.2.13) and

Exercise 4.2.2] that CapUλ ({x}) = 1/uU

λ (x, x) for any (λ, x) ∈ (0,∞) × U .

Lemma 4.12. Let x ∈ K, Λ be a partition of Σ and set Γ := {w ∈ Λ | x ∈ Kw}.Assume x ∈ Fw(V0) if #Γ = 1 and Γ = {w}, so that x ∈ Fw(V0) for any w ∈ Γ.Set U := {x}∪

⋃w∈Γ KI

w and Uq := {q}∪KI for q ∈ V0. Then for any λ ∈ (0,∞),

(4.11)(uUλ (x, x)

)−1=∑w∈Γ

γds−2w

uUF−1w (x)

γ2wλ

(F−1w (x), F−1

w (x)) .

Proof. Note that U and Uq, q ∈ V0, are open subsets of K by [23, Proposition1.3.6] and #V0 < ∞. If u ∈ FU and u(x) = 1 then u ◦ Fw ∈ F

UF−1w (x) and u ◦

Fw

(F−1w (x)

)= 1 for any w ∈ Γ, and conversely if uw ∈ F

UF−1w (x) and uw

(F−1w (x)

)=

1 for each w ∈ Γ, then u ∈ C(K) defined by u|Kw:= uw ◦ F−1

w for w ∈ Γ andu|K\U := 0 belongs to FU by (4.2) and satisfies u(x) = 1. Therefore using [11,

184 NAOTAKA KAJINO

Theorem 2.1.5-(i)] and (2.4), we see that for any λ ∈ (0,∞),(uUλ (x, x)

)−1= CapU

λ ({x}) = inf{Eλ(u, u) | u ∈ FU , u(x) = 1}

= inf

{∑w∈Γ

1

rwEγ2

wλ(u ◦ Fw, u ◦ Fw)

∣∣∣∣ u ∈ FU , u(x) = 1

}=∑w∈Γ

γds−2w inf

{Eγ2

wλ(uw, uw)∣∣∣ uw ∈ F

UF−1w (x) , uw

(F−1w (x)

)= 1}

=∑w∈Γ

γds−2w CapUF−1

w (x)

γ2wλ

(F−1w (x)

)=∑w∈Γ

γds−2w

uUF−1w (x)

γ2wλ

(F−1w (x), F−1

w (x)) ,

proving (4.11). �

Proof of Proposition 4.10. Set Γy := {ϕ(wy)w | w ∈ Γy}, Γz := {wyw |w ∈ Γz} and define ϕ : Γy → Γz by ϕ(ϕ(wy)w) := wyϕ(w) for w ∈ Γy, sothat rw = rϕ(w) for any w ∈ Γy. Also set U y := {y} ∪

⋃w∈Γy

KIw and U z :=

{z} ∪⋃

w∈ΓzKI

w. By y ∈ KIϕ(wy)

and z ∈ KIwy

we can choose partitions Λy,Λz

of Σ so that Γy = {w ∈ Λy | y ∈ Kw} and Γz = {w ∈ Λz | z ∈ Kw}, and inthe situations of (1) and (2) we have y ∈ Fw(V0) for any w ∈ Γy and z ∈ Fw(V0)

for any w ∈ Γz. Note that uUq

λ (a, b) = uUg(q)

λ (g(a), g(b)) for g ∈ G, q ∈ V0 and(λ, a, b) ∈ (0,∞) × Uq × Uq, where Uq := {q} ∪KI for q ∈ V0. Therefore recallingthat gw

(F−1ϕ(wy)w

(y))

= F−1wyϕ(w)(z) for each w ∈ Γy, we see from Lemma 4.12 that

for any λ ∈ (0,∞),(uU y

λ (y, y))−1

(4.12)

=∑w∈Γy

γds−2w

uUF−1w (y)

γ2wλ

(F−1w (y), F−1

w (y)) =

∑w∈Γy

γds−2ϕ(w)

uUF

−1ϕ(w)

(z)

γ2ϕ(w)

λ

(F−1ϕ(w)(z), F

−1ϕ(w)(z)

)=

⎧⎪⎨⎪⎩(uU z

λ (z, z))−1 −

∑w∈Γz\ϕ(Γy)

γds−2w

uUF−1w (z)

γ2wλ

(F−1w (z),F−1

w (z))for (1),

n(uU z

λ (z, z))−1

for (2).

(1) By Proposition 2.16, there exist c4.2, c4.3 ∈ (0,∞) such that for any x ∈ K,

(4.13) c4.2 ≤ λ1−ds/2uKλ (x, x) ≤ c4.3, λ ∈ [1,∞).

We easily see from Lemma 3.3, (2.11) and (4.13) that limλ→∞ uUλ (x, x)/uK

λ (x, x) =1 for any non-empty open subset U of K and any x ∈ U . It follows from this fact,(4.12) and (4.13) that lim infλ→∞ uK

λ (y, y)/uKλ (z, z) ≥ 1+c4.2c

−14.3#(Γz\ϕ(Γy)) > 1,

which immediately implies lim supt↓0 pt(y, y)/pt(z, z) > 1.

(2) (4.12) implies that pUz

t (z, z) = npUy

t (y, y) for any t ∈ (0,∞), from which theassertion is immediate since limt↓0 p

Ut (x, x)/pt(x, x) = 1 for any non-empty open

subset U of K and any x ∈ U by Lemma 3.3 and (2.11). �

Remark 4.13. As shown in the previous proof, in the situation of Proposition4.10-(1) it actually holds that lim infλ→∞ uK

λ (y, y)/uKλ (z, z) > 1. Unfortunately,

however, here we cannot conclude from this fact that lim inft↓0 pt(y, y)/pt(z, z) > 1,for p(·)(y, y) and p(·)(z, z) may not vary regularly at 0 and hence Tauberian theoremsfor the Laplace transform may not be applicable to them.


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c = π(12∞)

0 = π(1∞) 1 = π(2∞)

|c|2 = π(112∞) = π(21∞)

F2(c) = π(212∞)

Figure 3. Hata’s tree-like set (c = 0.4 + 0.3√−1) and the set V1

Now a simple application of Proposition 4.10-(1) yields the following theorem.Recall (4.4) for the definition of nx for x ∈ V∗.

Theorem 4.14. If nx = 1 for some x ∈ V∗\V0, then the conclusions of Theorem2.17 are valid for any regular harmonic structure (D, r) on L, where the set NRV

can be chosen independently of (D, r).

Proof. By x ∈ Vmxthere exist q ∈ V0 and wx ∈ Wmx

such that x = Fwx(q),

and {w ∈ Wmx| x ∈ Kw} = {wx} by nx = 1. On the other hand, by V0 = π(P) and

(2.1) we can choose ω ∈ P and v ∈ W∗ \ {∅} so that q = π(ω) and σv(ω) ∈ C. Thentaking τ ∈ W∗ such that Kτ ⊂ KI , which is possible by [23, Proposition 1.3.6], wesee that z := Fτv(q) = Fτ (π(σv(ω))) ∈ KI and that #{w ∈ W|τv| | z ∈ Kw} ≥ 2.Now for any regular harmonic structure (D, r) on L, Proposition 4.10-(1) easilyyields lim supt↓0 pt(Fτv(x), Fτv(x))/pt(Fwx

(z), Fwx(z)) > 1, and hence Theorem

2.17 applies with NRV determined solely by Fτv(x), Fwx(z) and π. �

At the last of this section, we apply Proposition 4.10 and Theorem 4.14 to someexamples.

Example 4.15. Let L = (K,S, {Fi}i∈S) be any one of the (N, l)-polygasketwith N, l ∈ N, N ≥ 4, l < N/2 in [21, Example 5.5], the snowflake and the Vicsekset (see Figure 2). Then the assumption of Theorem 4.14 is clearly satisfied andhence the conclusions of Theorem 2.17 hold for any regular harmonic structure onL.

Example 4.16 (Hata’s tree-like set). Following [23, Example 1.2.9], let c ∈C \ R satisfy |c|, |1 − c| ∈ (0, 1), set S := {1, 2} and define fi : C → C for i ∈ S byf1(z) := cz and f2(z) := (1 − |c|2)z + |c|2. Let K be the self-similar set associatedwith {fi}i∈S , i.e. the unique non-empty compact subset of C ∼= R2 that satisfiesK =

⋃i∈S fi(K), and set Fi := fi|K for i ∈ S. Then L := (K,S, {Fi}i∈S) is a

self-similar structure with K connected, P = {12∞, 1∞, 2∞} and V0 = {c, 0, 1}.Also F2(c) ∈ V1 \ V0 and nF2(c) = 1. L is called Hata’s tree-like set (see Figure 3).

Let r ∈ (0, 1), set r = (ri)i∈S := (r, 1− r2) and let D = (Dxy)x,y∈V0be the real

symmetric matrix given by Dc0 = −Dcc := 1/r, D01 = −D11 := 1, Dc1 := 0 andD00 := −1 − 1/r. Then (D, r) is a regular harmonic structure on L and, exceptfor constant multiples of D, any harmonic structure on L is of this form. NowTheorem 4.14 applies again and hence the conclusions of Theorem 2.17 are valid inthis case. Note that this case is beyond the reach of the author’s preceding result[21, Theorem 3.4], since G = {idK} by virtue of the following proposition.

186 NAOTAKA KAJINO

Proposition 4.17. Let L = (K,S, {Fi}i∈S) and (D, r) be as in Example 4.16and let R : K×K → [0,∞) be the resistance metric of the resistance form (E ,F) onK resulting from (D, r) by (4.2). If g : K → K is surjective and satisfies g(V0) = V0

and R(g(x), g(y)) = R(x, y) for any x, y ∈ K, then g = idK .

Proof. Since r−1R(F1(x), F1(y)) = (1 − r2)−1R(F2(x), F2(y)) = R(x, y) forany x, y ∈ K by K1 ∩K2 = {|c|2}, (4.2) and (2.4), an induction in m easily impliesthat supx∈K\K2m

R(0, x) < 1 for m ∈ N, so that 1 = maxx∈K R(0, x) is attained

only by x = 1. Then since R(F21(x), F21(y)) = r(1−r2)R(x, y) for any x, y ∈ K by(4.2) and (2.4) again, r(1−r2) = maxx∈K21

R(|c|2, x) is attained only by x = F2(c).Let GR be the collection of surjections g : K → K satisfying g(V0) = V0 and

R(g(x), g(y)) = R(x, y) for any x, y ∈ K, and let g ∈ GR. We first show g|V1= idV1

and g(Ki) = Ki, i ∈ S. It follows from R(c, 0) < R(0, 1) < R(c, 1) and g(V0) = V0

that g|V0= idV0

. Define γ : [0, 2] → K by γ(t) := (1 − t)c for t ∈ [0, 1] and γ(t) :=t− 1 for t ∈ [1, 2]. We easily see that R(c, γ(t)) = R(c, γ(s))+R(γ(s), γ(t)) for anys, t ∈ [0, 2] with s ≤ t, so that R(c, γ(·)) is strictly increasing. By K1∩K2 = {|c|2},a continuous path g ◦ γ : [0, 2] → K from c ∈ K1 to 1 ∈ K2 has to admit t ∈ (0, 2)such that g ◦ γ(t) = |c|2. Then R(c, |c|2) = R(c, g ◦ γ(t)) = R(c, γ(t)) and hencet = 1 + |c|2 by the strict monotonicity of R(c, γ(·)). Thus g(|c|2) = |c|2, and inparticular g defines a homeomorphism g|K\{|c|2} : K \ {|c|2} → K \ {|c|2}. Set

U :=⋃

m∈N∪{0} K12m2. Then since g(c) = c ∈ K1 \ {|c|2}, g(1) = 1 ∈ F2(U) and

K \{|c|2} consists of three connected components K1 \{|c|2}, F2(U) and F21(U), itfollows that g(K1\{|c|2}) = K1\{|c|2}, g(F2(U)) = F2(U) and g(F21(U)) = F21(U).Thus g(K1) = K1 and g(K2) = K2. Moreover, maxx∈K21

R(|c|2, x) is attained byg(F2(c)) ∈ K21 and hence g(F2(c)) = F2(c).

Now let m ∈ N and assume that g|Vm= idVm

for any g ∈ GR. Then for g ∈ GR

and i ∈ S, by g|V1= idV1

and g(Ki) = Ki we have gi := F−1i ◦ g ◦ Fi ∈ GR, hence

gi|Vm= idVm

and therefore g|Vm+1= idVm+1

. Thus g|V∗ = idV∗ for any g ∈ GR byinduction in m, which proves GR = {idK} since V∗ is dense in K. �

Example 4.18. Following [23, Example 4.4.9], let S := {1, 2, 3, 4} and define

fi : C → C for i ∈ S by f1(z) := 12 (z + 1), f2(z) := 1

2 (z − 1), f3(z) :=√−14 (z + 1)

and f4(z) :=√−14 (z− 1). Let K be the self-similar set associated with {fi}i∈S and

set Fi := fi|K , i ∈ S. Then L := (K,S, {Fi}i∈S) is a self-similar structure with Kconnected, P = {1∞, 2∞} and V0 = {−1, 1}. Defining g, h : C → C by g(z) := −zand h(z) := z, we easily see that Gs = {idK , g|K , h|K , gh|K}, and thus L is an affinenested fractal. Moreover, F3(1) =

√−1/2 ∈ V1 \ V0 and nF3(1) = 1.

Set D = (Dxy)x,y∈V0:=(−1 1

1 −1

), which is Gs-invariant, and let r = (ri)i∈S ∈

(0, 1)S be such that r1 +r2 = 1. Then (D, r) is clearly a regular harmonic structureon L and the conclusions of Theorem 2.17 hold by Theorem 4.14.

Next assume r1 = r2 = 12 and that r3 = r4 =

(12

)mfor some m ∈ N, so

that r is Gs-invariant and hence g|K ∈ Gs ⊂ G by [21, Proof of Theorem 4.5] and

[23, Corollary 3.8.21]. Set y := 0, z :=√−12 , Λy := {iw | i ∈ {1, 2}, w ∈ Wm} ∪

{ij | i ∈ {3, 4}, j ∈ S}, Λz := S and let Γy,Γz be as in Proposition 4.10. Then

Γy = {12m, 21m, 32, 41}, Γz = {3} and rw =(12

)m+1for any w ∈ Γy, from which

together with g|K ∈ G we can easily verify the assumptions of Proposition 4.10-(2) with ϕ(w) := 3, w ∈ Γy. Thus (2.14) is satisfied and hence the conclusion ofTheorem 2.18 is valid in this case.


5. Sierpinski carpets

In this last section, we apply Theorems 2.17 and 2.18 to the canonical heatkernel on generalized Sierpinski carpets, which are among the most typical examplesof infinitely ramified self-similar fractals and have been intensively studied e.g. in[1–6,16,18,19,25,31].

We fix the following setting throughout this section.

Framework 5.1. Let d, l ∈ N, d ≥ 2, l ≥ 2 and set Q0 := [0, 1]d. LetS ⊂ {0, 1, . . . , l − 1}d be non-empty, and for each i ∈ S define fi : Rd → Rd byfi(x) := l−1i + l−1x. Set Q1 :=

⋃i∈S fi(Q0), which satisfies Q1 ⊂ Q0. Let K

be the self-similar set associated with {fi}i∈S , i.e. the unique non-empty compactsubset of Rd that satisfies K =

⋃i∈S fi(K), and set Fi := fi|K for i ∈ S, so that

GSC(d, l, S) := (K,S, {Fi}i∈S) is a self-similar structure. Also let ρ : K × K →[0,∞) be the Euclidean metric on K given by ρ(x, y) := |x − y|, set df := logl #Sand let μ be the self-similar measure on L with weight (1/#S)i∈S .

Recall that df is the Hausdorff dimension of (K, ρ) and that μ is a constantmultiple of the df -dimensional Hausdorff measure on (K, ρ); see e.g. [23, Theorem1.5.7 and Proposition 1.5.8].

The following definition is essentially due to M. T. Barlow and R. F. Bass [5].

Definition 5.2 (Generalized Sierpinski carpets). GSC(d, l, S) is called a gen-eralized Sierpinski carpet if and only if S satisfies the following four conditions:

(GSC1) (Symmetry) f(Q1) = Q1 for any isometry f of Rd with f(Q0) = Q0.(GSC2) (Connectedness) Q1 is connected.

(GSC3) (Non-diagonality) intRd

(Q1∩

∏dk=1[(ik−εk)l

−1, (ik+1)l−1])

is either empty

or connected for any (ik)dk=1 ∈ Zd and any (εk)

dk=1 ∈ {0, 1}d.

(GSC4) (Borders included) {(x1, 0, . . . , 0) ∈ Rd | x1 ∈ [0, 1]} ⊂ Q1.

As special cases of Definition 5.2, GSC(2, 3, SSC) and GSC(3, 3, SMS) are calledthe Sierpinski carpet and the Menger sponge, respectively, where SSC := {0, 1, 2}2 \{(1, 1)} and SMS :=

{(i1, i2, i3) ∈ {0, 1, 2}3

∣∣ ∑3k=1 1{1}(ik) ≤ 1

}(see Figure 4).

We remark that there are several equivalent ways of stating the non-diagonalitycondition, as in the following proposition.

Proposition 5.3 ([20, §2]). Set |x|1 :=∑d

k=1 |xk| for x = (xk)dk=1 ∈ Rd.

Then (GSC3) is equivalent to any one of the following three conditions:

(ND)N intRd

(Q1∩

∏dk=1[(ik −1)l−m, (ik +1)l−m]

)is either empty or connected for

any m ∈ N and any (ik)dk=1 ∈ {1, . . . , lm − 1}d.

(ND)2 The case of m = 2 of (ND)N holds.

(NDF) For any i, j ∈ S with fi(Q0) ∩ fj(Q0) �= ∅ there exists {n(k)}|i−j|1k=0 ⊂ S

such that n(0) = i, n(|i − j|1) = j and |n(k) − n(k + 1)|1 = 1 for anyk ∈ {0, . . . , |i− j|1 − 1}.

Remark 5.4. Only the case of m = 1 of (ND)N was assumed in the originaldefinition of generalized Sierpinski carpets in [5, Section 2], but Barlow, Bass,Kumagai and Teplyaev [6] have recently realized that it is too weak for [5, Proofof Theorem 3.19] and has to be replaced by (ND)N (or equivalently, by (GSC3)).

Now in view of (NDF) in Proposition 5.3, (GSC2) and (GSC3) together implythat intRd Q1 is connected, so that Definition 5.2 turns out to be equivalent to thedefinition of generalized Sierpinski carpets in [6, Subsection 2.2].

188 NAOTAKA KAJINO

Figure 4. Sierpinski carpet, some other generalized Sierpinskicarpets with d = 2 and Menger sponge

In the rest of this section, we assume that L := GSC(d, l, S) = (K,S, {Fi}i∈S)is a generalized Sierpinski carpet. Then we easily see the following proposition.

Proposition 5.5. Set Sk,ε := {(in)dn=1 ∈ S | ik = (l−1)ε} for k ∈ {1, 2, . . . , d}and ε ∈ {0, 1}. Then P =

⋃dk=1(S

N

k,0 ∪ SN

k,1), V0 = V0 = K \ (0, 1)d and V∗∗ = V∗.

Analysis on generalized Sierpinski carpets was initiated by M. T. Barlow andR. F. Bass in [1]: they obtained a non-degenerate μ-symmetric diffusion X on Kin the case of d = 2 by taking a certain scaling limit of (a suitable subsequenceof) the reflecting Brownian motions X(m) on Qm :=

⋃w∈Wm

fw(Q0), where fw :=

fw1◦· · ·◦fwm

(f∅ := idRd) for w = w1 . . . wm ∈ W∗. Then they studied the diffusionX intensively in a series of papers [2–4] and extended their results to the case ofd ≥ 3 in [5]. On the other hand, Kusuoka and Zhou [31] also obtained a non-degenerate diffusion on K in the case of d = 2 by constructing a (homogeneouslyscaled self-similar) Dirichlet form on L2(K,μ) via a discrete approximation of K.It had been a long-standing problem to prove that the constructions in [1,5] and in[31] give rise to the same diffusion on K, until Barlow, Bass, Kumagai and Teplyaev[6] finally solved it by proving the uniqueness of a non-zero conservative symmetricregular Dirichlet form on L2(K,μ) possessing certain local symmetry properties.The following is a summary of the main results of [6].

Definition 5.6. (1) We define

(5.1) G0 := {f |K | f is an isometry of Rd with f(Q0) = Q0},which forms a subgroup of the group of homeomorphisms of K by virtue of (GSC1).

(2) Define ψ : Rd → Q0 by ψ((xk)

dk=1

):=(minn∈Z |xk−2n|

)dk=1

. For each w ∈ W∗,we set qw := Fw(0) and define the folding map ϕw : K → Kw into Kw by

(5.2) ϕw(x) := qw + l−|w|ψ(l|w|(x− qw)

),

so that ϕw|Kw= idKw

and ϕw ◦ ϕv = ϕw for any w, v ∈ W∗ with |w| = |v|.(3) For u ∈ L2(K,μ) and δ ∈ (0,∞), we define

(5.3) Jδ(u) := δ−df

∫K

∫Bδ(x,ρ)

(u(x) − u(y))2dμ(y)dμ(x).

Note that μ ◦ g = μ for any g ∈ G0. We set μ|A := μ|B(A) for A ∈ B(K). For

each w ∈ W∗, if u : Kw → [−∞,∞] is Borel measurable then∫K

|u ◦ ϕw|dμ =

(#S)|w| ∫Kw

|u|dμ, so that ϕ∗wu := u ◦ ϕw defines a bounded linear operator ϕ∗

w :

L2(Kw, μ|Kw) → L2(K,μ), which is called the unfolding operator from Kw.


Theorem 5.7 ([6, Theorem 1.2 and Subsection 4.7]). (1) There exists a unique(up to constant multiples of E) non-zero conservative symmetric regular Dirichletform (E ,F) on L2(K,μ) satisfying the following conditions:

(BBKT1) u ◦ ϕw ∈ F for any u ∈ F and any w ∈ W∗.(BBKT2) For any m ∈ N and any u ∈ F ,

(5.4) E(u, u) =1

(#S)m

∑w∈Wm

E(u ◦ ϕw, u ◦ ϕw).

(BBKT3) Let w, v ∈ W∗, |w| = |v| and g ∈ G0. If u ∈ L2(Kv, μ|Kv) and u◦ϕv ∈ F ,

then ugw,v := u◦Fv◦g◦F−1

w ◦ϕw ∈ F and E(ugw,v, u

gw,v) = E(u◦ϕv, u◦ϕv).

(2) (K,μ, E ,F) satisfies (CHK) and there exist dw ∈ [2,∞) and c5.1, c5.2 ∈ (0,∞)such that, with ds := 2df/dw, for any (t, x, y) ∈ (0, 1] ×K ×K,(5.5)

c5.1tds/2

exp

(−(ρ(x, y)dw

c5.1t

) 1dw−1

)≤ pt(x, y) ≤ c5.2

tds/2exp

(−(ρ(x, y)dw

c5.2t

) 1dw−1

).

(3) F = {u ∈ L2(K,μ) | lim supδ↓0 δ−dwJδ(u) < ∞}, and there exist c5.3, c5.4 ∈

(0,∞) such that for any u ∈ F ,

(5.6) c5.3E(u, u) ≤ lim supδ↓0

δ−dwJδ(u) ≤ supδ∈(0,∞)

δ−dwJδ(u) ≤ c5.4E(u, u).

Remark 5.8. The strict inequality dw > 2 holds if #S < ld. In the case ofd = 2, this estimate follows from [3, Proof of Proposition 5.2] (see also [4, (2.5)]),whereas for d ≥ 3 this fact is only stated in [5, Remarks 5.4-1.] without proof.

In fact, by virtue of [18, Proof of Proposition 5.1], we can also deduce fromTheorem 5.7-(1),(3) the following simpler characterization of (E ,F) although it ismore restrictive than that in Theorem 5.7-(1).

Proposition 5.9. (E ,F) is the unique (up to constant multiples of E) non-zeroconservative symmetric regular Dirichlet form on L2(K,μ) possessing the followingproperties:

(GSCDF1) If u ∈ F ∩C(K) and g ∈ G0 then u◦g ∈ F and E(u◦g, u◦g) = E(u, u).(GSCDF2) F ∩ C(K) = {u ∈ C(K) | u ◦ Fi ∈ F for any i ∈ S}.(GSCDF3) There exists r ∈ (0,∞) such that for any u ∈ F ∩ C(K),

(5.7) E(u, u) =∑i∈S

1

rE(u ◦ Fi, u ◦ Fi).

Moreover, dw = logl(#S/r) and ds = 2 log#S/r #S.

We need the following lemma, which easily follows by a direct calculation.

Lemma 5.10. Let w, v, τ ∈ W∗, |w| = |v|, εw,v = (εw,vk )dk=1 := ψ

(l|w|(qv − qw)

)and define fw,v : Rd → Rd by fw,v(x) :=

(εw,vk +(1−2εw,v

k )xk

)dk=1

for x = (xk)dk=1 ∈

Rd, so that gw,v := fw,v|K ∈ G0. Then ϕwτ ◦ Fv = Fw ◦ ϕτ ◦ gw,v.

Proof of Proposition 5.9. We first prove that (E ,F) as in Theorem 5.7-(1) possesses the stated properties. (GSCDF1) is immediate from (BBKT3) withw = v = ∅. We easily see from Theorem 5.7-(3) that u ◦ Fw ∈ F for any w ∈ W∗and any u ∈ F , and [18, Proof of Proposition 5.1] shows that u ∈ F whenever

190 NAOTAKA KAJINO

u ∈ C(K) and u ◦Fi ∈ F for any i ∈ S, proving (GSCDF2). (Note that [18, Proofof Proposition 5.1] for f ∈ C(K) is based only on Theorem 5.7-(3) and (NDF).)

(GSCDF3) is stated in [6, Theorem 1.2] without explicit proof. In fact, it canbe directly deduced from Theorem 5.7-(1),(3), as follows. Noting that u ◦ Fi ∈ Ffor i ∈ S and u ∈ F , define RE : F×F → R by (RE)(u, v) :=

∑i∈S E(u◦Fi, v◦Fi).

By (5.6) there exists c5.5 ∈ (0,∞) such that (RE)(u, u) ≤ c5.5E(u, u) for any u ∈ F ,and we can easily verify (BBKTk), k = 1, 2, 3 for (RE ,F) from those for (E ,F)and Lemma 5.10. It follows that (E +RE ,F) is a non-zero conservative symmetricregular Dirichlet form on L2(K,μ) satisfying (BBKTk), k = 1, 2, 3, and henceE + RE = θE for some θ ∈ (0,∞) by Theorem 5.7-(1). Since (E ,F) is non-zero,λ := θ− 1 ∈ [0,∞) and RE = λE . Furthermore take u ∈ F ∩C(K) \ {0} such thatsuppK [u] ⊂ KI . Then E(u, u) > 0 by Theorem 5.7-(3) and V0 �= ∅. For any w ∈ W∗,(Fw)∗u ∈ F by (GSCDF2), E

((Fw)∗u, (Fw)∗u

)= λ|w|E(u, u) by RE = λE , and

we easily see from E(u, u) > 0 and (5.6) that (λ#S/ldw)|w| ∈ [c5.3/c5.4, c5.4/c5.3].Letting |w| → ∞ yields λ = ldw/#S > 0, proving (GSCDF3), dw = logl(#S/r)and ds = 2 log#S/r #S with r := λ−1.

Next for the proof of the uniqueness, suppose that (E ′,F ′) is a non-zero con-servative symmetric regular Dirichlet form on L2(K,μ) with the stated properties.The regularity of (E ′,F ′) easily implies that u◦Fi ∈ F ′ for any i ∈ S and any u ∈ F ′

and that (GSCDF1) and (GSCDF3) with F ′ in place of F ′ ∩C(K) are valid. Fur-thermore we see from Lemma 5.10 with τ = ∅ and the assumed properties of (E ′,F ′)that for w ∈ W∗ and u ∈ L2(Kw, μ|Kw

), u ◦ϕw ∈ F ′ if and only if u ◦Fw ∈ F ′, andif u◦ϕw ∈ F ′ then E ′(u◦ϕw, u◦ϕw) = (#S/r)|w|E ′(u◦Fw, u◦Fw). Now it is imme-diate from these facts and (GSCDF1) that (E ′,F ′) satisfies (BBKTk), k = 1, 2, 3,and hence (E ′,F ′) = (θE ,F) for some θ ∈ (0,∞) by Theorem 5.7-(1). �

It follows from Proposition 5.9 that (L, μ, E ,F) is a homogeneously scaled self-similar Dirichlet space with weight (1/#S)i∈S and spectral dimension ds. Moreoverfor its associated scale S = {Λs}s∈(0,1] on Σ, we easily see that #Λ1

s,x ≤ 4d for any

(s, x) ∈ (0, 1]×K and that ρdw/2 is a (2/dw)-qdistance on K adapted to S, so that(L, μ, E ,F) satisfies (CUHK) by Theorem 5.7-(2).

Finally, we verify that Theorems 2.17 and 2.18 are applicable to (L, μ, E ,F) if#S < ld. Recall that L = GSC(d, l, S) = (K,S, {Fi}i∈S) is a generalized Sierpinskicarpet, that μ is the self-similar measure on L with weight (1/#S)i∈S and that(E ,F) is the Dirichlet form on L2(K,μ) as in Theorem 5.7 and Proposition 5.9.

Theorem 5.11. If #S < ld, then the conclusions of Theorems 2.17 and 2.18hold true for the continuous heat kernel p = pt(x, y) of (K,μ, E ,F).

Since (L, μ, E ,F) satisfies (CUHK), for the proof of Theorem 5.11 it sufficesto verify (2.14), which is an easy consequence of the following Proposition. Recallthat pU = pUt (x, y) denotes the Dirichlet heat kernel on U introduced in Lemma 3.2for a non-empty open subset U of K. Note that for any w, v ∈ W∗ with |w| = |v|,ϕv|Kw

= Fv ◦gv,w ◦F−1w by Lemma 5.10 and hence ϕv|Kw

: Kw → Kv is a surjectiveisometry with respect to the metric ρ.

Proposition 5.12. Let ε = (εk)dk=1 ∈ {0, 1}d, m ∈ N ∪ {0}, i = (ik)

dk=1 ∈

l−mZd, set Ri,εm :=

∏dk=1[ik − εkl

−m, ik + l−m], W i,εm := {w ∈ Wm | Kw ⊂ Ri,ε

m }and suppose W i,ε

m �= ∅. Also set U i,εm := K ∩ intRd Ri.ε

m , Uε := K ∩∏d

k=1(−εk, 1),

V ε0 := Uε ∩

∏dk=1[0, 1− εk], let τ ∈ W i.ε

m and let gτ ∈ G0 be such that Fτ ◦ gτ (Uε) =


Kτ ∩ U i,εm . Then with γ :=

√r/#S, for any (t, x, y) ∈ (0,∞) × Uε × Uε,

(5.8) pUε

t (x, y) = γmds

∑w∈W i,ε

m

pUi,ε

m

γ2mt

(Fτ ◦ gτ (x), (ϕτ |Kw

)−1 ◦ Fτ ◦ gτ (y)).

In particular, for any t ∈ (0,∞) and any (x, y) ∈ (Uε × V ε0 ) ∪ (V ε

0 × Uε),

(5.9) pUε

t (x, y) = (#W i,εm )γmdsp

Ui,εm

γ2mt

(Fτ ◦ gτ (x), Fτ ◦ gτ (y)

).

Note that V ε0 = K ∩

((0, 1){k∈{1,...,d}|εk=0} × {0}{k∈{1,...,d}|εk=1}) �= ∅; indeed,(

l−1(1 − εk))dk=1

∈ V ε0 by (GSC1) and (GSC4).

Proof. Recalling Definition 3.1, throughout this proof we regard FUi,εm

and

FUε as linear subspaces of L2(U i,εm , μ|Ui,ε

m

)and L2(Uε, μ|Uε), respectively, in the

natural manner. For w ∈ W i,εm , noting that ϕw(U i,ε

m ) = Kw ∩ U i,εm and μ|Ui,ε

m◦(

ϕw|Ui,εm

)−1= (#W i,ε

m )μ|Kw∩Ui,εm

, we set ϕ�wu := u◦ϕw|Ui,ε

mfor u : U i,ε

m → [−∞,∞],

so that it defines a bounded linear operator ϕ�w : L2

(U i,εm , μ|Ui,ε

m

)→ L2

(U i,εm , μ|Ui,ε

m

).

Then define Θ := Θi,εm := (#W i,ε

m )−1∑

w∈W i,εm

ϕ�w. We have Θ2 = Θ by ϕw ◦ϕw′ =

ϕw, w,w′ ∈ Wm. We claim that Θ is self-adjoint on L2(U i,εm , μ|Ui,ε

m

)and that

(5.10) Θ(FUi,ε

m

)⊂ FUi,ε

mand E(Θu, v) = E(u,Θv), u, v ∈ FUi,ε

m.

Indeed, let w ∈ W i,εm and let u ∈ F ∩C(K) satisfy suppK [u] ⊂ U i,ε

m . Then we have(u ◦ ϕw)1Ui,ε

m∈ C(K), suppK

[(u ◦ ϕw)1Ui,ε

m

]⊂ U i,ε

m ,((u ◦ ϕw)1Ui,ε

m

)◦ Fw′ = 0 for

w′ ∈ Wm \ W i,εm and

((u ◦ ϕw)1Ui,ε

m

)◦ Fw′ = u ◦ Fw ◦ gw,w′ ∈ F for w′ ∈ W i,ε

m by

Lemma 5.10, so that (u ◦ ϕw)1Ui,εm

∈ F ∩ C(K) by (GSCDF2) and

E((u ◦ ϕw)1Ui,ε

m, (u ◦ ϕw)1Ui,ε

m

)= (#W i,ε

m )r−mE(u ◦ Fw, u ◦ Fw) ≤ (#W i,εm )E(u, u).

These facts together with the regularity of (E ,F) easily implies ϕ�w

(FUi,ε

m

)⊂ FUi,ε

m

and hence Θ(FUi,ε

m

)⊂ FUi,ε

m. Moreover for u, v ∈ FUi,ε

m, by Lemma 5.10,

(#W i,εm )E(Θu, v) =

∑w∈W i,ε

m

E(ϕ�wu, v) =

∑w,w′∈W i,ε

m

1

rmE(u ◦ Fw ◦ gw,w′ , v ◦ Fw′),

which is seen to be equal to (#W i,εm )E(u,Θv) by the same calculation in the converse

direction and g−1w,w′ = gw,w′ = gw′,w. Thus (5.10) follows, and a similar calculation

also shows that Θ is self-adjoint on L2(U i,εm , μ|Ui,ε

m

). As a consequence, we can

easily verify that TUi,ε

mt Θ = ΘT

Ui,εm

t for any t ∈ (0,∞), in exactly the same way as[6, Proof of Proposition 2.21, (b) ⇒ (c)].

Next we set ιi,εm u := u◦ (g−1τ ◦F−1

τ ◦ϕτ )|Ui,εm

for u : Uε → [−∞,∞] and κi,εm u :=

u ◦ (Fτ ◦ gτ )|Uε for u : U i,εm → [−∞,∞], so that they define bounded linear oper-

ators ι := ιi,εm : L2(Uε, μ|Uε) → L2(U i,εm , μ|Ui,ε

m

)and κ := κi,ε

m : L2(U i,εm , μ|Ui,ε

m

)→

L2(Uε, μ|Uε). Clearly κι = idL2(Uε,μ|Uε ) and hence ι is injective. Similarly to the

proof of (5.10), we easily see κ(FUi,ε

m

)⊂ FUε , ι(FUε) ⊂ FUi,ε

m, hence

(5.11) ι−1(FUi,ε

m

)= FUε , and E(ιu, ιu) = (#W i,ε

m )r−mE(u, u), u ∈ FUε .

On the other hand, it follows by ϕw ◦ ϕw′ = ϕw, w,w′ ∈ Wm, that Θι = ι andικΘ = Θ, which together with the last assertion of the previous paragraph im-

ply that for any t ∈ (0,∞), TUi,ε

mt ι = T

Ui,εm

t Θι = ΘTUi,ε

mt ι = ικΘT

Ui,εm

t ι and hence

192 NAOTAKA KAJINO

TUi,ε

mt ι

(L2(Uε, μ|Uε)

)⊂ ι(L2(Uε, μ|Uε)

). Therefore

{ι−1T

Ui,εm

γ2mtι}t∈(0,∞)

is a well-

defined symmetric strongly continuous contraction semigroup on L2(Uε, μ|Uε), andthen in view of [11, Lemma 1.3.4-(i)], (5.11) means that its associated closed sym-

metric form is (EUε

,FUε). Thus TUε

t = ι−1TUi,ε

m

γ2mtι for any t ∈ (0,∞), which to-

gether with the uniqueness of pUε

immediately yields (5.8). Since Fτ ◦ gτ (Vε0 ) ⊂⋂

w∈W i,εm

Kw, (5.9) follows from (5.8) and the symmetry of pUε

t (x, y) and pUi,ε

mt (x, y)

in x, y. �

Proof of Theorem 5.11. We follow the notation of Proposition 5.12 in this

proof. Let ε := (1, 0, . . . , 0) ∈ {0, 1}d and 0 := (0, . . . , 0) ∈ Zd, so that W l−1ε,ε1 =

{0, ε} by (GSC4). By #S < ld and (GSC1), i− ε ∈ {0, 1, . . . , l − 1}d \ S for some

i ∈ S, and then W i,ε1 = {i}. Now for x ∈ V ε

0 , Fε(x), Fi(x) ∈ KI , and (5.9) implies

that 2pUl−1ε,ε

1t

(Fε(x), Fε(x)

)= p

Ui,ε1

t

(Fi(x), Fi(x)

)for any t ∈ (0,∞), from which

it follows that limt↓0 pt(Fi(x), Fi(x)

)/pt(Fε(x), Fε(x)

)= 2 by virtue of Lemma 3.3

and (5.5). Thus (2.14) holds and hence Theorems 2.17 and 2.18 apply. �

Acknowledgements. The author would like to thank Professor Jun Kigamifor having suggested to the author the problem of non-periodic oscillation of the on-diagonal heat kernels on self-similar fractals. The author also would like to thankDr. Mateusz Kwasnicki for a valuable comment on the statement of Theorem 2.18in an early version of the manuscript.

References

[1] Martin T. Barlow and Richard F. Bass, The construction of Brownian motion on theSierpinski carpet, Ann. Inst. H. Poincare Probab. Statist. 25 (1989), no. 3, 225–257 (English,with French summary). MR1023950 (91d:60183)

[2] Martin T. Barlow and Richard F. Bass, Local times for Brownian motion on the Sierpinskicarpet, Probab. Theory Related Fields 85 (1990), no. 1, 91–104, DOI 10.1007/BF01377631.MR1044302 (91j:60129)

[3] M. T. Barlow and R. F. Bass, On the resistance of the Sierpinski carpet, Proc. Roy. Soc.London Ser. A 431 (1990), no. 1882, 345–360, DOI 10.1098/rspa.1990.0135. MR1080496(91h:28008)

[4] Martin T. Barlow and Richard F. Bass, Transition densities for Brownian motion onthe Sierpinski carpet, Probab. Theory Related Fields 91 (1992), no. 3-4, 307–330, DOI10.1007/BF01192060. MR1151799 (93k:60203)

[5] Martin T. Barlow and Richard F. Bass, Brownian motion and harmonic analysis on Sier-

pinski carpets, Canad. J. Math. 51 (1999), no. 4, 673–744, DOI 10.4153/CJM-1999-031-4.MR1701339 (2000i:60083)

[6] Martin T. Barlow, Richard F. Bass, Takashi Kumagai, and Alexander Teplyaev, Uniquenessof Brownian motion on Sierpinski carpets, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3,655–701. MR2639315 (2011i:60146)

[7] Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpinski gasket, Probab.Theory Related Fields 79 (1988), no. 4, 543–623, DOI 10.1007/BF00318785. MR966175(89g:60241)

[8] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92,Cambridge University Press, Cambridge, 1989. MR990239 (90e:35123)

[9] William Feller, An introduction to probability theory and its applications. Vol. II., Secondedition, John Wiley & Sons Inc., New York, 1971. MR0270403 (42 #5292)

[10] Pat J. Fitzsimmons, Ben M. Hambly, and Takashi Kumagai, Transition density estimates forBrownian motion on affine nested fractals, Comm. Math. Phys. 165 (1994), no. 3, 595–620.MR1301625 (95j:60122)


[11] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetricMarkov processes, Second revised and extended edition, de Gruyter Studies in Mathematics,vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR2778606 (2011k:60249)

[12] Sheldon Goldstein, Random walks and diffusions on fractals, Percolation theory and er-godic theory of infinite particle systems (Minneapolis, Minn., 1984), IMA Vol. Math. Appl.,vol. 8, Springer, New York, 1987, pp. 121–129, DOI 10.1007/978-1-4613-8734-3 8. MR894545(88g:60245)

[13] Peter J. Grabner and Wolfgang Woess, Functional iterations and periodic oscillations forsimple random walk on the Sierpinski graph, Stochastic Process. Appl. 69 (1997), no. 1,127–138, DOI 10.1016/S0304-4149(97)00033-1. MR1464178 (98h:60104)

[14] Alexander Grigor’yan, Heat kernel upper bounds on fractal spaces, 2004, preprint.http://www.math.uni-bielefeld.de/~grigor/fkreps.pdf (accessed February 5, 2013)

[15] Alexander Grigor’yan, Jiaxin Hu, and Ka-Sing Lau, Comparison inequalities for heat semi-groups and heat kernels on metric measure spaces, J. Funct. Anal. 259 (2010), no. 10, 2613–2641, DOI 10.1016/j.jfa.2010.07.010. MR2679020 (2012c:58059)

[16] B. M. Hambly, Asymptotics for functions associated with heat flow on the Sierpinski carpet,Canad. J. Math. 63 (2011), no. 1, 153–180, DOI 10.4153/CJM-2010-079-7. MR2779136(2012f:35548)

[17] B. M. Hambly, V. Metz, and A. Teplyaev, Self-similar energies on post-critically fi-nite self-similar fractals, J. London Math. Soc. (2) 74 (2006), no. 1, 93–112, DOI10.1112/S002461070602312X. MR2254554 (2007i:31011)

[18] Masanori Hino, Upper estimate of martingale dimension for self-similar fractals, Probab.Theory Related Fields 156 (2013), no. 3-4, 739–793, DOI 10.1007/s00440-012-0442-3.MR3078285


[20] Naotaka Kajino, Remarks on non-diagonality conditions for Sierpinski carpets, Probabilisticapproach to geometry, Adv. Stud. Pure Math., vol. 57, Math. Soc. Japan, Tokyo, 2010,pp. 231–241. MR2648262 (2011c:28021)

[21] Naotaka Kajino, On-diagonal oscillation of the heat kernels on post-critically finite

self-similar fractals, Probab. Theory Related Fields 156 (2013), no. 1-2, 51–74, DOI10.1007/s00440-012-0420-9. MR3055252

[22] Naotaka Kajino and Alexander Teplyaev, Spectral gap sequence and on-diagonal oscillationof heat kernels, 2012, in preparation.


[24] Jun Kigami, Local Nash inequality and inhomogeneity of heat kernels, Proc. LondonMath. Soc. (3) 89 (2004), no. 2, 525–544, DOI 10.1112/S0024611504014807. MR2078700(2005f:47105)

[25] Jun Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem.Amer. Math. Soc. 199 (2009), no. 932, viii+94. MR2512802 (2010e:28007)

[26] Jun Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem.Amer. Math. Soc. 216 (2012), no. 1015, vi+132, DOI 10.1090/S0065-9266-2011-00632-5.MR2919892

[27] Jun Kigami and Michel L. Lapidus, Weyl’s problem for the spectral distribution of Laplacianson p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), no. 1, 93–125. MR1243717(94m:58225)

[28] Bernhard Kron and Elmar Teufl, Asymptotics of the transition probabilities of the simplerandom walk on self-similar graphs, Trans. Amer. Math. Soc. 356 (2004), no. 1, 393–414(electronic), DOI 10.1090/S0002-9947-03-03352-X. MR2020038 (2004k:60130)

[29] Takashi Kumagai, Estimates of transition densities for Brownian motion on nested frac-tals, Probab. Theory Related Fields 96 (1993), no. 2, 205–224, DOI 10.1007/BF01192133.

MR1227032 (94e:60068)[30] Shigeo Kusuoka, A diffusion process on a fractal, Probabilistic methods in mathematical

physics (Katata/Kyoto, 1985), Academic Press, Boston, MA, 1987, pp. 251–274. MR933827(89e:60149)

194 NAOTAKA KAJINO

[31] Shigeo Kusuoka and Zhou Xian Yin, Dirichlet forms on fractals: Poincare constant and resis-tance, Probab. Theory Related Fields 93 (1992), no. 2, 169–196, DOI 10.1007/BF01195228.MR1176724 (94e:60069)

[32] Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 (1990),no. 420, iv+128. MR988082 (90k:60157)

[33] Roberto Peirone, Existence of eigenforms on fractals with three vertices, Proc. Roy. Soc. Ed-inburgh Sect. A 137 (2007), no. 5, 1073–1080, DOI 10.1017/S0308210505001137. MR2359927

(2009e:28036)[34] Roberto Peirone, Existence of eigenforms on nicely separated fractals, Analysis on graphs

and its applications, Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI,2008, pp. 231–241. MR2459872 (2010h:31011)

[35] Roberto Peirone, Existence of self-similar energies on finitely ramified fractals, 2011, preprint.http://hal.archives-ouvertes.fr/docs/00/62/86/61/PDF/exist4.pdf (accessed February5, 2013)

Department of Mathematics, University of Bielefeld, Postfach 10 01 31, 33501

Bielefeld, Germany

Current address: Department of Mathematics, Graduate School of Science, Kobe University,Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan

E-mail address: [email protected]: http://www.math.kobe-u.ac.jp/HOME/nkajino/


Lattice Effects in the Scaling Limitof the Two-Dimensional Self-Avoiding Walk

Tom Kennedy and Gregory F. Lawler

Abstract. We consider the two-dimensional self-avoiding walk (SAW) in asimply connected domain that contains the origin. The SAW starts at theorigin and ends somewhere on the boundary. The distribution of the endpointalong the boundary is expected to differ from the SLE partition function pre-diction for this distribution because of lattice effects that persist in the scalinglimit. We give a precise conjecture for how to compute this lattice effect cor-rection and support our conjecture with simulations. We also give a preciseconjecture for the lattice corrections that persist in the scaling limit of theλ-SAW walk.

In his book Fractal Geometry of Nature [15], Mandelbrot observed from nu-merical simulations that the outer boundary of two-dimensional Brownian motionappeared to have the same fractal dimension as that of the self-avoiding walk. Thiswas surprising at the time, but later work [10] has confirmed the equivalence atthe continuum level. In fact, the precise nature of the convergence has been pre-dicted and these predictions have been confirmed numerically. However, we stilldo not know how to establish the continuum limit rigorously. This paper contin-ues the nonrigorous analysis by studying the nature of the lattice corrections of aself-avoiding walk in a domain with smooth boundaries that are not parallel to thecoordinate axes.

1. Introduction

Let D be a bounded, simply connected domain in the complex plane thatcontains 0. We are interested in the self-avoiding walk (SAW) in D starting at theorigin and ending on the boundary of D. It is defined as follows. We introduce alattice with spacing δ > 0, e.g., δZ2. A self-avoiding walk is a nearest neighbor walkon the lattice with the property that it does not visit any site more than once. To beprecise, let W(D, δ) denote the set of functions of the form ω : {0, 1, 2, · · · , n} → δZ2

where n is a positive integer; ||ω(i) − ω(i− 1)|| = δ for i = 1, 2, · · · , n; ω(i) �= ω(j)for 0 ≤ i < j ≤ n; ω(0) = 0; ω(j) ∈ D, j < n; ω(n) �∈ D. The integer n is thenumber of steps in the SAW, and from now on we will denote it by |ω|.

2010 Mathematics Subject Classification. Primary 82B41, 60J67 .Tom Kennedy’s research is supported by NSF grant DMS-0758649.Greg Lawler’s research is supported by NSF grant DMS-0907143.


195

196 TOM KENNEDY AND GREGORY F. LAWLER

Since W(D, δ) is finite, we can define a probability measure on W(D, δ) bytaking the probability of ω to be proportional to β|ω| where β > 0 is a parameter.So

P(ω) = PD,δ(ω) =β|ω|

Z(D)(1.1)

where the partition function Z(D) = Zδ(D) is defined by the requirement that thisbe a probability measure. One can consider this model for all β > 0, but it is mostinteresting for one particular value that makes the model critical, β = 1/μ, whereμ is the connective constant which we define next.

Let cN be the number of SAW’s in the lattice with N steps that start at0. (They are not constrained to lie in D.) It is known that this number growsexponentially with N in the sense that the following limit exists [14].

μ = limN→∞

c1/NN .(1.2)

The connective constant μ depends on the lattice. Nienhuis [16] predicted that for

the hexagonal lattice μ =√

2 +√

2, and this was recently proven by Duminil-Copinand Smirnov [4]. For the square and triangular lattices there are only numericalestimates of the value of μ. For the remainder of this paper we will take β = 1/μto make the model critical.

We also consider the analogy of the above definition with the ordinary randomwalk. The natural way to describe the random walk in D starting at 0 and endingon the boundary of D is to start a random walk at 0 and run it until it hits theboundary. Let S(D, δ) be the set of such walks. The probability of a particularsuch random walk ω is z−|ω| where z is the coordination number of the lattice, e.g.,z = 4 for the square lattice. We can consider this as a random walk starting atthe origin stopped when it leaves D, or just as a measure on S(D, δ) that assignsmeasure z−n to each walk of length n. In the SAW the connectivity constant μplays the role of the coordination number z for the random walk.

In both the SAW and the ordinary random walk we are interested in the scalinglimit in which the lattice spacing δ → 0. Let us first discuss the case of ordinaryrandom walk which is well understood.

For the ordinary random walk the scaling limit is Brownian motion starting at0 and stopped when it hits the boundary of D. The distribution of the endpointof the Brownian motion on the boundary is harmonic measure. The lattice effectsassociated to the definition of the first boundary point of the lattice walk disappearin the scaling limit. The key fact is that if a random walk or a Brownian motiongets very close to the boundary, then it will hit it soon. Therefore, if we couple arandom walk and a Brownian motion on the same probability space so the paths areclose, then the first time that the random walk hits the boundary will be close tothe first time that the Brownian motion hits the boundary. See [8, Section 7.7] fora precise statement. The argument there works for any simply connected domaineven with nonsmooth boundaries, and extends to finitely connected domains aswell. One does need to assume that the boundary is sufficiently large so that whenthe Brownian motion or random walk gets close, then it is very likely to hit it soon.

If the boundary of our domain D is a piecewise smooth curve, then harmonicmeasure is absolutely continuous with respect to arc length along the boundary[17]. We let hD(z) denote its density with respect to arc length; this is often called

LATTICE EFFECTS IN 2d SELF-AVOIDING WALK 197

the Poisson kernel (starting at 0). If f is a conformal map on D that fixes the originand such that the boundary of f(D) is also piecewise smooth, then the conformalinvariance of Brownian motion [13] implies that the density for harmonic measureon the boundary of f(D) is related to the density on the boundary of D by

(1.3) hD(z) = |f ′(z)|hf(D)(f(z)).

If D is simply connected and we take gD to be a conformal map of D onto theunit disc which fixes 0, then by symmetry hf(D)(f(z)) is just 1/2π, and so hD(z) =|g′D(z)|/2π. We emphasize that conformal invariance of harmonic measure impliesthat the exponent of |f ′(z)| in (1.3) equals one.

We now consider the SAW in D from the origin to the boundary of D. Herewe will assume that ∂D is piecewise smooth. In this case we have the followingconjecture.

• As δ → 0, the measures PD,δ converge to a probability measure PD onsimple paths from the origin to ∂D.

• The measure PD can be written as∫∂D

ρD(z)PD(0, z) |dz|,(1.4)

where ρD(z) is the density of a probability measure on ∂D and PD(0, z)is the probability measure associated to radial SLE8/3 from 0 to z in D.

• There exists a periodic function, l(θ), such that

ρD(z) = l(θ(z,D)) ρD(z)(1.5)

where θ(z,D) is the angle of the tangent to ∂D at z and ρD(z) is amultiple of the SLE8/3 partition function [7]. The function l(θ) and itsperiod depend on the lattice. For example, on the square lattice the periodis π/2. (We have only assumed the boundary is piecewise smooth, so thereare points were θ is not defined. This is not a problem since ρD(z) is adensity.)

• The density ρD(z) transforms under conformal maps by

ρD(z) = c|f ′(z)|5/8ρf(D)(f(z))(1.6)

The constant c is determined by the constraint that this be a probabilitydensity. (It depends on D and f .) In particular, if D is simply connected,and gD : D → D with gD(0) = 0, then

ρD(z) = c|g′D(z)|5/8(1.7)

(D denotes the unit disc centered at the origin.)

In the case of Z2 where the boundary of D is composed of horizontal and verticalline segments, this conjecture was made by Lawler, Schramm and Werner [10].Simulations on an infinite horizontal strip [3] give strong support to the conjecture.For such boundaries, l(θ(z,D)) is constant. The conjecture was reiterated in [9]where it was also conjectured for other domains “...after taking care of the locallattice effects.” Our conjecture makes precise the nature of the lattice effects interms of the lattice correction l(θ(z,D)). The function l(θ) will depend on just howwe define “ending on the boundary of D” and on the lattice type. In general, weuse ρ to denote densities that do not include the lattice effects that persist in thescaling limit, e.g., (1.6), and we use ρ to denote densities that do include the latticeeffects, e.g., (1.5).


Unfortunately, the Monte Carlo algorithms for simulating the above ensembleare local algorithms and so are not very efficient [14]. We have not attempted totest the conjecture for this ensemble by simulation. Instead we introduce anotherensemble that we can study with the pivot algorithm, a fast global Monte Carloalgorithm. Instead of stopping a walk at a boundary point, one chooses an infinitelength walk conditioned on the event that it crosses the boundary only once. We willrefer to this ensemble as the “cut-curve” ensemble since the boundary of the domaincuts the SAW into two SAW’s, one contained in D and one in the complement ofD. The above ensemble and the cut-curve ensemble are the same for the infinitestrip studied in [3], but for most domains this is not the case.

The scaling limits of the SAW and the loop-erased walk, which is obtainedby erasing loops from the ordinary random walk, are two cases of the Schramm-Loewner evolution (SLE). The discrete models can be considered as special cases ofthe λ-SAW. We review the λ-SAW in the next section, and we extend our conjectureto this case. (There is no precise conjecture in the literature on the nature of thelattice correction, and we think it is worthwhile to write it down.) After that weconsider the conjectured scaling limits of the two ensembles — walks stopped uponreaching the boundary and walks conditioned to hit the boundary only once. Inonly the κ = 8/3 (SAW) case do we expect an equivalence of these ensembles, andtherefore the tests we do here would not work for other values of κ.

In section two we return to the SAW and give explicit conjectures for the latticecorrection function l(θ) for the two lattice ensembles. In section three, we discusssimulations for the cut-curve ensemble including numerical calculation of the latticecorrection function. In the final section we summarize our results and discuss thelattice correction function l(θ) for other interpretations of the SAW ending on theboundary of the domain.

1.1. λ-SAW. The conjectures for the SAW and the results for the ordinarywalk (considered in terms of the loop-erasure of the paths) are particular cases ofconjectures for a model called the λ-SAW introduced by Kozdron and Lawler [6].Even though we are only testing the SAW conjecture, we will give the conjecturesfor the general model. There are two versions, chordal (boundary-to-boundary)and radial (boundary-to-interior); we will restrict our discussion here to the radialcase. (There is also an interior-to-interior case, but then there is no boundarylattice correction so it is not relevant for this paper.) As above, we assume thatD is a domain with piecewise smooth boundary. For convenience we use Z2 forour lattice, but the definition can be extended to other lattices. The parameter λcan be considered a free parameter, but we will set λ = −c/2 where c ≤ 1 denotescentral charge. (We will not define central charge in this paper and can just take itas a parameter.) The λ-SAW is a model that is conjectured to have a scaling limitof SLEκ where

κ =13 − c−

√(13 − c)2 − 144

3∈ (0, 4].

The (rooted) random walk loop measure is the measure on ordinary randomwalk loops that assigns weight (1/4)n n−1 to each loop of n > 0 steps. A loop is apath which begins and ends at the same point. The measure can also be consideredas a measure on unrooted loops by forgetting the root.

For each lattice spacing δ and each SAW ω as above, we let mD(ω) denote thetotal measure of the set of loops that lie entirely in D and share at least one site


with ω. If β > 0, the λ-SAW gives each SAW ω as above weight

q(ω) = qδ,D,c,β(ω) = β|ω| exp{−c

2mD(ω)

}.

The partition function is

Z(D; δ) = Z(D; δ, c, β) =∑

q(ω),

where the sum is over all SAWs on the lattice δZ2 that start at the origin andend at the boundary. (As noted before, there are several definitions for “ending atthe boundary”. In the discussion here, we fix one such definition and the lattice-dependent quantities depend on the choice.) It is conjectured that for each c, thereis a critical value β = βc such that Z(D; δ) follows a power law in δ as δ → 0. Weassume this conjecture and fix β at the critical value.

If c = 0, then this is the usual SAW model. If c = −2, then this is the loop-erased random walk (LERW) which is obtained by taking the ordinary random walkas above and erasing loops chronologically from the paths. The partition functionZ(D; δ) for the loop-erased walk is exactly the same as that for the usual randomwalk. (See [8, Chapter 9] for a discussion of this.) We state our conjectures interms of the boundary and interior scaling exponent for SLEκ:

b =6 − κ

2κ∈ [1/4,∞), b =

b (κ− 2)

4.

• There is a lattice correction function l(θ) that is continuous, strictly pos-itive and periodic.

• If ω is from 0 to ∂D, let l(ω) denote l(θ(z,D)) where z is the first pointon ∂D hit by a bond of ω and θ(z,D) is the angle of the tangent to ∂Dat z. Define the lattice-corrected weight by

q(ω) =q(ω)

l(ω).

• As δ → 0, the measure δ1−b−b q on paths converges to a nontrivial finitemeasure νD on simple paths from 0 to ∂D. It can be written as

νD =

∫∂D

ρD(z) ν#D (0, z) |dz|,

where ρD is a positive function and ν#D (0, z) is a probability measure onsimple paths starting at 0 and leaving D at z.

• If g is a conformal transformation with g(0) = 0 that is smooth on ∂D,then

(1.8) ρD(z) = |g′(z)|b |g′(0)|b ρg(D)(z).

• The probability measures ν#D (0, z), considered as measures on curves mod-ulo reparametrization, are conformally invariant. More specifically, ifγ : [0, t0) → D is a curve with γ(t0−) ∈ ∂D, let

σ(t) =

∫ t

0

|g′(γ(s))|d ds, d = 1 +κ

8,

and define g ◦ γ(t) by

g ◦ γ(σ(t)) = g(γ(t)), 0 ≤ t < t0.


If g ◦ μ#D(0, z) denotes the induced measure on curves on g(D), then

g ◦ ν#D(0, z) = ν#g(D)(0, z).

• If D is simply connected, then ν#D (0, z) is the reversal of radial SLEκ fromz to 0 in D with the natural parametrization. See [11] for the definitionof this parametrization. (This should also be true for multiply connectedD under the appropriate definition of SLEκ proposed in [7].)

These conjectures are a long way from being proved. Indeed, a special caseis the SAW model which is a notoriously difficult problem! However, we statethem here to see that the precise conjecture requires discussing a boundary latticecorrection; our conjecture is that the correction only depends on the angle of theboundary. We make a number of comments.

• The density ρD(z) is sometimes called the partition function for radialSLEκ. It is defined up to an arbitrary constant.

• The loop-erased walk (c = −2) is particularly nice because it is closelyrelated to the ordinary random walk. The partition function is the Pois-son kernel (density of harmonic measure) even for non-simply connecteddomains.

• For other values of c, if D is simply connected, the partition function isthe Poisson kernel raised to the bth power. However, this is not true formultiply connected domains.

• The loop terms mD(ω) can be considered as having three parts. The verysmall loops that occur away from the boundary contribute a microscopic(lattice dependent) part that affects the critical value of β. The largeloops contribute a macroscopic term that is seen in the scaling limit; thisis the Brownian loop measure as defined in [12]. Finally, there are thesmall loops that occur near the boundary. They give both a macroscopiceffect seen in the exponent b and a microscopic effect in the function l.The boundary effect is measured both in the number of walks that stayone one side of a line and in the measure of loops that go on the walks.We only see the first effect in the SAW (c = 0) case.

• If κ = 8/3, then b = 5/8, b = 5/48. There is a |g′(0)|b factor in (1.8) thatdoes not seem to appear in (1.6). However, it is implicitly there in thenormalization to make the measure a probability measure.

1.2. Cut-curve configurations. We will be testing a cut-curve ensemble forSAW’s. The case c = 0, which is what we use in this paper, is special for suchconfigurations and agrees with the bridge decomposition of restriction measures[1], but for the sake of completeness let us discuss the general case for c ≤ 1.Suppose D is a bounded, simply connected domain containing the origin whoseboundary ∂D is a smooth Jordan curve. Let D∗ be the unbounded component ofC\∂D. We consider two measures on simple curves from 0 to infinity that intersect∂D only once:

(1) Take “whole-plane” SLEκ and condition on the event that the curve in-tersects ∂D only once. This is conditioning on an event of probabilityzero, so one must define this in terms of a limit.

(2) Take independent copies of radial SLEκ in D and in D∗ and conditionthem to hit the same point in ∂D. Then concatenate the paths.


In order to see the difference, let us describe the lattice models that we expect toconverge to these measures. Since it is hard to talk about infinite walks, we willchoose a point z∗ ∈ D∗ with large absolute value. We take the scaling limit forfixed z∗ and then let z∗ go to infinity. If δ is a scaling factor, we abuse notationslightly and write z∗ for a point in δZ2 closest to z∗. We consider two measureson paths. For each δ, we let W(δ,D, z∗) be the set of SAW’s from 0 to z∗ thatonly cross the boundary of D once. More precisely, it is the set of SAW’s ω on δZ2

with ω(0) = 0, ω(|ω|) = z∗ for which there is only one bond that intersects ∂D.For ω in this set we also require that |ω(j)| ≤ |z∗|2 for all j ≤ |ω|. This constraint,which should become irrelevant in the limit that z∗ goes to infinity, makes the setW(δ,D, z∗) finite and so insures that the measures we are defining are finite.

We write zω, wω for the vertices in this bond with zω ∈ D,wω ∈ C \D, and wewrite ω = ωD ⊕ [zω, wω] ⊕ ω∗ where

ωD = [ω(0), . . . , zω], ω∗ = [wω, . . . , ω(|ω|)].Let β = βc be the critical value. Then we consider the following measures onW(δ,D, z∗):

(1) Each ω gets weight

q1(ω) = β|ω| exp{−c

2m(ω; z∗)

},

where m(ω; z∗) denotes the measure of loops staying in the ball of radius|z∗|2 that intersect ω.

(2) Each ω gets weight

q2(ω) = β|ω| exp{−c

2[mD(ωD) + mD∗(ω∗)]

}We can state our conjectures as follows. Let θ(ω) be the angle of the tangent to∂D at the point where ω hits ∂D.

(1) There exists a lattice correction function l1 such that we can take thescaling limit of the measure

q1(ω) =q1(ω)

l1(ω).

If we then take z∗ → ∞, we get whole plane SLEκ conditioned to hit ∂Donly once.

(2) There exists a lattice correction function l2 such that the scaling limit ofthe measure

q2(ω) =q2(ω)

l2(ω)

exists. If we then take z∗ to infinity, we get the measure given by

c

∫∂D

ρD(z) ρ∗D(z)[ν#D (0, z) ⊕ ν#D∗(z,∞)

]|dz|.

Here ρ∗D(z) is the density as in (1.8) for radial SLEκ in D∗ centered atinfinity.

For c = 0 (restriction measures), the two limits agree and this is what weuse for SAW in this paper. For other values of c we get different measures. Forexample if c = −2 (loop-erased walk), the first measure corresponds to loop-erased


walk conditioned to hit ∂D only once and the second measure corresponds to theloop-erasure of an ordinary walk conditioned to hit ∂D only once.

2. Lattice effects

The constraint that the SAW stays in D has both a macroscopic and a micro-scopic effect on the boundary density. The macroscopic effect is captured by theconjecture (1.6). The microscopic effect comes from the behavior near the endpointof the walk on the boundary of D. Consider a SAW that ends at z ∈ ∂D and con-sider the tangent line to the boundary at z. The constraint that the SAW stays inD implies that near z the SAW must stay on one side of this line. Loosely speaking,the number of SAW’s that end at z and stay on one side of this line depends onthe orientation of the line with respect to the lattice. The result is a factor l(θ)that depends on the angle of the tangent line with respect to the lattice, and so weobtain our conjecture (1.5).

The lattice correction function l(θ) depends on the type of lattice and on how weinterpret “ending on the boundary of D.” We will first discuss the interpretation weintroduced at the start of the introduction. We consider all SAW’s ω with ω(0) = 0,ω(i) ∈ D for i = 0, 1, 2, · · · , |ω| − 1 and ω(|ω|) /∈ D. (|ω| denotes the number ofsteps in ω.) So the last bond of the SAW intersects the boundary of D, and this isthe only bond in the SAW that intersects the boundary.

We can compute the lattice correction function l(θ) as follows. We give thedetails for the square lattice. Other lattices, e.g., the triangular or hexagonal, willrequire some modifications. Consider a bond that intersects the boundary ∂D. Letz be the endpoint of the bond that is in D. Let w be the point where the bondintersects the boundary (typically not a lattice site). Consider the tangent line tothe boundary at w. We need to count the number of SAW’s of a fixed length Nthat end at z and do not intersect this tangent line. This quantity will depend onthe angle of the tangent line with respect to the lattice. It will also depend on thedistance |w − z| and on whether the bond is horizontal or vertical. However, thesetwo factors are the only dependence on the bond.

Motivated by the above, we consider the vertical bond between (0, 0) and (0, 1).We fix an l in the interval [0, 1] and an angle θ. The parameter l plays the roleof the distance |w − z|. Let L be the line with polar angle θ passing through thepoint (0, l). We consider SAW’s with N steps ending at the origin which by reversalcan be considered as beginning at the origin. Let cN be the number of such walks,and let aN (l, θ) be the number of such walks that do not intersect the line. SoaN (l, θ)/cN is the probability that the N step SAW does not intersect the line. Weexpect that there exists a function pv(l, θ) such that

aN (l, θ)

cN∼ pv(l, θ)N−ρ, N → ∞

with ρ = 25/64 [10]. The actual exponent is not important here. The key is thatif l′, θ′ are two different values, then

aN (l, θ)

aN (l′, θ′)∼ pv(l, θ)

pv(l′, θ′), N → ∞.

We define

pv1(l, θ) = limN→∞

aN (l, θ)

cNNρ(2.1)


(We do not know how to prove this limit exists.) The superscript v on p indicatesthat we took the bond crossing the boundary to be vertical. We use the subscript1 to distinguish the quantities related to the lattice correction function for thisparticular ensemble from the ensemble that we will consider next. We let ph1 denotethe analogous quantity for a horizontal bond. The symmetry of the square latticeimplies that ph1(l, θ) = pv1(l, θ + π/2). (Note that ph1 (l, θ) and pv1(l, θ) have period πin θ.)

Now consider what happens as we move along the boundary. The angle θof the tangent will vary smoothly. The distance l will not. As long as tan(θ) isnot rational, l will be distributed uniformly between 0 and 1. So in the scalinglimit, averaging over an infinitesimal section of the boundary will be equivalent toaveraging l over [0, 1]. So we define

px1(θ) =

∫ 1

0

px1(l, θ) dl, x = h, v

The function px1(θ) captures the microscopic lattice effect caused by the con-straint that the SAW must stay on one side of a line as it approaches the boundary.There is another lattice effect that comes from the density of bonds that cross theboundary. Define bv(θ) to be the density of vertical bonds along a line with po-lar angle θ, i.e., the average number of vertical bonds that intersect the line perunit length. We have bv(θ) = | cos(θ)|. The density of horizontal bonds bh(θ) isbv(θ − π/2) = | sin(θ)|. The lattice correction function is then

l1(θ) = bv(θ)pv1(θ) + bh(θ)ph1(θ)(2.2)

and the boundary density for this first interpretation of ending on the boundary is

ρD,1(z) = c1|g′D(z)|5/8l1(θ(z,D)),(2.3)

whre gD is a conformal map of D onto D fixing the origin.

The Monte Carlo algorithms for simulating the above ensemble are local al-gorithms and so are not very efficient [14]. We have not attempted to test theconjecture for this ensemble by simulation. Instead we use the cut-curve ensembleintroduced earlier. We consider the infinite length SAW in the full plane startingat 0. We condition on the event that the SAW crosses the boundary of D exactlyonce. Consider a bond crossing the boundary, and let z be the endpoint insideD and w the endpoint outside of D. If we condition on the event that the SAWcontains this particular bond and this is the only bond in the SAW that crosses theboundary, then we have a SAW in D from 0 to z and a SAW in the exterior of Dfrom w to ∞. So the conjecture for the boundary density will be a product of twofunctions. The interior SAW from 0 to z gives a factor of ρD(z) with ρD(z) givenby (1.6). The exterior SAW from w to ∞ gives a factor of ρ∗D(z) given by

ρ∗D(z) = c|h′D(z)|5/8

where hD is the conformal map of D∗ onto the unit disc D with hD(∞) = 0. Ourconjecture for the density of the point where the SAW crosses the curve is

ρD,2(z) = ρD(z) ρ∗D(z) l2(θ(z,D)),(2.4)

where l2(θ) is the lattice correction function for the cut-curve ensemble. (Thesubscript 2 indicates that the quantities are for the cut-curve ensemble.)


We can compute l2(θ) as follows. Again, we restrict our attention to the squarelattice. For l ∈ [0, 1] and an angle θ, let L be the line with polar angle θ passingthrough (0, l). We consider SAW’s with N = 2n + 1 steps such that the middlebond is the bond between (0, 0) and (0, 1). Let dN be the number of such walks,and let bN (l, θ) be the number of such walks that only intersect the line once. (Ofcourse it must be the middle bond that has the intersection). (By translating ourSAW’s so that they start at 0, we see that dN = cN/2.) Note that bN (l, θ)/dNis the probability the N step SAW has no intersection with the line other thanthe middle bond. We can think of generating this N step SAW by first generatingtwo n step SAW’s which are independent and attach to (0, 0) and (0, 1) and thenkeeping them only if they mutually self avoid. The probability they are mutuallyself-avoiding goes as N1−γ with γ = 43/32. The probability that both of the n stepSAW’s do not intersect the line goes as N−2ρ with ρ = 25/64 [10]. Note that ifboth of them do not intersect the line, then they are mutually self-avoiding. Hencewe expect that there exists a function pv2(l, θ) such that

bN (l, θ)

dN∼ pv2(l, θ)N

−2ρ+γ−1

We define

pv2(l, θ) = limN→∞

bN (l, θ)

dNN2ρ−γ+1.

Then we define

pv2(θ) =

∫ 1

0

pv2(l, θ)dl

As in our derivation of the lattice correction function for the first ensemble, theintegral over l comes from averaging over bonds crossing a small section of theboundary. We define ph2 (l, θ) and ph2(θ) in the analogous way. As before we letbh(θ) and bv(θ) denote the densities of horizontal and vertical bonds along a linewith polar angle θ. The lattice correction function is then

l2(θ) = bv(θ)pv2(θ) + bh(θ)ph2(θ)(2.5)

In the above discussion we have considered SAW’s starting at an interior pointin the domain. The same discussion applies to an ensemble of SAW’s that startat a prescribed boundary point of D. The conjecture for the boundary densityagain transforms according to (1.6). With the starting point on the boundary thisdensity is not normalizable. We must restrict the endpoint of the SAW to a subset ofthe boundary that is bounded away from the starting point to get a normalizabledensity. A useful reference domain in this case is the upper half plane with thestarting point at 0. The unnormalized density for the harmonic measure is x−2 andso for the SAW it is x−5/4.

3. Simulations

In this section we study the cut-curve ensemble by Monte Carlo simulations.There are two types of simulations. We compute the lattice correction functionl2(θ) by simulation, and we do simulations of the SAW in two different geometriesto test the conjecture (2.4). We first discuss the computation of l2(θ).

Recall that for odd N , dN is the number of SAW’s with N steps such thatthe middle bond is the bond between (0, 0) and (0, 1). For l ∈ [0, 1] and an angle


θ, bN (l, θ) be the number of such SAW’s whose only intersection with the linethrough the point (0, l) at angle θ is in this middle bond. The ratio bN (l, θ)/dNis a probability and so may be computed as follows. We use the pivot algorithmto generate SAW’s with N steps that start at the origin and such that the middlebond is vertical. We then pick a point on this bond uniformly at random and takethe line with angle θ to go through this point. We test if the only intersection ofthe SAW with the line is through the middle bond. We find the fraction of thesamples that satisfy this condition and multiply it by N2ρ−γ+1. The result is anestimate of pv2(θ). Note that we have included the integral over l from 0 to 1 in thesimulation by randomly choosing l uniformly from [0, 1] for each sample.

We did this simulation for values of N ranging from 101 to 5001. For thesmaller values of N , one can clearly see finite N effects. As is always the case withsimulations of the SAW, the time required grows with N . However, in this simu-lation this is exacerbated by the fact that the probability the SAW only intersectsthe line once goes to 0, and we must multiply the probability we compute in thesimulation by N2ρ−γ+1. Even with 2 billion samples, our simulations for N = 2001and 5001 have significant statistical errors. The results for N = 501 and N = 1001differ by at most 0.05%, and we use N = 1001 in this paper. We generated 5 billionsamples for this case which took about 67 cpu-days on a rather old cluster with 2.4GHz cpu’s. Figure 1 shows the function pv2(θ). In all our figures we give the angleθ in degrees.

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

-100 -80 -60 -40 -20 0 20 40 60 80 100Theta

Figure 1. The function pv2(θ).

We now turn to the simulations to test conjecture (2.4). We use the pivotalgorithm to generate walks in the full plane with a constant number of steps N .We take N = 1, 000, 000. The lattice spacing and N are such that the size of theSAW is large compared to the domain D so that the SAW is effectively infinite.We condition on the event that the SAW intersects the boundary of D exactlyonce. The probability of this event goes so zero in the scaling limit, so we must


generate very large numbers of SAW’s to get good statistics. We use Clisby’s fastimplementation of the pivot algorithm using binary trees [2].

In our first test we take the domain for the cut-curve ensemble to be a disccentered at 0 where the SAW starts. We take the lattice spacing to be N−ν andtake the radius of the disc to be R = 0.2. (With R = 0.3 the effect of the finitelength of the SAW begins to be noticeable. At R = 0.4 it is quite noticeable.) Inthe simulation we sample the Markov chain every 1000 iterations and generate atotal of approximately 47 million samples. Just over 10% of these samples satisfythe condition that the SAW only intersects the boundary of the circle once, and wehave approximately 4.9 million samples of the boundary density.

In this geometry both ρD(z) and ρ∗D(z) are constant, so the prediction for theboundary density without lattice effects is just the uniform density. The angleθ(z,D) of the tangent line at z is equal to the polar angle θ of z mod 90 degrees.So if we think of the boundary density as a function of the polar angle θ, thenour conjecture (2.4) is that the boundary density is proportional to l2(θ). Figure2 shows the function l2(θ) and the boundary density we find in the simulation ofthe cut-curve ensemble. Both functions have a period of 90 degrees. We plot theboundary density as a function of θ mod 90. Both curves are normalized so thatthe total area under the curves is one.

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0 10 20 30 40 50 60 70 80 90Theta

Figure 2. The function l2(θ) (the curve) and the den-sity for the full plane SAW conditioned to hit a circleexactly once (the histogram). Note that the range ofthe vertical axis does not start at 0.

Figure 2 compares densities. Actually, the function plotted for the simulationof the cut-curve ensemble is a histogram. So the points plotted correspond to theaverage value of the density over a small interval. The simulations do not computedensities directly. Finding the density requires taking a numerical derivative, i.e.,


computing a histogram. We can avoid this extra source of numerical error byworking with cumulative distribution functions (cdf’s) rather than densities.

In figure 3 we study the cdf for the cut-curve ensemble using the unit disccentered at 0. We plot two curves. One is the cdf we find in the simulation of thecut-curve ensemble minus the cdf for the uniform density, i.e., the density givenby (1.7). (In this figure we have again taken advantage of the periodicity of theunderlying density functions.) This difference is small with the maximum beingslightly less than 2%, but it is clearly not zero. In the second curve we show thecdf for the simulation of the cut-curve ensemble minus the cdf corresponding to thedensity given by (2.4), i.e., corresponding to l2(θ). The difference is on the order of0.02%. The error bars in the figure are two standard deviations for the statisticalerrors, i.e., the error that comes from not running the Monte Carlo simulationforever. There are also errors in the simulation from two other sources - the finitelength of the SAW and the nonzero lattice spacing. We have studied the error fromthe finite length of the SAW by simulating the ensemble with several values of theradius of the disc. With R = 0.2 we believe that the error from the finite length ofthe SAW is much smaller than the statistical errors. The nonzero lattice spacingmeans that all our random variables are at a small scale discrete random variables.This is reflected in the slightly jagged nature of the second curve. The error fromthe nonzero lattice spacing appears to be comparable in size to the statistical error.Thus the difference between the cdf from the simulation and the cdf given by (2.4)is zero within the errors in our simulation. Figure 3 gives evidence that there areindeed lattice effects that must be taken into account in the boundary density andthat our conjecture correctly accounts for these lattice effects.

For our second test of conjecture (2.4), we consider the SAW in the upper halfplane, starting at the origin. We take the cut-curve to be a semi-circle centeredat the origin. Again, we take the lattice spacing to be N−ν and the radius of thesemi-circle disc to be R = 0.2. We sample the Markov chain every 1000 iterationsand generate a total of approximately 27 million samples. Approximately 13% ofthese samples satisfy the condition that the SAW only intersects the boundary ofthe circle once, and we have approximately 3.6 million samples of the boundarydensity for this geometry.

The ensemble consists of all SAW’s in the upper half plane which start at0 and only cross the semicircle once. In this geometry the arc length along thesemicircle equals the polar angle θ. So we will express densities as functions of θ. Asimple computation using the conformal map z+1/z shows that the interior densityρD(θ) is [sin(θ)]5/8. The exterior density ρ∗D(θ) is exactly the same. (This is just aconsequence of the symmetry of our geometry under the inversion z → −1/z.) Soour conjecture for the density along the cut-curve is proportional to

[sin(θ)]5/4 l(θ)(3.1)

The comparison of our simulation of the cut-curve ensemble cdf for the SAWand our conjecture is shown in figure 4. Again we plot two curves. For one curve wefind the cdf corresponding to just the density function [sin(θ)]5/4. This would be theconjectured cdf if there were no lattice effects. We plot the cdf for the simulationof the cut-curve ensemble minus the cdf corresponding to just [sin(θ)]5/4. Thisdifference is small (the max is on the order of 1.5%), but is clearly not zero. Forthe other curve we compute the cdf corresponding to our conjecture (3.1) with the


-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 10 20 30 40 50 60 70 80 90Polar angle

no lattice correctionwith lattice correction

Figure 3. The simulation of the cdf for the bound-ary density for the cut-curve ensemble for the first ge-ometry minus the theoretical prediction. The largercurve does not include the lattice correction function;the smaller curve does.

lattice effect and subtract this function from the simulation of the cdf for the cut-curve ensemble. The difference is on the order of 0.05% which is zero within theerrors in our simulation. This figure gives further evidence that there are indeedlattice effects that must be taken into account in the boundary density and thatour conjecture correctly accounts for these lattice effects.

4. Conclusions and future work

We have considered the ensemble of SAW’s in a simply connected domain con-taining the origin which start at the origin and end on the boundary. It has beennoted before that the conjecture for this boundary density will have lattice effectsthat persist in the scaling limit. We have conjectured that this lattice effect is givenby multiplying the density by a function l(θ(z,D)) where θ(z,D) is the angle ofthe tangent line to the boundary of D at the point z ∈ ∂D. The lattice correc-tion function l(θ) depends on the lattice and on how we interpret “ending on theboundary of D.” We have shown how to compute the lattice correction functionl(θ) for two particular interpretations. Our focus has been on the distribution ofthe endpoint of the SAW on the boundary, but we should remark that in light of(1.4) the lattice effects in this boundary density will produce lattice effects in theprobability measure PD. We have also extended this conjecture to the λ-SAW.

As we have noted before, there is no efficient way to simulate the naturalinterpretations of the ensemble of SAW’s in a domain which start at an interiorpoint and end on the boundary. We have circumvented this difficulty by introducingthe cut-curve ensemble which can be thought of as an ensemble of two SAW’s, one


-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 20 40 60 80 100 120 140 160 180Polar angle

no lattice correctionwith lattice correction

Figure 4. The simulation of the cdf for the bound-ary density for the cut-curve ensemble for the secondgeometry minus the theoretical prediction. The largercurve does not include the lattice correction function;the smaller curve does.

from the interior point to the boundary and the other from that boundary pointto ∞. There is another ensemble that is amenable to efficient simulation which isstudied in [5]. Given a domain D containing the origin the ensemble is defined asfollows. We assume the domain has the property that a ray from the origin onlyintersects the boundary of the domain in one point. For a SAW ω we let λ(ω) > 0be such that the endpoint of ω is on the boundary of λ(ω)D. In general the SAWneed not be inside the dilated domain λ(ω)D. Our ensemble consists of all SAW’sω of any length such that ω is contained in λ(ω)D. (One must introduce cutoffsto make this a finite measure.) In [5] we show how one can simulate this ensembleusing the ensemble of SAW’s of a fixed length.

Finally, it is natural to ask if there is an interpretation of “ending on the bound-ary of D” for which there are no lattice effects in the scaling limit. We speculatethat the following ensemble has this property. As before, let δ be the lattice spac-ing. Let ε > 0 and consider all SAW’s that start at the origin, stay inside D andend within a distance ε of the boundary of D. We let δ go to 0 first and then let ε goto zero. We conjecture that l(θ) is constant for this ensemble. Unfortunately, thedouble limit involved in this ensemble makes it difficult to simulate this ensemble.

References

[1] Tom Alberts and Hugo Duminil-Copin, Bridge decomposition of restriction measures, J.Stat. Phys. 140 (2010), no. 3, 467–493, DOI 10.1007/s10955-010-9999-3. MR2660337(2012b:60274)


[2] Nathan Clisby, Efficient implementation of the pivot algorithm for self-avoiding walks,J. Stat. Phys. 140 (2010), no. 2, 349–392, DOI 10.1007/s10955-010-9994-8. MR2659283(2011j:82036)

[3] Ben Dyhr, Michael Gilbert, Tom Kennedy, Gregory F. Lawler, and Shane Passon, The self-avoiding walk spanning a strip, J. Stat. Phys. 144 (2011), no. 1, 1–22, DOI 10.1007/s10955-011-0258-z. MR2820032 (2012m:60196)

[4] Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lat-

tice equals√

2 +√2, Ann. of Math. (2) 175 (2012), no. 3, 1653–1665, DOI 10.4007/an-

nals.2012.175.3.14. MR2912714[5] Tom Kennedy, Simulating self-avoiding walks in bounded domains, J. Math. Phys. 53 (2012),

no. 9, 095219, 12. MR2905801[6] Michael J. Kozdron and Gregory F. Lawler, The configurational measure on mutually avoid-

ing SLE paths, Universality and renormalization, Fields Inst. Commun., vol. 50, Amer. Math.Soc., Providence, RI, 2007, pp. 199–224. MR2310306 (2008m:60079)

[7] Gregory F. Lawler, Partition functions, loop measure, and versions of SLE, J. Stat. Phys.134 (2009), no. 5-6, 813–837, DOI 10.1007/s10955-009-9704-6. MR2518970 (2010i:60232)

[8] Gregory F. Lawler and Vlada Limic, Random walk: a modern introduction, CambridgeStudies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.MR2677157 (2012a:60132)

[9] G. Lawler, Schramm-Loewner evolution (SLE), Statistical mechanics, IAS/Park City Math.Ser., vol. 16, Amer. Math. Soc., Providence, RI, 2009, pp. 231–295. MR2523461 (2011d:60244)

[10] Gregory F. Lawler, Oded Schramm, and Wendelin Werner, On the scaling limit of planarself-avoiding walk, Fractal geometry and applications: a jubilee of Benoıt Mandelbrot, Part2, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 339–364.MR2112127 (2006d:82033)

[11] Gregory F. Lawler and Scott Sheffield, A natural parametrization for the Schramm-Loewnerevolution, Ann. Probab. 39 (2011), no. 5, 1896–1937, DOI 10.1214/10-AOP560. MR2884877

[12] Gregory F. Lawler and Wendelin Werner, The Brownian loop soup, Probab. Theory Re-lated Fields 128 (2004), no. 4, 565–588, DOI 10.1007/s00440-003-0319-6. MR2045953(2005f:60176)

[13] Paul Levy, Processus stochastiques et mouvement brownien, Suivi d’une note de M.Loeve. Deuxieme edition revue et augmentee, Gauthier-Villars & Cie, Paris, 1965 (French).MR0190953 (32 #8363)

[14] N. Madras and G. Slade, The Self-Avoiding Walk. Birkhauser (1996).[15] Benoit B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Co., San Francisco,

Calif., 1982. Schriftenreihe fur den Referenten. [Series for the Referee]. MR665254 (84h:00021)[16] Bernard Nienhuis, Exact critical point and critical exponents of O(n) models in two dimen-

sions, Phys. Rev. Lett. 49 (1982), no. 15, 1062–1065, DOI 10.1103/PhysRevLett.49.1062.MR675241 (84a:81035)

[17] F. and M. Riesz, Uber die Randwerte einer analytischen Funktion, Quatrieme Congres desMathematiciens Scandinaves, Stockholm, pp. 27-44 (1916).

Department of Mathematics, University of Arizona, Tucson, Arizona 85721-0089


Department of Mathematics, University of Chicago, 5734 S. University Avenue,

Chicago, Illinois 60637



The Casimir Effect on Laakso Spaces

Robert Kesler and Benjamin Steinhurst

Abstract. We explore the properties of an analog to the Casimir effect onLaakso spaces such as the dependence on the separation of the plates andboundary effects. We also mention some results on the influence of complexpoles in the spectral zeta function over finite approximations to Laakso spaces.

1. Introduction

With recent advances in fractal analysis there has become available a largeamount of information concerning Laplacians and their spectra over fractal spaces[4, 11, 20]. There has been progress in using this information to construct andanalyze analogs to physical systems, e.g. the behavior of a photon in a fractal [2]and other “physical” consequences [1,7,8,21]. The physical consequence of fractalgeometry that we explore in this paper is the Casimir effect [14,17].

In these works, the underlying space is typically a finitely ramified fractal witha symmetry condition; however, the spectrum of the Laplacian for these objects isgenerally not known exactly or only described as a scaled Julia set, which meansthe growth estimates for the eigenvalue counting functions must be used instead.Laakso spaces, whose exact spectrum the authors previously computed, enable us toavoid this complication. In [10] the authors also computed the exact eigenfunctionsof the Hamiltonian with a square well potential, the spectral zeta function for cer-tain defining sequences {jn}, and a Casimir effect on a 1 dimensional arrangement.This paper continues the analysis of the Casimir effect on Laakso spaces.

We begin with defining Laakso spaces in a convenient manner in Section 2 wherewe will also give an explicit description of the spectrum of a natural Laplacian onLaakso spaces. Following this is Section 3 where we discuss the general propertiesand calculations for spectral zeta functions over Laakso spaces. In Section 5 weobserve that the complex dimensions appear in a model which has only finite com-plexity and is in principle constructible as a physical object. In Section 6 we revisitthe authors’ earlier work in [10] and determine the strength of the Casimir effectin a Laakso space as a function of both the defining sequence {ji} and the distancebetween the plates. Lastly, in Section 6 we construct a 3+ dimensional arrange-ment involving a Laakso space and show that the Casimir pressure is proportionalto the inverse fourth power of the separation distance. This power has the same

2010 Mathematics Subject Classification. Primary 81Q35, 28A80.Key words and phrases. Casimir effect, spectral zeta function, zeta regularization, Laakso

spaces.


211

212 ROBERT KESLER AND BENJAMIN STEINHURST

exponent as the classical Casimir effect between two parallel uncharged conductingplates in R3. What makes this result unusual is that this exponent is not equal tothe spectral dimension, ds, plus an integer as been seen on the few fractal domainswhich have been explored [1,2,6]. This reflects the fact that Laplacians on Laaksospaces are truly 1−dimensional rather than ds−dimensional operators. The rapidgrowth in the eigenvalue counting function is due more to the geometry of Laaksospaces and their graph approximations than the nature of the Laplacian.

Acknowledgements: We thank Christopher Kauffman, Amanda Parshall,and Evelyn Stamey for their work on the foundational counting arguments that areso often used in this paper as well as Erik Akkermans and Alexander Teplyaev fortheir useful comments and challenges.

2. Laakso spaces

These spaces were introduced in [12], and the spectral theory on them wasdeveloped in [10,15,19]. We will use the construction indicated in [3] and spelledout in detail in [19]. Let {ji} be a sequence of positive integers such that

(2.1) limn→∞

(n∏

i=1

ji

)1/n

= r.

The Laakso space construction can proceed for integer sequences without the exis-tence of the above limit, but one gives up a nice formula for the Hausdorff dimension.Define

(2.2) In =

n∏i=1

ji Ln =

{m

In: m = 1, . . . , In − 1

},

where Ln will be the locations of “wormholes” of lever n or lower and Ln \ Ln−1

the locations of the new “wormholes” at level n. We will write I = [0, 1] and K fora Cantor set.

Set F0 = I, G = {0, 1}, and Bn = Ln \ Ln−1 ⊂ I. Let φ1,0 : F0 × G → F0

be the projection onto the interval F0. Define F1 = F0 ×G/φ−11,0(B1). Inductively

construct φn,n−1 and Fn. Notice that there are also naturally defined projectionsφn,m : Fn → Fm. Let μn be the probability measure on Fn that is inherited fromLebesgue measure on F0.

Proposition 2.1. The system (Fn, φn,n−1, μn) is a projective system of mea-sure spaces.

It is important to note that Fn is a metric graph as it is a collection of linesegments of all of length I−1

n at nodes whose locations have a coordinate in I takenfrom Ln. It is this particularly regular structure for the Fn that we will use in theabsence of any strict geometric self-similarity.

Definition 2.1. The projective limit of Fn is a Laakso space with data {ji}.This is written as lim← Fn = L. There are also associated projections Φn : L → Fn

such that φn,m ◦ Φn = Φm for all m ≤ n.

For more on projective limits of measure spaces see [9].

Theorem 2.1. For any choice of {ji} such that r exists the correspondingLaakso space L is a complete geodesic metric measure space with Hausdorff dimen-

sion 1 + log(r)log(2) .

THE CASIMIR EFFECT ON LAAKSO SPACES 213

That the presented construction gives a Laakso space is Lemma 4.6.1 in [18].The properties of the Laakso spaces are proved in [12].

Let Δn be the self-adjoint Laplacian on Fn acting as − d2

dx2e

where xe is a co-

ordinate on each line segment. Then the domain of Δn is taken to be the closureof all continuous functions on Fn that are twice differentiable when restricted toeach interval and satisfy Kirchoff matching conditions at all vertices. This forcesNeumann boundary conditions at the degree one vertices that form the boundaryof Fn.

Since φn,m maps Fn onto Fm we can by composition use φn,m to map functionsover Fm to functions over Fn by the convention φ∗

n,mf = f ◦ φn,m. The samedefinition is used for Φ∗

n as well.

Proposition 2.2. For m < n, φ∗n,mDom(Δm) ⊂ Dom(Δn).

Theorem 2.2 ([15]). There exists a self-adjoint Laplacian Δ on L such thatΔΦ∗

nf = Φ∗nΔnf for all f ∈ Dom(Δn) and for all n with domain

(2.3) Dom(Δ) =

∞⋃n=0

Φ∗nDom(Δn).

Furthermore

σ(Δ) =

∞⋃k=0

{π2k2} ∪∞⋃

n=1

∞⋃k=0

{(k + 1/2)2π2I2n} ∪∞⋃

n=1

∞⋃k=1

{k2π2I2n}

∪∞⋃n=2

∞⋃k=1

{k2π2I2n} ∪∞⋃n=2

∞⋃k=1

{k2π2I2n

4

}(2.4)

with respective multiplicities

(2.5) 1, 2n, 2n−1(jn − 2)In−1, 2n−1(In−1 − 1), 2n−2(In−1 − 1).

Recall that In =∏n

j=1 ji. The method of calculating the spectrum is basedon the fact that an eigenfunction of Δn can be localized between wormholes sinceinterior wormholes are degree four vertices and it is possible for the function to beconstant on two of the incoming edges and non-constant on the other two and stillsatisfy the Kirchoff matching conditions that all incoming first derivatives sum tozero. The spectrum is then determining by breaking down the graphs, Fn into sub-graphs on which such eigenfunctions are supported and the multiplicities countedby counting the number of each of these subgraphs. These counting arguments willbe revisited later in this paper.

3. Spectral Zeta Functions

Definition 3.1. Let Δ be a self-adjoint positive-definite operator with a dis-crete spectrum λi and multiplicities gi. Then the spectral zeta function is defined,where convergent as

(3.1) ζΔ(s) =

∞∑i=1

giλsi

.

We will also denote the analytic continuation of this function as ζΔ(s).


If one considers the Laplacian on [0, 1] with Dirichlet boundary conditions thespectrum is {k2π2}∞k=1 and all of multiplicity one. Then

(3.2) ζ(s) =

∞∑k=1

1

(kπ)2s=

1

π2sζR(2s),

where ζR(s) is the Riemann zeta function which has a meromorphic continuation tothe whole complex plane. For the Laplacian on the Laakso space with data ji = 2

ζΔL(s) =

ζR(2s)

π2s

(4(22s−1 + 1)

4s(42 − 4)+

6(22s−1 − 1)

4s(4s − 2)+

2s+1 − 2 + 22s

4s

),(3.3)

which then also has a meromorphic continuation to the complex plane with polesat known locations. In [10] a formula for the spectral zeta function of any Laaksospace with periodic data {ji} is given as a rational complex valued function timesthe Riemann zeta function. A feature of the spectral zeta functions on Laaksospaces that does not appear in the interval case is the existence of poles for ζΔL

(s)off of the real axis. These are referred to as complex dimensions [13]. In [4] theresidues of ζΔL

(s) are used to calculate the leading terms of the Weyl asymptoticsfor Δ which for Laakso spaces with periodic ji have a log-periodic oscillating termof leading order.

Theorem 3.1. Suppose that F∞ is constructed in such a way that F0 ⊂ Rd iscompact with non-empty interior and φ0(Bi) induces a self-similar cell structure onF0. Further assume that the Bi have empty interior. Then the associated spectralzeta function will have a tower of simple poles above the spectral dimension. Fur-thermore, if the spectral zeta function over F0 is meromorphic on the entire complexplane so is the spectral zeta function over F∞.

Proof. As shown in [19] the projective limit construction in this case willyield a non-negative, real, and discrete spectrum. Because F1 can be realized as anassembly of identical pieces that are scaled copies of F0 that overlap only on theset B1, which has empty interior, Dom(Δ1) = φ∗

1(Dom(Δ0)) ⊕ F1 where F1 arethe eigenfunctions that are orthogonal to φ∗

iDom(Δ0) in L2(F1). By a geometricalargument these eigenfunctions are piecewise defined as eigenfunctions on scaledcopied of F0 with suitable matching conditions to assure the orthogonality. See[15] for the case of Laakso spaces. Thus σ(Δi|F1

) = c1σ(Δ0) for some c1. By theself-similarity of the cell structure the constant c1 is the same for all n not justn = 1. This gives rise to a geometric series over n whose summation has a seriesof simple poles over the spectral dimension of L. Such series will be the topic ofSection 5. Another series of eigenfunctions could occur due to Neumann boundaryconditions but these will also have the same scaling and will merely provide anothertower of simple poles.

The spectral zeta function over F∞ is the sum over n of scaled copied of thespectral zeta function over F0 plus a finite number of bootstrap terms. Becausethis is actually the same geometric summation as in the previous paragraph thesum is a meromorphic function after regularization if and only if each term ismeromorphic. �

4. Casimir Effect

The Casimir effect arising between conductors and the quantum vacuum canbe viewed as a consequence of vacuum zero-point energy. Simply put, displacing


conductors generates new boundary conditions for the quantized vacuum, which inturn alters the zero-point energy and gives rise to a negative energy gradient. Ithas been experimentally verified [16] that two parallel uncharged conducting platesexperience an attractive pressure given by

|PC | =π2c�

240d4.

That such an attraction has its origins in relativistic quantum mechanics is reflectedby the appearance of both Planck’s quantum mechanical � and the relativistic c.However, the direction of the Casimir pressure generally depends on the geometryof the conductors with which one is working. While planes and cylinders undergoattraction, a spherical shell exhibits self-repulsion, and Laakso spaces demonstrateboth attraction and repulsion [5].

In computing the Casimir effect on Laakso spaces, we take the vacuum ex-pectation of a self-adjoint Hamiltonian operator which represents the quantizedelectromagnetic field and whose spectrum yields the permissible energies for thesystem. In particular, as our boundary conditions depend on on some displacementparameter d, we obtain

Evac(d) = 〈0|H(d)|0〉 ∝∑

λ∈σ(Δ)

ωλ(d) ∝∑

λ∈σ(Δ)

√λ(d) ∝ ζL(d)(−1/2)

where ζΔL(d)(−1/2) is interpreted in the sense of meromorphic continuation. The

Casimir pressure will therefore be proportional to a derivative of the Laakso spec-tral zeta function evaluated at −1/2. In [10], the authors looked at the Casimireffect on jn = j Laakso spaces that arose from conducting plates attached at nodesin the F1 graph approximation and placed symmetrically about the center. Ateach point of intersection with L, the conducting plates imposed Dirichlet con-ditions and Kirchoff matching conditions were maintained at the other nodes forelements of the domain of Δ. Following the above outline, the modified spectralzeta function subject to these new conditions was computed. To make sense of theenergy gradient, the plates were allowed to move symmetrically from their originallocations compressing and stretching the underlying space in a natural way. Mov-ing plates closer together compressed the interior space and stretched the exterior.Conversely, moving the plates away from each other stretched the interior spaceand compressed the exterior.

5. Finite Approximations to Laakso Spaces

In this section we consider the Casimir effect on Fm with Laplacian Δm inthe case of two perfectly conducting plates placed at opposite ends of the unitinterval. These boundary conditions are simply the Dirichlet boundary conditions.By truncating the counting arguments mentioned in Section 2 we see that thespectrum of Δm is given by

σ(Δm) =∞⋃k=1

{π2k2} ∪m⋃

n=1

∞⋃k=0

{(k + 1/2)2π2I2n} ∪m⋃

n=1

∞⋃k=1

{k2π2I2n}

∪m⋃

n=2

∞⋃k=1

{k2π2I2n} ∪m⋃

n=2

∞⋃k=1

{k2π2I2n

4

}(5.1)


with multiplicities

(5.2) 1, 2n, 2n−1(jn − 2)In−1, 2n−1(In−1 − 1), 2n−2(In−1 − 1).

Recall that In =∏n

i=1 ji. The introduction of the plates enforces Dirichlet bound-ary conditions on the eigenfunctions. The eigenfunctions whose eigenvalues are{(k + 1/2)2π2I2n} are those localized between the boundary and the closest worm-holes in Fn with Neumann boundary conditions on the boundary and Dirichletconditions at the wormholes. When the Dirichlet boundary conditions are imposedthese eigenvalues change to {k2π2I2n} for k ≥ 1 for all n ≥ 1 with the same multi-plicities. The spectral zeta function for Δm with these new boundary conditions,denoted ζm(s), can be calculated in the case where ji = j and m ≥ 4 to be

ζm(s) =ζR(2s)

π2s

[1 +

2 − (2 + 4s)j−2s + (2 + 4s)2mj−2sm

j2s − 2

+2j − 2 + (2 + 4s)j1−2s + (1 − 2j − 1

24s)(2j)mj−2sm

j2s − 2j

].(5.3)

The observed poles arise from the use of the summation formula for geometricseries

(5.4)m∑

n=1

rn = r1 − rm

1 − r,

which are not removable due to the interactions between the several sums beingtaken simultaneously. Since the poles found are outside of the domain of conver-gence for the summation over k these poles can only be approached through ananalytic continuation and so we can choose to represent the summations using thisformula since the extended functions will agree on an open subdomain and henceeverywhere.

Proposition 5.1. For all T−periodic sequences {ji} the spectral zeta functioncorresponding to Δn on Fn for n ≥ 3T have towers of complex poles.

Proof. It has already been seen for constant sequences ji. For periodic se-quences with longer periods, the summation methods are augmented by summingover an individual period then summing over all periods. The requirement that nis large enough for three periods is so that the geometric aspect of the summationover n is fully present. �

6. Casimir Effect on L

This is the original setting in which the authors considered a Casimir effectin [10]. In addition to the boundary of the Laakso space playing the part of onepair of plates with Neumann boundary conditions we insert another symmetricallyplaced pair of plates in the interior so that the dependence of Casimir force on theseparation of the plates can be explored. In this section we will consider for the sakeof simplicity Laakso spaces with constant sequences ji = j. The interior plates willbe places symmetrically at level one wormhole locations. That is their location inthe unit interval will be taken from L1. Set X0 to be one half the distance betweenthe interior plates, this gives the distance of the plates from the “center” of theLaakso space. Let Z be the number of nodes between the plates in the F1 graphapproximation of L. See Figure 1 for an example.


a b

Figure 1. The F1 graph for j = 5 where the interior plates areplaced at nodes a and b. Here X0 = 3

5 × 12 = 3

10 and Z = 3.

σ(Δ′) =∞⋃

k=1

{[kπ

2X0

]2}

∪∞⋃

k=0

{[(k + 1/2)π

(1− 2X0)/2

]2}

∪∞⋃

k=0

{[(k + 1/2)π

j − (Z + 1)

1− 2X0

]2}

∪∞⋃

k=1

{[kπ

j − (Z + 1)

1− 2X0

]2}

∪∞⋃

k=1

{[kπ

Z + 1

2X0

]2}

∪∞⋃

n=2

∞⋃k=0

{[In(k + 1/2)π

1− Z+1j

1− 2X0

]2}

∪∞⋃

n=2

∞⋃k=1

{[Inkπ

1− Z+1j

1− 2X0

]2}∪

∞⋃n=2

∞⋃k=1

{[Inkπ

1− Z+1j

2(1− 2X0)

]2}

∪∞⋃

n=2

∞⋃k=1

{[Inkπ

Z + 1

2jX0

]2}

∪∞⋃

n=2

∞⋃k=1

{[Inkπ

Z + 1

4jX0

]2}

.

and multiplicities are listed in the same order

1) 1;2) 2;3) 2;4) j − Z − 3;5) Z + 1;6) 2n;7) (1− Z+1

j)In−12

n−1(j − 2) + 2n−1(1− Z+1j

)In−1;

8) 2n−2[(1− (Z+1j

)In−1 − 1]− 2n−2;

9) Z+1j

In−12n−1(j − 2) + 2n−1 Z+1

jIn−1 + 2n−1;

10) 2n−2[Z+1j

In−1 − 1].

Figure 2. Recall that In =∏n

i=1 ji.

Definition 6.1. Given a Laakso space with ji = j and a symmetrically locatedpair of plates whose location is determined by j and a chosen Z. The operator Δ′

on L2(L, μ) acts as Δ but with domain determined by imposing Dirichlet boundaryconditions at the interior plates.

Theorem 6.1 ([10] Theorem 4.2). The operator Δ′ is self-adjoint and hasspectrum with multiplicities as given in Figure 2.

Using this it is a tedious but straightforward task to calculate ζΔ′(s). Since wewill be interested in how ζΔ′(s) varies as j, Z, and X0 are varied we will make thedependence explicit by writing ζΔ′(s) = ζj,X0,Z(s).


ζj,X0,Z(s) =

∞∑k=0

2

[(2k + 1)π/(1− 2X0)]2s+

∞∑k=1

1

[kπ/(2X0)]2s

+∞∑

k=1

(j − Z − 3)

[jkπ (1−(Z+1)/j)1−2X0

]2s+

∞∑k=1

Z + 1

[kπ(Z + 1)/(2X0)]2s

+

∞∑n=1

∞∑k=0

2n

[In(k + 1/2)π(1− Z+1j

)/(1− 2X0)]2s

+

∞∑n=2

∞∑k=1

(1− Z+1j

)2n−1In−1(j − 2) + 2n−1(1− (Z + 1)/j)In−1

[Inkπ(1−(Z+1)/j)

(1−2X0)]2s

+∞∑

n=2

∞∑k=1

2n−2[(1− (Z + 1)/j)In−1 − 1]− 2n−2

[Inkπ(1− (Z + 1)/j)/[2(1− 2X0)]]2s

+

∞∑n=2

∞∑k=1

(Z + 1)/j[2n−1In−1(j − 2) + 2n−1In−1] + 2n−1

[kπIn(Z + 1)/(2jX0)]2s

+∞∑

n=2

∞∑k=1

(Z + 1)/j[2n−2In−1]− 2n−2

[Inkπ/(4jX0)]2s.


i=1 ji.

Figure 4. Values of the Casimir force plotted for j = 256 andZ ranging between 1 and 125. Notice that for large Z we see aninteraction between the plates and the boundary of the Laaksospace while for small Z see a similar interaction between the twoplates.

Corollary 6.1. Given a Laakso space with ji = j, and conducting platesplaced according to X0 and Z we have the spectral zeta function given in Figure 3.


FC(j, Z) ∝ d

dxζj,x,Z

(−1

2

)|x=X0

=(j − (Z + 1))

24(1− 2j)(1− 2X0)2− (j − (Z + 3))(j − (Z + 1))

12(1− 2X0)2

−j3(j − 2)(1− Z+1

j)2

12(1− 2j2)(1− 2X0)2−

(1− Z+1j

)j2(j − (Z + 1))

24(1− 2X0)2(1− 2j2)

+(1− Z+1

j)j2

24(1− 2X0)2(1− 2j)+

(Z + 1)2

48X20

+(Z + 1)2(j − 2)

24X20 (1− 2j2)

+j(Z + 1)2

96(1− 2j2)X20

+1

6(1− 2X0)2−

j2(1− Z+1j

)

24(1− 2j)(1− 2X0)2

+j(Z + 1)

96X20 (1− 2j)

−j2(1− Z+1

j)

12(1− 2X0)2(1− 2j)+

1

48X20

+(Z + 1)j

48X20 (1− 2j)


i=1 ji.

Proposition 6.1. The Casimir energy of a Laakso space given by ji = j andplates positioned according to X0 and Z is proportional to ζj,X0,Z(−1/2) and theself-exerted Casimir force due to this energy is proportional to Figure 5.

Proof. See [10] for details. �

To see how Casimir force can vary with Z for a given j see Figure 4. Inspectingthe expression in Figure 5 it is readily apparent that the force depends on the plateseparation X0 as X−2

0 instead of the expected X−ds−10 . The parameter Z represents

how many cells separate the plates, a sort of geometric distance. The dependenceon Z is easily seen to also be quadratic in Figure 5. Recall that Laakso spaces all

have walk dimension dw = 2 so ds = dh = 1 + log(r)log(2) . The reason this happens is

that Δ′, like Δ, is a truly one-dimensional operator so that the rapid growth in theeigenvalue counting function is due to the geometry of Laakso spaces rather thandimensionality of the Laplacian.

7. A Higher Dimensional Case

7.1. The 3+ dimensional model. Let Lj be the Laakso space representedby the sequence jn = j, let K be the Cantor set, and let I = [0, 1]. Then modifythe configuration in [10] by considering the space

Lj × R2 = [(I ×K)/ ∼] × R2

and attaching two conducting plates P1, P2 ⊂ Lj × R2 where

P1 = [(0 ×K)/ ∼] × R2;P2 = [(1 ×K)/ ∼] × R2.

The Laplacian ΔLj×R2 in this context takes the form

ΔLj×R2 = ΔLj− ∂2

∂x21

− ∂2

∂x22

where ΔLjis the non-negative definite self-adjoint Laplacian on Lj such that

ΔΦ∗nf = Φ∗

nΔnf for all f ∈ Dom(Δn). The plates enforce Dirichlet boundary


conditions on ΔLjas above. Moreover, the generalized spectrum for ΔLj×R2 is

described by

σ(ΔLj×R2) =: σLj×R2 = {λσ + k2x1+ k2x2

: λσ ∈ σ(ΔLj)},

and, as usual, the zero-point energy associated with Lj × R2 is the ground stateexpectation of the Hamiltonian for the corresponding quantized electromagneticfield, i.e.

ECas = �c ζΔLj×R2(−1/2).

Now stretch the Laakso space Lj = ([0, 1]×K)\ ∼ to Lj(d) := ([0, d]×K)\ ∼ which

distorts the eigenvalues of ΔLjto those of ΔLj(d) by λ �→ λ

d2 . Moreover, stretch

Lj × R2 to Lj(d) × R2 by leaving the Euclidean components untouched. Then thegeneralized energy density between two parallel plates separated a distance d apartis computed as:

ECas(s, j, d) := �c limL→∞

∑λ∈σ(ΔLj(d)

)

∑k1,k2∈Z+

1

L2

(λ +

(πk1L

)2

+

(πk2L

)2)−s

=�c

π2

∑λ∈σ(ΔLj

)

∫ ∞

0

∫ ∞

0

dk1dk2

(λ

d2+ k21 + k22

)−s

=�c

4(1 − s)π

∑λ∈σ(ΔLj

)

(λ

d2

)1−s

=�c

4(1 − s)πd2−2sζLj

(s− 1),

where

ζΔLj(s) =

ζR(2s)

π2s

[1 +

(j − 2) + (2 + 4s)j1−2s

j2s − 2j− (2 + 4s)j−2s − 2

j2s − 2

].(7.1)

Hence, ECas(s, j, d) has a meromorphic continuation via its connection to theRiemann zeta function ζR(s). In particular, when s = −1/2,

ECas(−1/2, j, d) =�c

6πd3ζΔLj

(−3/2).(7.2)

Lastly, let

PCas(j, d) := −∂ECas(s, j, d)

∂d|s=−1/2,d=d(7.3)

and

PCas(j) := PCas(j, 1).(7.4)

Proposition 7.1. Two conducting plates attached at the boundary of Lj ×R2

as described in the set up will experience a pressure given by

PCas(j)

=�cπ2

240

[1 +

2j3

1 − 2j3+

17

8·(

j7

1 − 2j4− j6

1 − 2j3

)+

(j4

1 − 2j4− 2j3

1 − 2j4

)]=

�cπ2

240

[8 − 16j3 − 8j4 + 15j6 + j7

8 − 16j3 − 16j4 + 32j7

],

where a positive signed pressure indicates an attractive force.


Proof. Computing the Casimir pressure involves eigenvalue counting argu-ments similar to the ones made in [15] except that Neumann boundary conditionsare replaced by Dirichlet boundary conditions at the boundary (0 × K)/ ∼ and(1 × K)/ ∼. We construct the full spectrum of ΔLj

by exploiting orthogonalityrelations between eigenfunctions in different quantum graph approximations. De-composing each quantum graph Fn into loops, V’s, and crosses, note that the V’sare the only shapes whose localized eigenfunctions are altered by the new boundaryconditions. Once the new spectrum with multiplicities is found, substitute into theexpression for Casimir energy, regularize the sum, and take derivatives with re-spect to displacement to obtain the result. Detailed calculations which imply thosenecessary for this proposition are included in the proof of Proposition 7.3. �

By combining (7.1)-(7.4), it is clear that the second factor in (7.5) arises directlyfrom the second factor in (7.1). This leads to a delicate point in taking the limit asj → ∞. For Laakso spaces generated by the constant sequences j, the metrics onLj converge pointwise as functions on I×K to the metric on I with no dependenceon the K-coordinate. Because of this it is reasonable to expect that the strengthof the Casimir pressure should also approach that expected on I × R2 which is

|PC | = �cπ2

240 . If one naively takes the limit j → ∞ in (7.5) an extra factor of 132

is acquired from the rational function in j that appears in (7.5). However if thelimit j → ∞ is taken before evaluating (7.1) at s = −3/2 this factor of 1

32 does notappear and the limiting strength of the Casimir pressure matches that expectedfrom the classical scenario. The authors endorse the second interpretation as thecorrect one because it implies a continuity at j = ∞ of the Casimir effect whichmatches the existence of the limiting metric space, i.e. Lj → [0, 1].

Corollary 7.1. Two conducting plates attached at the boundary of Lj × R2

and then stretched to a distance d from one another will experience a Casimirpressure

PCas(j, d) =PCas(j)

d4.

Recall that the stretched Laakso space Lj(d) is where scaling is only in thedirection of the unit interval in the construction Lj = (I ×K) / ∼.

Proof. This follows immediately from the fact that multiplying the displace-ment by a factor of d means λs → λs

d2 ∈ σ(ΔLj(d)) for every λs ∈ σ(ΔLj). �

In particular, the power law governing the Casimir pressure as a function ofdisplacement is independent of the spectral dimension of the Laakso space Lj .

7.2. Casimir Pressure as a function of {ji}.

Proposition 7.2. Let L be the Laakso space represented by some N-periodicsequence {ji}. Then two conducting plates attached at the boundary of L×R2 yieldan unnormalized Casimir energy density given by


ECas(s, {ji}, d) = �c limL→∞

∑λ∈σ(ΔLj (d))

∑k1,k2∈Z+

1

L2

(λ +

(πk1L

)2

+

(πk2L

)2)−s

= − �c

4(1 − s)π

[ ∞∑k=1

(kπ

d

)2−2s

+∞∑

n=1

∞∑k=1

2n(kπInd

)2−2s

+

∞∑n=1

∞∑k=1

2n−1In−1(jn − 2)

(kπInd

)2−2s

+17

16

∞∑n=2

∞∑k=1

2n−1 (In−1 − 1)

(kπInd

)2−2s].

Proof. This more general case follows from Equation 7.1 and eigenvalue count-ing arguments similar to the ones made in [15] for periodic Laakso spaces. �

Proposition 7.3. Let L be the Laakso space represented by some N-periodicsequence {ji}. Then two conducting plates attached at the boundary of L×R2 willexperience a Casimir pressure given by

PCas({ji}, d) =�cπ2

240d4

⎡⎣1 +15

32

⎛⎝ N∑i=1

∏k≤i

2j3k

⎞⎠( 1

1 − r3N2N

)

+1

2

(N∑i=1

2j4k

)(1

1 − r4N2N

)

− 15

32

(N∑i=1

∏k≤i 2j

4k

ji

)(1

1 − r4N2N

)]where a positive sign indicates an attractive force.

Proof. By Proposition 7.2,

ECas(s, {ji}, d) = − �c

4(1 − s)π

[ ∞∑k=1

(kπ

d

)2−2s

+

∞∑n=1

∞∑k=1

2n(kπInd

)2−2s

+

∞∑n=1

∞∑k=1

2n−1In−1(jn − 2)

(kπInd

)2−2s

+17

16

∞∑n=2

∞∑k=1

2n−1 (In−1 − 1)

(kπInd

)2−2s]

= − �c

4(1 − s)π(I + II + III + IV ).

Compute

I = −π2−2sζR(2− 2s)

d2−2s


II =∞∑

n=1

∞∑k=1

2n(kπInd

)2−2s

=π2−2sζR(2− 2s)

d2−2s

∞∑n=1

2nI2−2sn

=π2−2sζR(2− 2s)

d2−2s

( ∞∑l=0

2Nlr(2−2s)Nl

)⎛⎝ N∑

i=1

∏k≤i

2j2−2sk

⎞⎠

=π2−2sζR(2− 2s)

d2−2s

1

1− r(2−2s)N2N

⎛⎝ N∑

i=1

∏k≤i

2j2−2sk

⎞⎠

III =

∞∑n=1

∞∑k=1

2n−1In−1(jn − 2)

(kπInd

)2−2s

=1

2

π2−2sζR(2− 2s)

d2−2s

1

1− r(3−2s)N2N

⎛⎝ N∑

i=1

∏k≤i

2j3−2sk

⎞⎠

− 1

2

π2−2sζR(2− 2s)

d2−2s

2

1− r3N2N

⎛⎝ N∑

i=1

∏k≤i

2j3−2sk

ji

⎞⎠

IV =17

16

∞∑n=2

∞∑k=1

2n−1 (In−1 − 1)

(kπInd

)2−2s

=17

32

(π2−2sζR(2− 2s)

d2−2s

)1

1− r(3−2s)N2N

⎛⎝ N∑

i=1

∏k≤i

2j3−2sk

ji

⎞⎠

− 17

32

(π2−2sζR(2− 2s)

d2−2s

)1

1− r(2−2s)N2N

⎛⎝ N∑

i=1

∏k≤i

2j2−2sk

⎞⎠ .

Putting it all together,

ECas(s, {ji}, d) = − �c

4(1 − s)π(I + II + III + IV )

= −�cπ1−2sζR(2 − 2s)

4(1 − s)d2−2s

⎡⎣1 +15

32

⎛⎝ N∑i=1

∏k≤i

2j2−2sk

⎞⎠( 1

1 − r(2−2s)N2N

)⎤⎦−�cπ1−2sζR(2 − 2s)

4(1 − s)d2−2s

⎡⎣1

2

⎛⎝ N∑i=1

∏k≤i

2j3−2sk

⎞⎠( 1

1 − r(3−2s)N2N

)⎤⎦+�cπ1−2sζR(2 − 2s)

4(1 − s)d2−2s

[15

32

(N∑i=1

∏k≤i 2j

3−2sk

ji

)(1

1 − r(3−2s)N2N

)]

and


PCas({ji}, d) = − ∂ECas(s, {ji}, d)∂d

∣∣∣∣s=−1/2

=�cπ2

240d4

⎡⎣1 +15

32

⎛⎝ N∑i=1

∏k≤i

2j3k

⎞⎠( 1

1 − r3N2N

)

+�cπ2

240d4

⎡⎣1

2

⎛⎝ N∑i=1

∏k≤i

2j4k

⎞⎠( 1

1 − r4N2N

)⎤⎦− �cπ2

240d4

[15

32

(N∑i=1

∏k≤i 2j

4k

ji

)(1

1 − r4N2N

)].

�

References

[1] Eric Akkermans, Gerald V. Dunne, and Alexander Teplyaev. Physical consequences of com-plex dimensions of fractals. EPL, 2009.

[2] Eric Akkermans, Gerald V. Dunne, and Alexander Teplyaev. Thermodynamics of photons onfractals. Phys. Rev. Lett., 105(23):230407, Dec 2010.

[3] Martin T. Barlow and Steven N. Evans, Markov processes on vermiculated spaces, Ran-dom walks and geometry, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 337–348.MR2087787 (2006b:60164)

[4] Matthew Begue, Levi Devalve, David Miller, and Benjamin Steinhurst, Spectrum and heatkernel asymptotics on general Laakso spaces, Fractals 20 (2012), no. 2, 149–162, DOI10.1142/S0218348X12500144. MR2950203

[5] Antoine Canaguier-Durant, Romain Guerout, Paulo A Maia Neta, Astrid Lambrecht, andSerge Reynaud. The casimir effect in the sphere-plane geometry. In Proceedings of thee 10thInternational Conference “Quantum Field Theory Under the Influence of External Condi-tions”.

[6] Joe Chen. Statistical mechanics of Bose gase in Sierpinski carpets. Submitted to Comm. Math.Phys, arXiv:1202.1274.

[7] Gerald V. Dunne. Heat kernels and zeta functions on fractals. Invited Contribution to theJPhysA Special Issue in honour of J.S. Dowker’s 75th Birthday.

[8] Edward Fan, Zuhair Khandker, and Robert S. Strichartz, Harmonic oscillators on infiniteSierpinski gaskets, Comm. Math. Phys. 287 (2009), no. 1, 351–382, DOI 10.1007/s00220-008-0633-z. MR2480752 (2011f:35059)

[9] John G. Hocking and Gail S. Young, Topology, 2nd ed., Dover Publications Inc., New York,1988. MR1016814 (90h:54001)

[10] Christopher J. Kauffman, Robert M. Kesler, Amanda G. Parshall, Evelyn A. Stamey, andBenjamin A. Steinhurst, Quantum mechanics on Laakso spaces, J. Math. Phys. 53 (2012),no. 4, 042102, 18, DOI 10.1063/1.3702099. MR2953261


[12] T. J. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincare inequal-ity, Geom. Funct. Anal. 10 (2000), no. 1, 111–123, DOI 10.1007/s000390050003. MR1748917(2001m:30027)

[13] Michel L. Lapidus and Machiel van Frankenhuijsen, Fractal geometry, complex dimensionsand zeta functions, Springer Monographs in Mathematics, Springer, New York, 2006. Geom-etry and spectra of fractal strings. MR2245559 (2007j:11001)

[14] Philippe A. Martin and Pascal R. Buenzli. The casimir effect. Acta Physica Polonica B, 2006.[15] Kevin Romeo and Benjamin Steinhurst, Eigenmodes of the Laplacian on some Laakso spaces,

Complex Var. Elliptic Equ. 54 (2009), no. 6, 623–637, DOI 10.1080/17476930903009584.MR2537259 (2010h:81095)


[16] Marcus J. Sparnaay. Measurements of attractive forces between flat plates. Physica, 24(6-10):751–764, 1958.

[17] Larry Spruch. Retarded, or casimir, long-range potentials. Physics Today, 1986.[18] Benjamin Steinhurst, Diffusions and Laplacians on Laakso, Barlow-Evans, and other frac-

tals, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–University of Connecticut.MR2753167

[19] Benjamin Steinhurst and Alexander Teplyaev. Spectral analysis and Dirichlet forms on

Barlow-Evans fractals. in preparation.[20] Robert S. Strichartz, Differential equations on fractals, Princeton University Press, Princeton,

NJ, 2006. A tutorial. MR2246975 (2007f:35003)[21] Robert S. Strichartz, A fractal quantum mechanical model with Coulomb potential, Com-

mun. Pure Appl. Anal. 8 (2009), no. 2, 743–755, DOI 10.3934/cpaa.2009.8.743. MR2461574(2010c:81086)

Department of Mathematics, Cornell University, Ithaca, New York 14850


Department of Mathematics, Cornell University, Ithaca, New York 14850



The Decimation Method for Laplacians on Fractals:Spectra and Complex Dynamics

Nishu Lal and Michel L. Lapidus

In memory of Benoıt Mandelbrot

Abstract. In this survey article, we investigate the spectral properties offractal differential operators on self-similar fractals. In particular, we discussthe decimation method, which introduces a renormalization map whose dy-namics describes the spectrum of the operator. In the case of the boundedSierpinski gasket, the renormalization map is a polynomial of one variable onthe complex plane. The decimation method has been generalized by C. Sabotto other fractals with blow-ups and the resulting associated renormalizationmap is then a multi-variable rational function on a complex projective space.Furthermore, the dynamics associated with the iteration of the renormalizationmap plays a key role in obtaining a suitable factorization of the spectral zetafunction of fractal differential operators. In this context, we discuss the worksof A. Teplyaev and of the authors regarding the examples of the bounded andunbounded Sierpinski gaskets as well as of fractal Sturm–Liouville differentialoperators on the half-line.

Contents

1. Introduction2. The bounded Sierpinski gasket2.1. Spectral properties of the Laplacian on the Sierpinski gasket3. Generalization of the decimation method3.1. The fractal Sturm–Liouville operator3.2. The eigenvalue problem3.3. The renormalization map and the spectrum of the operator4. An infinite lattice based on the Sierpinski gasket5. Factorization of the spectral zeta functionReferences

2010 Mathematics Subject Classification. Primary 28A80, 31C25, 32A20, 34B09, 34B40,34B45, 37F10, 37F25, 58J15, 82D30; Secondary 30D05, 32A10, 94C99.

Key words and phrases. Analysis on fractals, Laplacians on the bounded and unboundedSierpinski gasket, fractal Sturm–Liouville differential operators, self-similar measures and Dirichletforms, decimation method, renormalization operator and its iterates, single and multi-variablecomplex dynamics, spectral zeta function, Dirac delta hyperfunction, Riemann zeta function.

The work of the author was partially supported by the US National Science Foundationunder the research grant DMS-1107750, as well as by the Institut des Hautes Etudes Scientifiques(IHES) where the second author was a visiting professor in the Spring of 2012 while part of thispaper was written.


227

228 NISHU LAL AND MICHEL L. LAPIDUS

1. Introduction

From the probabilistic point of view, the Laplacian on the Sierpinski gasket SGwas introduced independently by S. Goldstein in [18] and S. Kusuoka in [27] (anda little later, by M. Barlow and E. Perkins in [6]), as the generator of the semigroupassociated with Brownian motion on SG. (See, e.g., [4] and [5] for early reviews ofthe subject of diffusions and random walks on self-similar fractals.) However, fromthe point of view of analysis, which will be our main concern here, the Laplaceoperator was first defined by J. Kigami [23] for the Sierpinski gasket and was laterextended in [24] to a class of self-similar fractals, called the post critically finitesets (p.c.f. sets). (See, e.g., [25] and [51] for a detailed exposition.) The Laplacianon a p.c.f. set is defined as the limit of a sequence of Laplacians of finite graphsthat approximate the fractal. Following the work of the physicists R. Rammal andG. Toulouse [40,41], M. Fukushima and T. Shima [17,50] studied the eigenvalueproblem associated with the Laplacian on the Sierpinski gasket and introduced thedecimation method in order to give an explicit construction of the set of eigenvalues.The decimation method, described in §2 of the present paper, is a process throughwhich we find the spectrum of the Laplacian on a fractal (in a certain class of self-similar sets) via the iteration of a rational function of a single complex variable,called the renormalization map. In the case of the finite (or bounded) Sierpinskigasket, this rational map is a polynomial on the complex plane and its dynamics isnot too difficult to understand in order to analyze the spectrum of the Laplacian.

Later on, C. Sabot ([43]–[47]) generalized the decimation method to Laplaciansdefined on a class of finitely-ramified self-similar sets with blow-ups. First, in[43,44,47], he studied fractal Sturm–Liouville operators on the half-line, viewedas a blow-up of the self-similar unit interval, and discovered that the correspondingdecimation method then involves the dynamics of a rational map which is no longera function of a single complex variable but is instead defined on the two-dimensionalcomplex projective space; see §3. (It therefore arises from a homogeneous rationalfunction of three complex variables.) This rational map is initially defined on aspace of quadratic forms associated with the fractal and its construction involvesthe notion of trace of a symmetric matrix on a (finite) subset of this space; see §4.From the point of view of multi-variable complex dynamics, the set of eigenvalues(or spectrum) is best understood in terms of an invariant curve under the iterationof the rational map.

Using as a model the unbounded (or infinite) Sierpinski gasket (the so-calledSierpinski lattice, see §4 and Figure 4), Sabot [46] used Grassmann algebras in orderto construct the renormalization map for other lattices based on (symmetric) finitelyramified self-similar sets. The idea is that one embeds the space of symmetricmatrices in a Grassmann algebra in order to analyze the operation of taking thetrace on a (suitable) finite subset. In some sense, this enables one to linearizethis operation. The polynomial associated with the classical bounded Sierpinskigasket, initially introduced in the work of Rammal and Toulouse [40,41] and laterrigorously formalized in [17,50], can then be recovered from the renormalizationmap associated with the unbounded Sierpisnki gasket.

Finally, in recent work, Lal and Lapidus [28] have studied the spectral zetafunction of the Laplacian on a suitable self-similar set and established a factorizationformula for the associated spectral zeta function in terms of a certain hyperfunction,a geometric zeta function and a zeta function associated with the iteration of a

LAPLACIANS ON FRACTALS AND COMPLEX DYNAMICS 229

Figure 1. The (bounded) Sierpinski gasket SG.

renormalization map, which is a multi-variable rational map acting in a complexprojective space. This latter work in [28] extends to several complex variables anearlier factorization formula due to Teplyaev [54,55], itself extending the secondauthor’s factorization formula [30, 31] for the spectral zeta function of a fractalstring (see also [33, 34, 36] for various applications of, and motivations for, thelatter factorization). We survey some of these results in the last part of this paper;see §5.

In closing this introduction, we mention that the work of [28] described in §5focuses on two different models, namely, fractal Sturm–Liouville differential oper-ators on the half-line (as in §3) and the infinite (or unbounded) Sierpinski gasket(as in §4). In each of these cases, the Dirac delta hyperfunction plays a key role inthe rigorous mathematical formulation of the factorization results.

2. The bounded Sierpinski gasket

The (bounded or finite) Sierpinski gasket (SG) is generated by the iteratedfunction system (IFS) consisting of three contraction mappings Φj : R2 → R2

defined by

(2.1) Φj(x) =1

2(x− qj) + qj

for j = 0, 1, 2, where q0, q1, q2 are the vertices of an equilateral triangle. (See Figure1.) Note that each Φj has a unique fixed point, namely, qj . The Sierpinski gasket isthe unique (nonempty) compact subset of R2 such that SG = Φ1(SG) ∪ Φ2(SG)∪Φ3(SG).

For each integer m ≥ 0, denote by Γm the mth level finite graph approximatingSG, and let Γ0 be the complete graph on V0 := {q1, q2, q3}. The set Vm of verticesof Γm is defined recursively as follows:

Vm :=2⋃

j=0

Φj(Vm−1),


for each integer m ≥ 1.Let �2(Vm) be the space of real-valued functions on Vm, equipped with the

standard inner product (u, v) =∑

x∈Vmu(x)v(x). The discrete Laplacian on �2(Vm)

(or the finite graph Laplacian on Γm) is defined by

Δmf(x) =1

4

∑y∼x

f(x) − f(y),

where x ∈ Γm and the sum is extended over all neighbors y of x in the graph Γm.We then define the Laplacian Δ = Δμ on SG as the following limit of rescaled

finite-difference operators:

Δf(x) = limm→∞

5mΔmf(x).

The factor 5 is the product of the scaling factor 3 for the natural Haursdorff measureon SG and the renormalized factor 5

3 for the energy. Here and thereafter, when wewrite Δ = Δμ, the subscript μ refers to the natural self-similar (or equivalently, inthis case, Hausdorff) probability measure on SG.

The graph energy on each Vm (m ≥ 0) is defined by

Em(u, u) =

(3

5

)−m∑x∼y

(u(x) − u(y))2,

which does not change under the process of harmonic extension, and the graphenergy on SG is then defined by

E(u, u) = supm≥0

Em(u, u) = limm→∞

Em(u, u)

(this limit always exists in [0,∞] since the sequence {Em(u, u)}∞m=0 is nondecreas-ing). In the sequel, we also write E(u) = E(u, u), in short, to refer to this quadraticform. (By definition, u belongs to the domain of E if and only if E(u) < ∞.) Theassociated bilinear form E(u, v) is then defined by polarization.

Suppose u is a function defined on V0, with values at each of the three verticesof V0 denoted by a, b, and c; see Figure 2. We want to extend u to V1 in such away that it minimizes the energy.

Let u be the harmonic extension of u to V1 and denote by x, y and z the valuesof u to be determined at each of the three vertices of V1 \ V0; see Figure 2. Since,by definition, u minimizes E1(v) subject to the constraint v = u on V0, we can takethe partial derivatives with respect to x, y, z and set them equal to zero to obtain

4x = b + c + y + z

4y = a + c + x + z

4z = a + b + x + y.

These equations express the mean value property of a (discrete) harmonic func-tion, according to which the function value at each of the junction points is theaverage of the function values of the four neighboring points in the graph. We canuse the matrix representation of these equations⎛⎝ 4 −1 −1

−1 4 −1−1 −1 4

⎞⎠ ⎛⎝ xyz

⎞⎠ =

⎛⎝ b + ca + ca + b

⎞⎠


Figure 2. The values of the harmonic extension u at each vertexin V1. The values a, b, c at each vertex in V0 are prescribed, whereasthe values x, y, z at each vertex in V1 \V0 are uniquely determinedby requiring that the energy be minimized in the passage from V0

to V1.

in order to obtain the following solutions:

x =1

5a +

2

5b +

2

5c

y =2

5a +

1

5b +

2

5c

z =2

5a +

2

5b +

1

5c.

The harmonic extension u therefore satisfies the 15 − 2

5 rule. See [23], [25],[51]. (More generally, this rule also holds for the harmonic extension on each m-cell of SG.) More specifically, iterating this process to each finite graph Vm, onethen defines u on the countable set of vertices V ∗ := ∪m≥0Vm and finally, extendsit to all of SG, by continuity and using the density of V ∗ in SG. The resultingfunction, still denoted by u, is called the harmonic extension of u. According toDefinition 2.1 below (which is also a theorem), it is the unique harmonic function(i.e., Δu = 0) such that u = u on V0. Equivalently, it is the unique minimizer ofthe energy functional E = E(v) subject to the constraint v = u on V0.

Definition 2.1. A harmonic function on SG (with boundary value u on V0) isa continuous function whose restriction to any Γm or Vm is the harmonic extensionof u. In other words, it is the unique solution of the following Poisson problem:Δv = 0, v = u on V0. It must therefore necessarily coincide with the harmonicextension u of u to all of SG.

2.1. Spectral properties of the Laplacian on the Sierpinski gasket.The Laplacian operators on p.c.f. self-similar fractals are defined similarly via asuitable approximation. To study the spectrum of the Laplacian, we consider theequation −Δu = λu, where u is a continuous function. The spectrum of the Lapla-cian on the Sierpinski gasket was first studied in detail by the physicists R. Rammaland G. Toulouse [40,41]. Later on, M. Fukushima and T. Shima [17,50] gave aprecise mathematical description of the eigenvalues and the eigenfunctions. Stillin the case of the Sierpinski gasket, Rammal and Toulouse discovered interesting


relations between the spectrum of the discrete Laplace operator and the iterationof a polynomial of one complex variable, R = R(z) := z(5 − 4z). More precisely,for any m ≥ 0, if λ is an eigenvalue of −Δm+1 on Γm+1, then λ(5 − 4λ) is aneigenvalue of −Δm on Γm. Thus, the relationship between the eigenvalues of theLaplacians on one graph and it successor can be described by a quadratic equation,λm = λm+1(5 − 4λm+1) = R(λm+1). The restriction to Vm of any eigenfunctionbelonging to λm+1 is an eigenfunction belonging to λm. The relationship betweenthe eigenvalues λm and λm+1 of −Δm and −Δm+1, respectively, can be found bycomparing the corresponding eigenvalue problem for a point common to both Vm

and Vm+1.

Theorem 2.2 (Fukushima and Shima, [17], [50]).

(i) If u is an eigenfunction of −Δm+1 with eigenvalue λ (that is, −Δm+1u = λu),and if λ /∈ B, then −Δm(u|Vm

) = R(λ)u|Vm, where B := { 1

2 ,54 ,

32} is the set

of ‘forbidden’ eigenvalues and u|Vmis the restriction of u to Vm.

(ii) If −Δmu = R(λ)u and λ /∈ B, then there exists a unique extension w of u toVm+1 such that −Δm+1w = λw.

At any given level m, there are two kind of eigenvalues of −Δm, called the initialand continued eigenvalues. The continued eigenvalues arise from the spectrum of−Δm−1 via the decimation method (described in Theorem 2.2), and the remainingeigenvalues are called the initial eigenvalues. The forbidden eigenvalues { 1

2 ,54 ,

32} in

Theorem 2.2 have no predecessor, i.e., they are the initial eigenvalues. Furthermore,the exclusion of the eigenvalue 1

2 can be explained by showing that 12 is an eigenvalue

of −Δm only for m = 1. (See Figure 3.)Given that the eigenvalues of −Δ0 are {0, 3

2}, we consider the inverse images

of 0 and 32 under R (that is, their images under R−(z) = 5−

√25−16z8 and R+(z) =

5+√25−16z8 , the two inverse branches of the quadratic polynomial R(z) = z(5−4z)),

to obtain the eigenvalues of −Δ1. The continuation of this process generates theentire set of eigenvalues for each level. The diagram provided in Figure 3 illustratesthe eigenvalues associated with each graph Laplacian −Δm, for m = 0, 1, 2, ..., interms of the inverse iterates of the polynomial R.

The spectrum of the Sierpinski gasket is the renormalized limit of the spectraof the graph Laplacians −Δm. More specifically, each eigenvalue satisfying theequation −Δμu = λu can be written as

(2.2) λ = limm→∞

5mλm,

for a sequence {λm}∞m=m0such that λm = λm+1(5 − 4λm+1) = R(λm) for all

m ≥ m0 and for some smallest integer m0 (which is allowed to depend on λ). Notethat for m > m0, λm does not coincide with any of the forbidden eigenvalues in{ 12 ,

54 ,

34}, whereas λm0

belongs to the set { 12 ,

54 ,

34}. Furthermore, the values λm are

determined by the solutions of λm = λm+1(5 − 4λm+1): λm+1 = 5+εm√25−16λm

8 ,where εm = ±1, provided that the limit in (2.2) exists. The limit λ only existsif εm = −1 for all but finitely many integers m. In that case, λ is an eigenvalueof −Δμ where, as before, Δ = Δμ denotes the Laplacian on SG. And conversely,every eigenvalue of −Δμ can be obtained in this manner.

In the next section, we will discuss a generalization (due to Sabot [44, 45])of the decimation method to rational functions of several complex variables, whenpresenting the case of fractal Sturm–Liouville differential operators. The extended


−Δ0

−Δ1

−Δ

−Δ 2

3

−Δ4

Operators Eigenvalues

54

54

54

54

[1]0

[1]0

[1]0

[1]0

0

2[2]3

34

[2]

+[2] [2]

[2] [2]

23 [3]

34

[3]12

R(3/4)+[3]

R(3/4)[3]

32

[6]

34

[6]21

R(3/4)+[6]

R(3/4)[6]

45 [1]

+R(5/4)

[1] [1]

R(5/4) 32

[15]

3

[15]4

12

12

54

[4]

R(5/4) R(5/4)+[4] [4]

R R(3/4)+ +R R(3/4)+

[1]

R (3/4) R (3/4)

Figure 3. The eigenvalues of SG are obtained via the decimationmethod as limits of inverse images of the renormalization map R.At each level, the two branches of the tree correspond to the twoinverse branches of the quadratic polynomial R, denoted by R+

and R−. The number in the bracket represents the multiplicity ofthe corresponding eigenvalue.

(multi-variable) decimation method is not valid for all self-similar fractals, in gen-eral. However, it does apply to a large class of symmetric p.c.f. self-similar fractals(whereas the original single-variable decimation method only applied to a rather re-stricted and difficult to characterize class of symmetric finitely ramified and hence,p.c.f., self-similar fractals). Therefore, this generalization is a very significant exten-sion of the original (single-variable) decimation method, for which it also providesa nice geometric and algebraic interpretation (see §4).

In fact, mathematically, the truly beautiful underlying structure of the extendeddecimation method is only revealed by considering the multi-variable case, even inthe original setting of a single complex variable. (The latter one-variable caseshould really be formulated in terms of two complex variables or equivalently, onthe complex projective line, and via phase space symplectic geometry or Grassmannalgebras, in terms of two conjugate variables. In hindsight, the reduction to a singlevariable is simply a confusing, albeit convenient, artifact in this situation.)

Remark 2.3. The interested reader can find in [2, 3] some detailed and rel-atively elementary computations pertaining to the decimation method and the as-sociated renormalization map (viewed only as a rational function of one complexvariable) in the case of certain examples of finitely ramified, symmetric self-similarfractals.


3. Generalization of the decimation method

3.1. The fractal Sturm–Liouville operator. C. Sabot, in a series of papers([43]–[47]), extended the decimation method to Laplacians defined on a class of(symmetric) finitely ramified (really, p.c.f.) self-similar sets with blow-ups. Thisextension involves the dynamics of rational functions of several complex variables.We discuss the prototypical example he studied, fractal Laplacians on the blow-upI<∞> = [0,∞) of the unit interval I = I<0> = [0, 1]. From now on, we will assumethat

(3.1) 0 < α < 1, b = 1 − α, δ =α

1 − α, and γ =

1

α(1 − α).

Consider the contraction mappings from I = [0, 1] to itself given by

Ψ1(x) = αx, Ψ2(x) = 1 − (1 − α)(1 − x),

and the unique self-similar measure m on [0, 1] such that for all f ∈ C([0, 1]) (thespace of continous functions on I = [0, 1]),

(3.2)

∫ 1

0

fdm = b

∫ 1

0

f ◦ Ψ1dm + (1 − b)

∫ 1

0

f ◦ Ψ2dm.

Here, I is viewed as the unique self-similar set (in the sense of [21]) associated withthe iterated function system {Ψ1,Ψ2}:(3.3) I = Ψ1(I) ∪ Ψ2(I).

Define H<0> = − ddm

ddx , the free Hamiltonian with Dirichlet boundary condi-

tions on [0, 1], by H<0>f = g on the domain{f ∈ L2(I,m) : ∃g ∈ L2(I,m), f(x) = cx+d+

∫ x

0

∫ y

0

g(z)dm(z)dy, f(0) = f(1) = 0

}.

The operator H<0> is the infinitesimal generator associated with the Dirichlet form(a,D) given by

a(f, g) =

∫ 1

0

f ′g′dx, for f, g ∈ D,

where

D = {f ∈ L2(I,m) : f ′ ∈ L2(I, dx)}.As can be easily checked, the Dirichlet form a satisfies the similarity equation

(3.4) a(f) = α−1a(f ◦ Ψ1) + (1 − α)−1a(f ◦ Ψ2),

where we denote the quadratic form a(f, f) by a(f). (See, e.g., [13] for an exposi-tion.)

The idea is that viewing the unit interval I as a self-similar set, as in (3.3)above, we construct an increasing sequence of intervals I<n>, for n = 0, 1, 2, ..., byblowing-up the initial unit interval by the scaling ratio α−n. Hence, we can extendall the objects involved, (m, a, H<0>), from I = I<0> to I<n> = Ψ−n

1 (I) = [0, α−n],which can be expressed as a self-similar set as follows:

I<n> =⋃

i1,...,in

Ψi1...in(I<n>),

where (i1, ..., in) ∈ {1, 2}n. Here, we have set Ψi1...in = Ψin ◦ ... ◦ Ψi1 .


More precisely, for each n ≥ 0, we define the self-similar measure m<n> onI<n> by ∫

I<n>

fdm<n> = (1 − α)−n

∫I

f ◦ Ψ−n1 dm,

for all f ∈ C(I<n>). Similarly, the corresponding differential operator, H<n> =− d

dm<n>

ddx on I<n> = [0, α−n], can be defined as the infinitesimal generator of the

Dirichlet form (a<n>,D<n>) given by

a<n>(f) =

∫ α−n

0

(f ′)2dx = αna(f ◦ Ψ−n1 ), for f ∈ D<n>,

where

D<n> = {f ∈ L2(I<n>,m<n>) : f ′ exists a.e. and f ′ ∈ L2(I<n>, dx)}.We define H<∞> as the operator − d

dm<∞>

ddx with Dirichlet boundary condi-

tions on I<∞> = [0,∞). It is clear that the (projective system of) measures m<n>

give rise to a measure m<∞> on I<∞> since for any f ∈ D<n> with support con-tained in [0, 1], a<n>(f) = a(f) and

∫I<n>

fdm<n> =∫Ifdm. Furthermore, we

define the corresponding Dirichlet form (a<∞>,D<∞>) by

a<∞>(f) = supn≥0

a<n>(f |I<n>) = lim

n→∞a<n>(f |I<n>

), for f ∈ D<∞>,

where

D<∞> = {f ∈ L2(I<∞>,m<∞>) : supn

a<n>(f |I<n>) < ∞}.

Clearly, a<∞> satisfies a self-similar identity analogous to Equation (3.4) and itsinfinitesimal generator is H<∞>.

3.2. The eigenvalue problem. The study of the eigenvalue problem

(3.5) H<n>f = − d

dm<n>

d

dxf = λf

for the Sturm–Liouville operator H<n>, equipped with Dirichlet boundary condi-tions on I<n>, revolves around a map ρ, called the renormalization map, which isinitially defined on a space of quadratic forms associated with the self-similar setI<n> (or with a corresponding finite graph, see Remark 3.1 below) and then, viaanalytic continuation, on C3 as well as (by homogeneity) on P2(C), the complexprojective plane. (More precisely, the renormalization map is associated to thepassage from I<n−1> to I<n> or, equivalently, from I<0> to I<1>.) As will beexplained in §3.3, the propagator of the above differential equation (3.5) is veryuseful in producing this rational map, initially defined on C3, and later on, as thepolynomial map

(3.6) ρ([x, y, z]) = [x(x + δ−1y) − δ−1z2, δy(x + δ−1y) − δz2, z2],

defined on the complex projective plane P2(C). Here, [x, y, z] denotes the homo-geneous coordinates of a point in P2(C), where (x, y, z) ∈ C3 is identified with(βx, βy, βz) for any β ∈ C, β �= 0. Clearly, in the present case, ρ is a homogeneousquadratic polynomial.

(Note that if ρ is viewed as a map from P2(C) to itself, then one should writemore correctly,

ρ([x : y : z]) = [x(x + δ−1y) − δ−1z2 : δy(x + δ−1y) − δz2 : z2],


where [u : v : w] = [u, v, w] denote the homogeneous coordinates of a generic pointin the projective plane P2 = P2(C). We will use this notation in §4.)

As we shall see in §3.3, the spectrum of the fractal Sturm–Liouville operatorH<∞>, as well as of its finite graph (or rather, here, bounded interval, see Remark3.1 below) approximations H<n> (n = 0, 1, 2, ...), is intimately related to the iter-ation of ρ. The spectrum of H<0> and of H<n> (n = 1, 2, ...) is discrete for anyvalue of α in (0, 1) (and hence, for any γ ≥ 4). (See, e.g., [8], [12], [13], [14], [15].)However, in the sequel, we will focus our attention on the case where α ≤ 1

2 (or

equivalently, δ ≤ 1). In that case, the spectrum of H<∞> is pure point for α < 12

(i.e., γ > 4), but absolutely continuous for α = 12 (i.e., for γ = 4). Furthermore,

observe that since α ∈ (0, 1) and γ = 1α(1−α) (see Equation (3.1)), we always have

γ ≥ 4 (and in particular, γ > 1), independently of the above assumption accordingto which α ≤ 1

2 . Finally, note that γ = 4 if and only if α = 12 (i.e., δ = 1), an

interesting special case which will be investigated at the end of §5.

Remark 3.1. In [44] (see also [45] and [46]), is also provided a description ofthe renormalization map in terms of lattice or finite graph (rather than of boundedself-similar interval) approximations of the half-line. By necessity of concision, wewill not discuss this matter further in this paper. The method, however, is veryanalogous to the one described in part of §4 below in the case of the infinite latticeSG<∞> based on the bounded Sierpinski gasket SG. (In some definite sense, thebounded self-similar interval I = [0, 1] would play the role here of SG, while thehalf-line I<∞> = [0,∞) would be a substitute for SG<∞>.)

3.3. The renormalization map and the spectrum of the operator. Therenormalization map ρ is a function from the complex projective plane P2(C) toitself which is induced by the above homogeneous polynomial map from C3 to itself;see Equation (3.6) and the discussion surrounding it. As we shall see later on, thespectrum of the operator H<∞> defined in §3.1 is intimately connected with theiteration of the renormalization map. Following [44], we next explain how theexplicit expression for the renormalization map given in Equation (3.6) above canbe derived by studying the propagator for the eigenvalue problem associated withthe operator.

We define the propagator Γλ(s, t) for the eigenvalue problem − ddm<∞>

ddxf = λf

associated with the operator H<∞> on I<∞> = [0,∞) as a time evolution functionwhich, for each 0 ≤ s ≤ t, is a 2 × 2 matrix with nonzero determinant such thatthe solution of the equation satisfies[

f(t)f ′(t)

]= Γλ(s, t)

[f(s)f ′(s)

],

where f ′ denotes the derivative of f .Using the self-similarity relations (3.2) and (3.4) satisfied by the measure m

and the Dirichlet form a, respectively, and recalling that γ is given by Equation(3.1), we obtain Γ<n>,λ = Dαn ◦Γγnλ ◦Dα−n for the propagator Γ<n>,λ associated

with the eigenvalue problem − ddm<n>

ddxf = λf , where

Dαn :=

[1 00 αn

].

In order to derive the expression of the renormalization map stated in Equation(3.6), we will consider the case when n = 1. Thus, we have Γ<1>,λ = Dα ◦ Γγλ ◦


Dα−1 . Let

Γλ :=

[a(λ) b(λ)c(λ) d(λ)

].

We will proceed with the following calculations:

Γ<1>,λ =

[1 00 α

] [a(γλ) b(γλ)c(γλ) d(γλ)

] [1 00 α−1

]=

[a(γλ) α−1b(γλ)αc(γλ) d(γλ)

].

On the other hand, we have

Γ<1>,λ = Γλ(1, α−1) ◦ Γλ(0, 1) = Dδ ◦ Γλ ◦Dδ−1 ◦ Γλ

=

[1 00 δ

] [a(λ) b(λ)c(λ) d(λ)

] [1 00 δ−1

] [a(λ) b(λ)c(λ) d(λ)

]=

[a(λ)2 + δ−1b(λ)c(λ) a(λ)b(λ) + δ−1d(λ)b(λ)δa(λ)c(λ) + c(λ)d(λ) δb(λ)c(λ) + d(λ)d(λ)

].

Using the fact that Γλ ∈ SL2(C) (the special complex linear group of 2×2 matrices)and hence, that a(λ)d(λ) − b(λ)c(λ) = 1, we see that the two diagonal entries canbe rewritten as

a(λ)2+δ−1b(λ)c(λ) = a(λ)

[a(λ)+δ−1

(d(λ)a(λ) − 1

a(λ)

)]= a(λ)(a(λ)+δ−1d(λ))−δ−1,

δb(λ)c(λ)+d(λ)d(λ) = δd(λ)

[a(λ)d(λ) − 1

d(λ)+δ−1d(λ)

]= δd(λ)(a(λ)+δ−1d(λ))−δ.

We initially define the renormalization map ρ : C2 → C2 in terms of the abovediagonal entries as

ρ(x, y) = (x(x + δ−1y) − δ−1, δy(x + δ−1y) − δ)

and the map φ : C → C2 as

φ(λ) =

[a(λ)d(λ)

].

Note that ρ ◦ φ(λ)) = φ(γλ) for all λ ∈ C.We now go back to the 2-dimensional complex projective space P2 = P2(C),

and note that any point [x, y, z] ∈ P2 is equivalent to [xz ,yz , 1] for z �= 0. We can

therefore represent P2 by

P2 = {(q1, q2, 1) : (q1, q2) ∈ C2} ∪ {[x, y, 0] : (x, y) ∈ C2}.We can then naturally define the map ρ, now viewed as a polynomial map from thecomplex projective plane P2 = P2(C) to itself, as follows:

(3.7) ρ([x, y, z]) = [x(x + δ−1y) − δ−1z2, δy(x + δ−1y) − δz2, z2],

which is in agreement with Equation (3.6).In light of the above discussion, the invariant curve φ can be viewed as a map

φ : C → C3 defined by φ(λ) = (a(λ), b(λ), 1) and satisfying the functional equation

(3.8) ρ ◦ φ(λ) = φ(γλ),

for all λ ∈ C.Next, we study the spectrum of the eigenvalue equation (3.5), as well as of

its counterpart for n = ∞. An attractive fixed point x0 of ρ is a point such thatρx0 = x0 and for any other point x in some neighborhood of x0, the sequence


{ρnx}∞n=0 converges to x0. The basin of attraction of a fixed point is contained inthe Fatou set of ρ. For δ > 1, x0 = [0, 1, 0] is an attractive fixed point of ρ. The set

(3.9) D := {[x, y, z] : x + δ−1y = 0}is part of the Fatou set of ρ since it is contained in the basin of attraction of x0. (Forvarious notions of higher-dimensional complex dynamics, we refer to the surveysprovided in [10] and [11].) The set D and the invariant curve φ of ρ togetherdetermine the spectrum of H<n> and of H<∞>. Moreover, the set of eigenvalues(i.e., here, the spectrum) can be described by the set

(3.10) S := {λ ∈ C : φ(γ−1λ) ∈ D},the ‘time intersections’ of the curve φ(γ−1λ) with D. It turns out that S is countablyinfinite and contained in (0,∞). We write S = {λk}∞k=1, with λ1 ≤ λ2 ≤ ... ≤ λk ≤..., each eigenvalue being repeated accordingly to its multiplicity. (It also turns outthat the λk’s are all simple, in this case.) Furthermore, following [28], we call Sthe generating set for the spectrum of H<n>, with n = 0, 1, ...,∞.

Let Sp = γpS, for each p ∈ Z. Recall that we are assuming throughout thatα ≤ 1

2 (i.e., δ ≤ 1). It follows that not only the spectrum of H<n> (for n = 0, 1, 2, ...)

is discrete but unless α = 12 (i.e., δ = 1 or equivalently, γ = 4), so is the Dirichlet

spectrum of H<∞>. Furthermore, the spectrum of H<n> and of H<∞> can bededuced from that of H<0>, as will be seen in the next theorem. Finally, recallthat we always have γ ≥ 4 and hence, γ > 1.

Theorem 3.2 (Sabot, [47]). The spectrum of H<0> on I = I<0> is equalto⋃∞

p=0 Sp, while (if α < 12 ) the spectrum of H<∞> on I<∞> = [0,∞) is equal

to⋃∞

p=−∞ Sp.1 Furthermore, for any n ≥ 0, the spectrum of H<n> is equal to⋃∞

p=−n Sp. Moreover, for n = 0, 1, ...,∞, each eigenvalue of H<n> is simple. (In

particular, each λj ∈ S has multiplicity one.)

The diagram associated with the set of eigenvalues of the operator H<∞> canbe represented as follows:

......

......

γ−2λ1 γ−2λ2 γ−2λ3 γ−2λ4 · · ·γ−1λ1 γ−1λ2 γ−1λ3 γ−1λ4 · · ·λ1 λ2 λ3 λ4 · · ·γλ1 γλ2 γλ3 γλ4 · · ·γ2λ1 γ2λ2 γ2λ3 γ2λ4 · · ·

......

......

Sabot’s work ([43]–[47]) has sparked an interest in generalizing the decimationmethod to a broader class of fractals and therefore, to the iteration of rationalfunctions of several complex variables.

For each k ≥ 1, we denote by fk the solution of the equation H<∞>f = λkffor λk ∈ S. In other words, fk is an eigenfunction of H<∞> associated withthe eigenvalue λk ∈ S. (Note that fk is uniquely determined, up to a nonzeromultiplicative constant which can be fixed by a suitable normalization.)

1If α = 12, then the spectrum of H<∞> is given by the closure of ∪∞

p=−∞Sp.


Theorem 3.3 (Sabot, [47]). Assume that α < 12 .

(i) Given any k ≥ 1 and given p ∈ Z, if fk is the normalized solution of the

equation H<∞>f = λkf for λk ∈ S, then fk,p := fk ◦ Ψ−p1 is the solution of

the equation H<∞>f = λk,pf , where λk,p := γpλk and p ∈ Z is arbitrary.

(ii) Moreover, if fk,p = fk ◦ Ψ−p1 is the solution of the equation H<∞>f = λk,pf ,

then fk,p,<n> := fk,p|I<n>, the restriction of fk,p to I<n>, is the solution of

the equation H<n>f = λk,pf .

Finally, for each n = 0, 1, ...,∞, {fk,p,<n> : k ≥ 1, p ≥ −n} is a complete set ofeigenfunctions of H<n> in the Hilbert space L2(R+,m<∞>), where R+ = I<∞> =[0,∞).

4. An infinite lattice based on the Sierpinski gasket

In this section, we will show that the polynomial induced by the decimationmethod in the case of the classical (bounded) Sierpinski gasket SG can be recoveredfrom the infinite lattice SG(∞) based on the (bounded) Sierpinki gasket.

We start with a self-similar set F = {1, 2, 3}, the vertices of an equilatraltriangle, and construct an increasing sequence of finite sets F<n> by blowing-upthe initial set F = F<0>. For instance, F<1> is defined as the union of three copiesof F . Namely, F = ∪3

i=1F<1>,i. The unbounded set F<∞>, called the infinite

Sierpinski gasket and also denoted by SG(∞) (see Figure 4), is the countable setdefined as the union of all the finite sets F<n> = Φ−1

i1◦ Φ−1

i2◦ ... ◦ Φ−1

in(F ), where

(i1, i2, ...in) ∈ {0, 1, 2}n, n ≥ 0, and (Φ0,Φ1,Φ2) are the contraction mappings,expressing the self-similarity of the set F :

F<∞> =

∞⋃n=0

F<n>.

Similarly, we can define a sequence of operators H<n> on F<n>, and the Laplaceoperator H<∞> on F<∞> can be viewed as a suitable limit of the operators H<n>.The operators H<n> arise from the Dirichlet forms A<n> defined on �(F<n>) =RF<n> , the space of real-valued functions on F<n>. To construct the Laplacian onthe sequence F<n>, define the Dirichlet form by

A<n>(f(x)) =∑y∼x

(f(y) − f(x))2,

where f ∈ RF<n> and the sum runs over all the neighbors y of x in the finite graphassociated with F<n>.

The operator H<n> on L2(F<n>, b<n>) is the infinitesimal generator of theDirichlet form defined by < A<n>f, g >= −

∫H<n>fgdb<n>, where b<n> is the

positive measure on F<n> which gives a mass of 1 to the points in ∂F<n> and 2 tothe points of F<n>\∂F<n>, where ∂F<n> denotes the boundary of the graph F<n>.The sequence H<n> is uniformly bounded and we can use it to define the operatorH<∞> on L2(F<∞>, b<∞>), where b<∞> is the positive measure on F<∞> definedas a suitable limit of the measures b<n>.

Sabot [45] has shown that the extended decimation method which he estab-lished for these fractals naturally involved the complex dynamics of a renormaliza-tion map of several complex variables.

Let G be the group of symmetries acting on the finite lattices F<n>, namelyS3, which is also the natural symmetry group of the equilateral triangle and of


Figure 4. An infinite Sierpinski gasket, F<∞> = SG(∞).

F = {1, 2, 3}. We denote by SymG the space of symmetric (linear) operatorson CF which are invariant under G. For every Q ∈ SymG, we can construct asymmetric operator Q<1> on CF<1> as a sum of the copies of Q on F<1>,i (wherewe recall that F<1>,1, F<1>,2 and F<1>,3 are the three copies of F<1>):

Q<1> =

3∑i=1

Q<1>,i.

We define a rational map T : SymG → SymG by

(4.1) T (Q) = (Q<1>)|∂F<1>,

the trace of Q on the ‘boundary’ of F<1>. In general, using the interpretation ofsymmetric matrices, it is difficult to analyze the map T , which involves the notion oftrace on a subset. (Clearly, one can also view Q as a quadratic form on CF ; hence,the notation.) To avoid the complication of taking the trace of a symmetric matrixon a subset, the space of symmetric matrices can be embedded into a Grassmannalgebra, in which case the map T becomes linear.

Let E and E be two linear subspaces of CF with canonical basis (ηx)x∈F and(ηx)x∈F , respectively, and such that CF = E⊕E. We define the Grassmann algebraassociated with this (orthogonal direct sum) decomposition of CF by∧

(E ⊕ E) =

2|F |⊕k=0

(E ⊕ E)∧k,

where (E ⊕ E)∧k is the k-fold antisymmetric tensor product of E ⊕ E with itself,and |F | = 3 is the cardinality of F . We consider the subalgebra of the Grassmannalgebra generated by the monomials containing the same number of variables η andη, namely,

A =

|F |⊕k=0

E∧k ∧ E∧k.

We can embed SymG into A via the injection map

SymG → AQ �→ exp(ηQη),


where ηQη =∑

i,j∈F Qi,j ηiηj and Q := (Qi,j)i,j∈F , a symmetric matrix. We

denote by P(A) the projective space associated with A. Let the correspondingcanonical projection be π : A → P(A). The closure of the set of points of the formπ(exp(ηQη) for Q ∈ SymG, denoted by LG, is a smooth submanifold of P(A) ofdimension dim(SymG). (We refer, for example, to [42] and [56] for an introductionto Grassmann algebras and projective geometry.)

With T defined by (4.1), we define the following homogeneous polynomial mapof degree 3 (recall that η2 = η2 = 0):

R : A → Aexp(ηQη) �−→ det((Q<1>)|F<1>\∂F<1>

exp(ηTQη).

This map induces a map g on the projective space LG = P1 × P1 such thatgn(π(x)) = π(Rn(x)) for x ∈ π−1(LG) and n ≥ 0. (Here, P1 = P1(C) = C ∪ {∞}is the complex projective line, also known as the Riemann sphere.) Since LG is acompactification of SymG, the map g : LG → LG is an extension of the trace mapT : SymG → SymG from SymG to LG.

More precisely, note that CF can be decomposed into a sum of two irreduciblerepresentations: CF = W0 ⊕ W1, where W0 is the subspace of constant functionsand W1 is its orthogonal complement. Let p|W0

and p|W1denote the orthogonal pro-

jections of CF onto W0 and W1, respectively. Via the isomorphism between SymG

and C2 (recall that F has cardinality 3), every element Q ∈ SymG can be writtenas Qu0,u1 = u0p|W0

+ u1p|W1, where (u0, u1) ∈ C2. Then, the renormalization map

T is defined by

T (u0, u1) =

(3u0u1

2u0 + u1,u1(u0 + u1)

5u1 + u0

).

This map induces another map on P1, denoted by g and given by

g([z0 : z1]) = [z0(5z1 + z0) : (2z0 + z1)(z0 + z1)].

(Here, [w0 : w1] denotes the homogeneous coordinates of a generic point of theprojective line P1 = P1(C).) Indeed, upon the substitution z = u0

u1, the map T

gives rise to

g(z) =u0u1

2u0+u1

u1(u0+u1

5u1+u0

=u0u1

2u0 + u1

5u1 + u0

u1(u0 + u1)

=u0u

21(5 + u0

u1)

u1(2u0

u1+ 1)u2

1(u0

u1+ 1)

=z(5 + z)

(2z + 1)(z + 1).

The dynamic of the map g plays an essential role in the study of the spectrum ofthe associated symmetric operators. After having made the additional change ofvariable v = 3z

1−z , we obtain the following equation

3v(2v + 5)

(3v + 3)(2v + 3),

from which we recover the polynomial p(v) = v(2v + 5). This polynomial plays asignificant role in the case of the bounded Sierpinski gasket (as shown in §2) and


was introduced (by completely different methods) in the initial work of Rammal[40] and of Rammal and Toulouse [41]. We note that the dynamics of rationalmaps in higher dimension is hidden in the one-dimensional case of the Sierpinskigasket.

Remark 4.1. A thorough discussion of the symplectic and supersymmetric as-pects of the new methods developed in [45] (as well as, implicitly, in [43] and [44])to extend to the multi-variable case the classic decimation method, is provided in[46]. In particular, beside supersymmetry (which is translated mathematically bythe presence of Grassmann algebras and variables), geometric quantization and theassociated momentum map in symplectic geometry play an important role in thiscontext.

Remark 4.2. A good review of many of the rigorously established propertiesof the Laplacian on various realizations of “the” (deterministic) Sierpinski gasketSG(∞) can be found in [53], both in the case of discrete and continuous spectra.(We only discuss the case of discrete spectra in the present paper; see, however,Remark 5.14 below.) Moreover, in [45], the emphasis is on the study of the spectralproperies of random (rather than deterministic) realizations of “the” infinite (orunbounded) Sierpinski gasket SG(∞).

5. Factorization of the spectral zeta function

In this section, we show that the spectral zeta function of the Laplacian definedon a (suitable) finitely ramified self-similar set or on an infinite lattice based on thisfractal can be written in terms of the zeta function associated with the renormaliza-tion map. We will focus here on the case of the Laplacian on the bounded Sierpinskigasket SG (as in §2 and [55]) or on the infinite Sierpinski gasket SG(∞) (as in §4and [28]), as well as on the case of fractal Sturm–Liouville differential operatorson the half-line I<∞> = [0,∞), viewed as a blow-up of the self-similar intervalI = [0, 1] (as in §3 and [28]).

Definition 5.1. The spectral zeta function of a positive self-adjoint operatorL with compact resolvent (and hence, with discrete spectrum) is given (for Re(s)sufficiently large) by

(5.1) ζL(s) =∞∑j=1

(κj)−s/2,

where the positive real numbers κj are the eigenvalues of the operator written innonincreasing order and counted according to their multiplicities.

A. Teplyaev ([55], see also [54]), motivated by the known identity for fractalstrings (see Remark 5.3 below), has studied the spectral zeta function of the Lapla-cian on SG and, in the process, has explored interesting connections between thespectral zeta function and the iteration of the polynomial induced by the decimationmethod.

Theorem 5.2 (Teplyaev, [55]). The spectral zeta function of the Laplacian onSG is given by(5.2)

ζΔμ(s) = ζR, 34

(s)5−

s2

2

(1

1 − 3 · 5− s2+

3

1 − 5−s2

)+ζR, 54

(s)5−s

2

(3

1 − 3 · 5− s2− 1

1 − 5−s2

),


where R(z) := z(5 − 4z) and

(5.3) ζR,z0(s) := limn→∞

∑z∈R−n{z0}

(cnz)−s2

is the polynomial zeta function of R, defined for Re(s) > 2 log 2log c (where c := 5 =

R′(0)). Furthermore, there exists ε > 0 such that ζΔμ(s) has a meromorphic con-

tinuation for Re(s) > −ε, with poles contained in

{2inπlog 5 ,

log 9+2inπlog 5 : n ∈ Z

}.

Remark 5.3. Theorem 5.2 extends to the present setting of analysis on certainfractals [25] (or ‘drums with fractals membrane’, see, e.g., [7], [9], [13]–[17], [20],[26]–[28], [31]–[32], [35], [40]–[45], [50], [51], [53]–[55]) and provides a dynamicalinterpretation of the factorization formula obtained by the second author in [30],[31] for the spectral zeta function ζν(s) of the Dirichlet Laplacian associated with afractal string (a one-dimensional drum with fractal boundary [29]–[36]):

(5.4) ζν(s) = ζ(s) · ζg(s),where ζ(s) is the classic Riemann zeta function and ζg(s) is (the meromorphic con-tinuation of ) the geometric zeta function of the fractal string (whose poles are calledthe complex dimensions of the string and help describe the oscillations intrinsic tothe geometry and the spectrum of the string, see [36]). See also ([36], §1.3) fora discussion of Equation (5.4) and, for example, [30], [31], [33], [34], along withmuch of [36], for various applications of this factorization formula. Finally, wenote that in [55], ζ(s) is reinterpreted as the polynomial zeta function of a certainquadratic polynomial of a single complex variable and hence, the factorization for-mula (5.4) can also be given a complex dynamical interpretation (in terms of theiteration of the renormalization map). This issue was revisited in [28], and, asa result, formula (5.4) can also be interpreted in terms of multi-variable complexdynamics. (See [28], along with Theorem 5.12 below.)

Remark 5.4. In [9], shortly after the completion of [54], [55], G. Derfel, P.Grabner and F. Vogl, working independently on this question and motivated in partby the results and conjectures of [26] and [31], have obtained another interpretationof the geometric factor of the factorization formulas (5.2) and (5.4), expressed interms of the multiplicities of the eigenvalues. In the process, they have shown thatthe spectral zeta function ζΔμ

(s) has a meromorphic continuation to all of C. More-over, they have proved further cases of a conjecture of [26], [31], according to whichthe asymptotic second term in the spectral counting function of the Laplacian on theSierpinski gasket SG and other lattice self-similar fractals is truly oscillatory (orequivalently, the corresponding periodic function obtained in [26] is not constant).[Compare with analogous conjectures and results stated or obtained in [29]–[36] fordrums with fractal boundary (instead of drums with fractal membrane).]

Later on, in [28], we extended this result about the factorization of the spectralfunction of the Laplacian on the finite (or bounded) gasket SG to the infinite (orunbounded) Sierpinski gasket SG(∞) and to the renormalization maps of severalcomplex variables associated with fractal Sturm–Liouville operators.

The Dirac delta hyperfunction on the unit circle T is defined as δT = [δ+T, δ−

T] =

[ 11−z ,

1z−1 ]. It consists of two analytic functions, δ+

T: E → C and δ−

T: C\E → C,

where E = {z ∈ C : |z| < 1 + 1N } for a large natural number N . In other words,


a hyperfunction on T can be viewed as a suitable pair of holomorphic functions,one on the unit disk |z| < 1, and one on its exterior, |z| > 1. (See, for example,([19], §1.3) and ([37], §3.3.2) for a discussion of various changes of variables in ahyperfunction. See also those two books [19], [37], along with [48], [49], the originalarticles by M. Sato, for an overview of the theory of hyperfunctions. Moreover, see[52] for a detailed discussion of δT and, more generally, of hyperfunctions on theunit circle T.)

Theorem 5.5 ([28], Lal and Lapidus). The spectral zeta function ζΔ(∞) of theLaplacian Δ(∞) on the infinite Sierpinski gasket SG(∞) is given by

(5.5) ζΔ(∞)(s) = ζΔμ(s)δT(5−

s2 ),

where δT is the Dirac hyperfunction on the unit circle T and ζΔμis the spectral

zeta function of the Laplacian on the finite (i.e., bounded) Sierpinski gasket SG, asgiven and factorized explicitly in Equation (5.2) of Theorem 5.2.

In the case of the Sturm–Liouville operator on the half-line, we introduce amulti-variable analog of the polynomial zeta function occurring in Equation (5.3)of Theorem 5.2.

Definition 5.6 ([28]). We define the zeta function of the renormalization mapρ to be

(5.6) ζρ(s) =

∞∑p=0

∑{λ∈C: ρp(φ(γ−(p+1)λ))∈D}

(γpλ)−s2 ,

for Re(s) sufficiently large.

Remark 5.7. Recall from §3 that in the present situation of fractal Sturm–Liouville operators, the renormalization map ρ is given by Equation (3.6) and thatthe ‘renormalization constant’ (or ‘scaling factor’ ) γ is given by Equation (3.1).(Also, see the functional equation (3.8) defining the invariant curve φ, as wellas Equations (3.9) and (3.10) defining D and the generating set S, respectively.)Furthermore, recall from the end of §3.2 that we assume that α ≤ 1

2 (i.e., δ ≤ 1)

in order to ensure the discreteness of all the spectra involved,2 and that we alwayshave γ ≥ 4; in particular, γ > 1, and γ = 4 if and only if α = 1

2 .

We consider the factorization formulas associated with the spectral zeta func-tions of the sequence of operators H<n> = − d

dm<n>

ddx , starting with H<0> on [0, 1],

which converges to the Sturm–Liouville operator H<∞> on [0,∞). In light of Def-inition 5.1 and Theorem 3.2, given any integer n ≥ 0, the spectral zeta ζH<n>

(s) of

H<n> on [0, α−n] is initially given by ζH<n>(s) =

∑λ∈S

∑∞p=−n(γpλ)−

s2 , for Re(s)

large enough.

Theorem 5.8 ([28]). The zeta function ζρ(s) of the renormalization map ρ isequal to the spectral zeta function of H<0>,

ζH<0>(s) =

∑λ∈S

∞∑p=0

(γpλ)−s2 ,

2Except when α = 12, in which case the spectrum of H<∞> is continuous.


or its meromorphic continuation thereof : ζρ(s) = ζH<0>(s). (An expression for

ζH<0>(s) is given by the n = 0 case of Proposition 5.9 just below ; see Equation

(5.8) of Remark 5.10.)

Proposition 5.9 ([28]). For n ≥ 0 and Re(s) sufficiently large, we have

(5.7) ζH<n>(s) =

(γn)s2

1 − γ− s2ζS(s),

where ζS(s) is the geometric zeta function of the generating set S. Namely, ζS(s) :=∑∞j=1(λj)

− s2 (for Re(s) large enough) or is given by its meromorphic continuation

thereof.

Remark 5.10. In particular, in light of Theorem 5.8, we deduce from the n = 0case of Proposition 5.9 that

(5.8) ζρ(s) = ζH<0>(s) =

1

1 − γ− s2ζS(s).

In the case of the operator H<∞>, the asymptotic behavior of the spectrumled us naturally to using the notion of delta hyperfunction. Indeed, according toTheorem 3.2, a part of the spectrum of H<∞> tends to 0 while another part tendsto ∞. If one mechanically applies Definition 5.1, one then deduces that ζH<∞>

(s)is identically equal to zero, which is clearly meaningless. Indeed, the geometricfactor in the factorization of ζH<∞>

(s) is equal to the sum of two geometric series(converging for Re(s) > 0 and for Re(s) < 0, respectively), and this sum is itselfidentically equal to zero (except for the fact that one of the two terms in the sumis not well defined, no matter which value of s ∈ C one considers). Fortunately,there is a satisfactory resolution to this apparent paradox. More specifically, wehave discovered that the geometric part of the product formula for the spectral zetafunction ζH<∞>

can be expressed in terms of the Dirac delta hyperfunction δT onthe unit circle.

Theorem 5.11 ([28]). Assume that α < 12 . The spectral zeta function ζH<∞>

is factorized as follows :

(5.9) ζH<∞>(s) = ζS(s) · δT(γ− s

2 ) = ζρ(s)(1 − γ− s2 )δT(γ− s

2 ).

Furthermore, we have shown in [28] that the zeta function associated with therenormalization map coincides with the Riemann zeta function ζ(s) for a specialvalue of α. When α = 1

2 (or equivalently, δ = 1 and so γ = 4), the self-similar mea-sure m coincides with Lebesgue measure on [0, 1] and hence, the ‘free Hamiltonian’H = H<0> coincides with the usual Dirichlet Laplacian on the unit interval [0, 1].

Recall that ζρ(s) = ζH<0>(s) (by Theorem 5.8) and that the spectrum of H<0>

(i.e., of the Dirichlet Laplacian on [0, 1]) is discrete and given by π2j2, for j = 1, 2, ....Hence, ζρ(s) = ζH<0>

(s) = π−sζ(s), where ζ(s) is the Riemann zeta function.

Theorem 5.12 ([28]). When α = 12 , the Riemann zeta function ζ is equal (up

to a trivial factor) to the zeta function ζρ associated with the renormalization mapρ on P2(C). More specifically, we have

(5.10) ζ(s) = πsζρ(s) =πs

1 − 2−sζS(s),

where ζρ is given by Definition 5.6 (or its mermorphic continuation thereof ) andthe polynomial map ρ : P2(C) → P2(C) is given by Equation (3.6) with α = 1

2 (and


hence, in light of (3.1), with δ = 1 and γ = 4):

(5.11) ρ([x, y, z]) = [x(x + y) − z2, y(x + y) − z2, z2].

This is an extension to several complex variables of A. Teplyaev’s result [55],which states that the Riemann zeta function can be described in terms of the zetafunction of a quadratic polynomial of one complex variable (as defined by Equation(5.3)).

Remark 5.13. An interesting open problem is to determine what happens ifwe consider ζH<∞>

(s) instead of ζH<0>(s), still when α = 1

2 . In that case, thespectrum of H<∞> is continuous and a suitable interpretation has to be found forζH<∞>

(s), in terms of a properly defined spectral density of states. We leave theinvestigation of this open problem for a future work. (See also Remark 5.14 justbelow.)

Remark 5.14. It would be interesting to obtain similar factorization formu-las for more general classes of self-similar fractals and fractal lattices, both in thepresent case of purely discrete spectra or in the mathematically even more chal-lenging case of purely continuous (or mixed continuous and discrete) spectra. Thelatter situation will require an appropriate use of the notion of density of states, offrequent use in condensed matter physics (see, e.g., [1], [38], [39]) and briefly dis-cussed or used mathematically in various related settings (involving either discreteor continuous spectra) in, e.g., [17], [20], [26] and especially, [43], [44], [45].

References

[1] Shlomo Alexander and Raymond Orbach, Density of states on fractals: fractons, J. PhysiqueLettres 43 (1982).

[2] Neil Bajorin, Tao Chen, Alon Dagan, Catherine Emmons, Mona Hussein, Michael Khalil,Poorak Mody, Benjamin Steinhurst, and Alexander Teplyaev, Vibration modes of 3n-gaskets and other fractals, J. Phys. A 41 (2008), no. 1, 015101, 21, DOI 10.1088/1751-8113/41/1/015101. MR2450694 (2010a:28008)

[3] Neil Bajorin, Tao Chen, Alon Dagan, Catherine Emmons, Mona Hussein, MichaelKhalil, Poorak Mody, Benjamin Steinhurst, and Alexander Teplyaev, Vibration spec-tra of finitely ramified, symmetric fractals, Fractals 16 (2008), no. 3, 243–258, DOI10.1142/S0218348X08004010. MR2451619 (2009k:47098)

[4] Martin T. Barlow, Random walks and diffusion on fractals, in: Proc. Internat. CongressMath. (Kyoto, 1990), vol. II, Springer-Verlag, Berlin and New York, 1991, pp. 1025–1035.

[5] Martin T. Barlow, Diffusions on fractals, Lectures on Probability Theory and Statistics

(Saint-Flour, 1995), Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 1–121,DOI 10.1007/BFb0092537. MR1668115 (2000a:60148)

[6] Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpinski gasket, Probab.Theory Related Fields 79 (1988), no. 4, 543–623, DOI 10.1007/BF00318785. MR966175(89g:60241)

[7] Michael V. Berry, Distribution of modes in fractal resonators, in: Structural Stability inPhysics (Proc. Internat. Symposia Appl. Catastrophe Theory and Topological Concepts inPhys., Inst. Inform. Sci., Univ. Tubingen, Tubingen, 1978), Springer Ser. Synergetics, vol. 4,Springer, Berlin, 1979, pp. 51–53, DOI 10.1007/978-3-642-67363-4 7. MR556688

[8] Erik J. Bird, Sze-Man Ngai, and Alexander Teplyaev, Fractal Laplacians on the unit inter-val, Ann. Sci. Math. Quebec 27 (2003), no. 2, 135–168 (English, with English and Frenchsummaries). MR2103098 (2006b:34192)



[10] John Erik Fornæss, Dynamics in Several Complex Variables, CBMS Regional ConferenceSeries in Mathematics, vol. 87, Published for the Conference Board of the MathematicalSciences, Washington, DC, 1996. MR1363948 (96j:32033)

[11] John Erik Fornæss and Nessim Sibony, Complex dynamics in higher dimension. I, Asterisque222 (1994), 5, 201–231. Complex Analytic Methods in Dynamical Systems (Rio de Janeiro,1992). MR1285389 (95i:32036)

[12] Uta Freiberg, Analytical properties of measure geometric Krein-Feller-operators on the

real line, Math. Nachr. 260 (2003), 34–47, DOI 10.1002/mana.200310102. MR2017701(2004j:28010)

[13] Uta Freiberg, A survey on measure geometric Laplacians on Cantor like sets, Arab. J. Sci.Eng. Sect. C Theme Issues 28 (2003), no. 1, 189–198 (English, with English and Arabicsummaries). Wavelet and Fractal Methods in Science and Engineering, Part I. MR2030736

[14] Uta Freiberg, Spectral asymptotics of generalized measure geometric Laplacians on Cantorlike sets, Forum Math. 17 (2005), no. 1, 87–104, DOI 10.1515/form.2005.17.1.87. MR2110540(2005k:28012)

[15] Uta Freiberg and Martina Zahle, Harmonic calculus on fractals—a measure geometricapproach. I, Potential Anal. 16 (2002), no. 3, 265–277, DOI 10.1023/A:1014085203265.MR1885763 (2002k:28011)

[16] Takahiko Fujita, A fractional dimension, self-similarity and a generalized diffusion operator,in: Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985), Academic Press,Boston, MA, 1987, pp. 83–90. MR933819 (89m:47042)

[17] Masatoshi Fukushima and Tadashi Shima, On a spectral analysis for the Sierpinski gasket,Potential Anal. 1 (1992), no. 1, 1–35, DOI 10.1007/BF00249784. MR1245223 (95b:31009)

[18] Sheldon Goldstein, Random walks and diffusions on fractals, (Minneapolis, Minn., 1984),IMA Vol. Math. Appl., vol. 8, Springer, New York, 1987, pp. 121–129, DOI 10.1007/978-1-4613-8734-3 8. MR894545 (88g:60245)

[19] Urs Graf, Introduction to Hyperfunctions and Their Integral Transforms: An Applied andComputational Approach, Birkhauser Verlag, Basel, 2010. MR2640737 (2012a:46074)

[20] Kumiko Hattori, Tetsuya Hattori, and Hiroshi Watanabe, Gaussian field theories on generalnetworks and the spectral dimensions, Progr. Theoret. Phys. Suppl. 92 (1987), 108–143, DOI

10.1143/PTPS.92.108. MR934668 (89k:81118)[21] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5,

713–747, DOI 10.1512/iumj.1981.30.30055. MR625600 (82h:49026)[22] Akira Kaneko, Introduction to the Theory of Hyperfunctions, Mathematics and its Applica-

tions, Kluwer, Dordrecht, 1988.[23] Jun Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math. 6 (1989),

no. 2, 259–290, DOI 10.1007/BF03167882. MR1001286 (91g:31005)[24] Jun Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335

(1993), no. 2, 721–755, DOI 10.2307/2154402. MR1076617 (93d:39008)[25] Jun Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge

University Press, Cambridge, 2001. MR1840042 (2002c:28015)[26] Jun Kigami and Michel L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians

on p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), no. 1, 93–125. MR1243717(94m:58225)

[27] Shigeo Kusuoka, A diffusion process on a fractal, in: Probabilistic Methods in MathematicalPhysics (Katata/Kyoto, 1985), Academic Press, Boston, MA, 1987, pp. 251–274. MR933827(89e:60149)

[28] Nishu Lal and Michel L. Lapidus, Hyperfunctions and spectral zeta functions of Laplacianson self-similar fractals, J. Phys. A 45 (2012), no. 36, 365205, 14 pp., DOI 10.1088/1751-8113/45/36/365205. MR2967908

[29] Michel L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partialresolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), no. 2, 465–529,

DOI 10.2307/2001638. MR994168 (91j:58163)[30] Michel L. Lapidus, Spectral and fractal geometry: from the Weyl-Berry conjecture for the

vibrations of fractal drums to the Riemann zeta-function, in: Differential Equations andMathematical Physics (Birmingham, AL, 1990), Math. Sci. Engrg., vol. 186, AcademicPress, Boston, MA, 1992, pp. 151–181, DOI 10.1016/S0076-5392(08)63379-2. MR1126694(93f:58239)


[31] Michel L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractalmedia and the Weyl-Berry conjecture, in: Ordinary and Partial Differential Equations, vol.IV (Dundee, 1992), Pitman Res. Notes Math. Ser., vol. 289, Longman Sci. Tech., Harlow,1993, pp. 126–209. MR1234502 (95g:58247)

[32] Michel L. Lapidus, Fractals and vibrations: can you hear the shape of a fractal drum?, Fractals3 (1995), no. 4, 725–736, DOI 10.1142/S0218348X95000643. Symposium in Honor of BenoitMandelbrot (Curacao, 1995). MR1410291 (97h:58167)

[33] Michel L. Lapidus and Helmut Maier, The Riemann hypothesis and inverse spectralproblems for fractal strings, J. London Math. Soc. (2) 52 (1995), no. 1, 15–34, DOI10.1112/jlms/52.1.15. MR1345711 (97b:11111)

[34] Michel L. Lapidus and Carl Pomerance, The Riemann zeta-function and the one-dimensionalWeyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3) 66 (1993), no. 1, 41–69, DOI 10.1112/plms/s3-66.1.41. MR1189091 (93k:58217)

[35] Michel L. Lapidus and Machiel van Frankenhuijsen (eds.), Fractal Geometry and Applica-tions: A Jubilee of Benoit Mandelbrot, Proc. Sympos. Pure Math., 72, Amer. Math. Soc.,Providence, R.I., 2004. (Part 1: Analysis, Number Theory and Dynamical Systems; Part 2:Multifractals, Probability and Statistical Mechanics.)

[36] Michel L. Lapidus and Machiel van Frankenhuijsen, Fractal Geometry, Complex Dimensionsand Zeta Functions: Geometry and Spectra of Fractal Strings, Springer Monographs in Math-ematics, Springer, New York, 2006 (Second Revised and Enlarged Edition, 2013). MR2245559(2007j:11001)

[37] Mitsuo Morimoto, An Introduction to Sato’s Hyperfunctions, Translations of MathematicalMonographs, vol. 129, American Mathematical Society, Providence, RI, 1993. Translated andrevised from the 1976 Japanese original by the author. MR1243638 (96e:32008)

[38] Tsuneyoshi Nakayama and Kousuke Yakubo, Fractal Concepts in Condensed Matter Physics,Springer-Verlag, Berlin, 2003.

[39] Alexander I. Olemskoi, Fractals in Condensed Matter Physics, I. Khalatnikov (ed.), Phys.Rev. vol. 18, Part I, Gordon & Breach, London, 1996.

[40] Rammal Rammal, Spectrum of harmonic excitations on fractals, J. Physique 45 (1984), no. 2,191–206. MR737523 (85d:82101)

[41] Rammal Rammal and Gerard Toulouse, Random walks on fractal structures and percolationcluster, J. Physique Lettres 44 (1983), L13–L22.

[42] Joseph J. Rotman, Advanced Modern Algebra, Graduate Studies in Mathematics, vol. 114,American Mathematical Society, Providence, RI, 2010. Second edition [of MR2043445].MR2674831

[43] Christophe Sabot, Density of states of diffusions on self-similar sets and holomorphic dynam-

ics in Pk: the example of the interval [0, 1], C. R. Acad. Sci. Paris Ser. I Math. 327 (1998),no. 4, 359–364, DOI 10.1016/S0764-4442(99)80048-2 (English, with English and French sum-maries). MR1649959 (2000c:32064)

[44] Christophe Sabot, Integrated density of states of self-similar Sturm-Liouville operators andholomorphic dynamics in higher dimension, Ann. Inst. H. Poincare Probab. Statist. 37(2001), no. 3, 275–311, DOI 10.1016/S0246-0203(00)01068-2 (English, with English andFrench summaries). MR1831985 (2002i:34161)

[45] Christophe Sabot, Spectral properties of self-similar lattices and iteration of rational maps,Mem. Soc. Math. Fr. (N.S.) 92 (2003), vi+104 (English, with English and French summaries).MR1976877 (2004d:82016)

[46] Christophe Sabot, Electrical networks, symplectic reductions, and application to the renor-malization map of self-similar lattices, in: Fractal Geometry and Applications: A Jubilee ofBenoıt Mandelbrot. Part 1, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence,RI, 2004, pp. 155–205. MR2112106 (2005m:34202)

[47] Christophe Sabot, Spectral analysis of a self-similar Sturm-Liouville operator, Indiana

Univ. Math. J. 54 (2005), no. 3, 645–668, DOI 10.1512/iumj.2005.54.2490. MR2151229(2006c:34060)

[48] Mikio Sato, Theory of hyperfunctions, Sugaku 10 (1958), 1–27. (Japanese)[49] Mikio Sato, Theory of hyperfunctions I & II, J. Fac. Sci. Univ. Tokyo, Sec. IA, 8 (1959),

139–193 & 8 (1960), 387–437.[50] Tadashi Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. In-

dust. Appl. Math. 13 (1996), no. 1, 1–23, DOI 10.1007/BF03167295. MR1377456 (97f:28028)


[51] Robert S. Strichartz, Differential Equations on Fractals: A Tutorial, Princeton UniversityPress, Princeton, NJ, 2006. MR2246975 (2007f:35003)

[52] Yoshiko Taguchi, A characterization of the space of Sato-hyperfunctions on the unit circle,Hiroshima Math. J. 17 (1987), no. 1, 41–46. MR886980 (88g:46057)

[53] Alexander Teplyaev, Spectral analysis on infinite Sierpinski gaskets, J. Funct. Anal. 159(1998), no. 2, 537–567, DOI 10.1006/jfan.1998.3297. MR1658094 (99j:35153)

[54] Alexander Teplyaev, Spectral zeta function of symmetric fractals, in: Fractal Geometry and

Stochastics III, Progr. Probab., vol. 57, Birkhauser, Basel, 2004, pp. 245–262. MR2087144(2005h:28028)


[56] Ernest B. Vinberg, A Course in Algebra (English transl. of the 2001 Russian edn.), GraduateStudies in Mathematics, vol. 56, American Mathematical Society, Providence, RI, 2003.

Department of Mathematics, Pomona College, Claremont, California 91711


Department of Mathematics, University of California, Riverside, California 92521-

0135



The Current State of Fractal Billiards

Michel L. Lapidus and Robert G. Niemeyer

In memory of Eugene Gutkin

Abstract. If D is a rational polygon, then the associated rational billiardtable is given by Ω(D). Such a billiard table is well understood. If F is aclosed fractal curve approximated by a sequence of rational polygons, thenthe corresponding fractal billiard table is denoted by Ω(F ). In this paper, we

survey many of the results from [LapNie1-3] for the Koch snowflake fractalbilliard Ω(KS) and announce new results on two other fractal billiard tables,namely, the T -fractal billiard table Ω(T ) (see [LapNie6)] and a self-similarSierpinski carpet billiard table Ω(Sa) (see [CheNie)].

We build a general framework within which to analyze what we call asequence of compatible orbits. Properties of particular sequences of compatibleorbits are discussed for each prefractal billiard Ω(KSn), Ω(Tn) and Ω(Sa,n),for n = 0, 1, 2 · · · . In each case, we are able to determine a particular limitingbehavior for an appropriately formulated sequence of compatible orbits. Sucha limit either constitutes what we call a nontrivial path of a fractal billiardtable Ω(F ) or else a periodic orbit of Ω(F ) with finite period. In our examples,F will be either KS, T or Sa. Several of the results and examples discussed

in this paper are presented for the first time.

We then close with a brief discussion of open problems and directions forfurther research in the emerging field of fractal billiards.

Contents

1. Introduction2. Rational billiards2.1. Translation surfaces and properties of the flow

2010 Mathematics Subject Classification. Primary: 28A80, 37D40, 37D50; Secondary:28A75, 37C27, 37E35, 37F40, 58J99.

Key words and phrases. Fractal billiard, polygonal billiard, rational (polygonal) billiard, lawof reflection, unfolding process, flat surface, translation surface, geodesic flow, billiard flow, it-erated function system and attractor, self-similar set, fractal, prefractal approximations, Kochsnowflake billiard, T -fractal billiard, self-similar Sierpinski carpet billiard, prefractal rational bil-liard approximations, sequence of compatible orbits, hook orbits, (eventually) constant sequencesof compatible orbits, footprints, Cantor points, smooth points, elusive points, periodic orbits,periodic vs. dense orbits.

The work of the first author was partially supported by the National Science Foundationunder the research grants DMS-0707524 and DMS-1107750, as well as by the Institut des HautesEtudes Scientifiques (IHES) in Bures-sur-Yvette, France, where he was a visiting professor whilethis paper was written. The work of R. G. Niemeyer was partially supported by the NationalScience Foundation under the MCTP grant DMS-1148801, while a postdoctoral fellow at theUniversity of New Mexico, Albuquerque.


251

252 M. L. LAPIDUS AND R. G. NIEMEYER

2.2. Unfolding a billiard orbit and equivalence of flows3. The fractals of interest3.1. Cantor sets3.2. The Koch curve and Koch snowflake3.3. The T -fractal3.4. Self-similar Sierpinski carpets4. Prefractal (rational) billiards4.1. A general structure4.2. The prefractal Koch snowflake billiard4.3. The T -fractal prefractal billiard4.4. A prefractal self-similar Sierpinski carpet billiard5. Fractal billiards5.1. A general framework for Ω(KS), Ω(T ) and Ω(Sa)5.2. The Koch snowflake fractal billiard5.3. The T -fractal billiard5.4. A self-similar Sierpinski carpet billiard6. Concluding remarksReferences

1. Introduction

This paper constitutes a survey of a collection of results from [LapNie1,LapNie2,LapNie3] as well as the announcement of new results on the T -fractalbilliard table Ω(T ) (see [LapNie6]) and a self-similar Sierpinski carpet billiardtable Ω(Sa) (see [CheNie]).

In §§2 and 3, we survey the necessary background material for understandingthe remainder of the article. More specifically, in §2, we introduce the notionof a rational polygonal billiard, a translation surface determined from a rationalpolygonal billiard and discuss the consequence of a dynamical equivalence betweenthe billiard flow and the geodesic flow.1 This dynamical equivalence allows us toexpress an orbit of a rational billiard table as a geodesic on an associated translationsurface, and vice-versa, with the added benefit of being able to determine thereflection in certain types of vertices of a rational billiard table. Furthermore, in§3, we provide additional background material from the subject of fractal geometrynecessary for understanding the construction of the Koch snowflake KS, T -fractalT ,2 and a Sierpinski carpet Sa, as well as particular orbits and nontrivial paths.

We then combine the background material presented in §§2 and 3 to analyze theprefractal billiard tables Ω(KSn), Ω(Tn) and Ω(Sa,n), for n = 0, 1, 2 · · · . We beginby providing a general language for prefractal billiards and subsequently focus ondetermining sufficient conditions for what we are calling a sequence of compatibleperiodic orbits. While §§4.2–4.4 contain specific results and specialized definitions,there is an over-arching theme that is more fully developed in §5.

1The references [VoGaSt,Gut1,MasTa,Sm,Ta1,Ta2,Vo,Zo] provide an excellent surveyof the various topics in the field of mathematical billiards, as well as specific results pertinent tothe theory of rational polygonal billiards and associated translation surfaces or flat surfaces.

2The T -fractal T was previously studied in a different context in [AcST].

FRACTAL BILLIARDS 253

In addition to providing a general language within which to analyze a fractalbilliard, we discuss in §§5.2–5.4 how one can determine well-defined orbits of Ω(KS),Ω(T ) and Ω(Sa), as well as nontrivial paths of Ω(KS) and Ω(T ) that connect twoelusive points of each respective billiard. Relying on the main result of [Du-CaTy],the second author and Joe P. Chen have shown that it is possible to determine aperiodic orbit of a self-similar Sierpinski carpet billiard Ω(Sa); additional resultsand proofs are forthcoming in [CheNie], but a synopsis is provided in §§4.4 and5.4.

Many of the results in §§4 and 5 are being announced for the first time. Specif-ically, §§4.3 and 5.3 contain new results on the prefractal T -fractal billiard Ω(Tn)and the T -fractal billiard Ω(T ) (see [LapNie6]); §§4.4 and 5.4 contain new resultsfor a prefractal Sierpinski carpet billiard Ω(Sa,n) and self-similar Sierpinski car-pet billiard Ω(Sa), where a is the single underlying scaling ratio (see [CheNie]).As these sections constitute announcements of new results on the respective pre-fractal and fractal billiards, we will provide in future papers [CheNie,LapNie4,LapNie5,LapNie6] detailed statements and proofs of the results given therein.Given the nature of the subject of fractal billiards, we will close with a discussion ofopen problems and possible directions for future work, some of which are to appearin [CheNie] and [LapNie4,LapNie5,LapNie6].

2. Rational billiards

In this section, we will survey the dynamical properties of a billiard ball as ittraverses a region in the plane bounded by a closed and connected polygon. In thelatter part of this article, we will remove the stipulation that the boundary be apolygon and focus on billiard tables having boundaries that are fractal or containingsubsets that are fractal (while still being simple, closed and connected curves in theplane).

Under ideal conditions, we know that a point mass making a perfectly elasticcollision with a C1 surface (or curve) will reflect at an angle which is equal to theangle of incidence, this being referred to as the law of reflection.

Consider a compact region Ω(D) in the plane with simple, closed and connectedboundary D. Then, Ω(D) is called a planar billiard when D is smooth enough toallow the law of reflection to hold, off of a set of measure zero (where the measure istaken to be the arc length measure on D). Though the law of reflection implicitlystates that the angles of incidence and reflection be determined with respect to thenormal to the line tangent at the basepoint, we adhere to the equivalent conventionin the field of mathematical billiards that the vector describing the position andvelocity of the billiard ball (which amounts to the position and angle, since we areassuming unit speed) be reflected in the tangent to the point of incidence.3 Thatis, employing such a law in order to determine the path on which the billiard balldeparts after impact essentially amounts to identifying certain vectors. Such anequivalence relation is denoted by ∼ and, in the context of a polygonal billiard, isdiscussed below in more detail.

3This is equivalent to reflecting the incoming vector through the normal to the tangent at thepoint of collision in the boundary. We continue with the convention established in the text, sinceit is more convenient in the context of polygonal billiards. Moreover, the fact that the equivalencerelation on the phase space is defined in terms of the convention we have adopted, necessitatesus continuing with this convention; see [Sm] for a formal discussion of the equivalence relationdefined on the phase space Ω(D)× S1.


For the remainder of the article, unless otherwise indicated, when D is a simple,closed, connected and piecewise smooth curve so as to allow the law of reflectionto hold off finitely many points, we assume D is a closed and connected polygon.In such a case, we will refer to Ω(D) as a polygonal billiard.

One may express the law of reflection in terms of equivalence classes of vectorsby identifying two particular vectors that form an equivalence class of vectors inthe unit tangent bundle corresponding to the billiard table Ω(D); see Figure 1.(See [Sm] for a detailed discussion of this equivalence relation on the unit tangentbundle Ω(D) × S1.)

Denote by S1 the unit circle, which we let represent all the possible directions(or angles) in which a billiard ball may initially move. To clearly understand howone forms equivalence classes from elements of Ω(D) × S1, we let (x, θ), (y, γ) ∈Ω(D)×S1 and say that (x, θ) ∼ (y, γ) if and only if x = y and one of the followingis true:

(1) x = y is not a vertex of the boundary D and θ = γ;(2) x = y is not a vertex of the boundary D, but x = y is a point on a

segment si of the polygon D and θ = ri(γ), where ri denotes reflection inthe segment si;

(3) If x = y is a vertex of D, then we identify (x, θ) with (y, g(γ)) for everyg in the group generated by reflections in the two adjacent sides having x(or y) as a common vertex.

For now, we shall denote by [(x, θ)] the equivalence class of (x, θ), relative to theequivalence relation ∼.

The collection of vertices of Ω(D) forms a set of zero measure (when we take ourmeasure to be the arc-length measure on D), since there are finitely many vertices.

The phase space for the billiard dynamics is given by the quotient space (Ω(D)×S1)/ ∼. In practice, one restricts his or her attention to the space (D × S1)/ ∼.The billiard flow on (D×S1)/ ∼ is determined from the continuous flow on Ω(D)×S1)/ ∼ as follows. Let x0 be an initial basepoint, θ0 be an initial direction andϕt(x

0, θ0) be a flow line corresponding to these initial conditions in the phase space(Ω(D) × S1)/ ∼. The values tj for which ϕtj (x

0, θ0) ∈ (D × S1)/ ∼ constitute the

return times (i.e., times at which ϕt(x0, θ0) returns to the section, or intersects it in

a non-tangential way). Then, the discrete map f tj (x0, θ0) constitutes the sectionmap. In terms of the configuration space, f tj (x0, θ0) constitutes the point andangle of incidence in the boundary D. Since Ω(D) is the billiard table and we areinterested in determining the collision points, it is only fitting that such a map becalled the billiard map. More succinctly, we denote f tj by f j and, in general, sucha map is called the Poincare map and the section is called the Poincare section.Furthermore, the obvious benefit of having a visual representation of f j(x0, θ0)in the configuration space is exactly why one restricts his or her attention to thesection (D × S1)/ ∼. Specifically, all one really cares about in the end, from theperspective of studying a planar billiard, are the collision points, which are clearlydetermined by the billiard map.

In order to understand how one determines the next collision point and directionof travel, we must further discuss the billiard map fD. As previously discussed,fD : (D × S1)/ ∼→ (D × S1)/ ∼, where the equivalence relation ∼ is the oneintroduced above. More precisely, if θ0 is an inward pointing vector at a basepointx0, then (x0, θ0) is the representative element of the equivalence class [(x0, θ0)].


x x

ϴ

ϴ

0

0

11

Figure 1. A billiard ball traverses the interior of a billiard andcollides with the boundary. The velocity vector is pointed outwardat the point of collision. The resulting direction of flow is found byeither reflecting the vector through the tangent or by reflecting theincidence vector through the normal and reversing the direction ofthe vector. We use the former method in this paper.

The billiard map then acts on (D×S1)/ ∼ by mapping [(xk, θk)] to [(xk+1, θk+1)],where xk and xk+1 are collinear in the direction determined by θk and where θk+1

is the reflection of angle θk through the tangent at xk+1. In general, we havefkD[(x0, θ0)] = [(xk, θk)], for every k ≥ 0.

Remark 2.1. In the sequel, we will simply refer to an element [(xk, θk)] ∈(Ω(D) × S1)/ ∼ by (xk, θk), since the vector corresponding to θk is inward point-ing at the basepoint xk. So as not to introduce unnecessary notation, when wediscuss the billiard map fFn

corresponding to the nth prefractal billiard Ω(Fn)approximating a fractal billiard Ω(F ), we will simply write fFn

as fn. When dis-cussing the discrete billiard flow on (Ω(Fn) × S1)/ ∼, the kth point in an orbit(xk, θk) ∈ (Ω(Fn) × S1)/ ∼ will instead be denoted by (xkn

n , θknn ), in order to keep

track of the space such a point belongs to (namely, with our present convention,(Ω(Fn) × S1)/ ∼). Specifically, kn refers to the number of iterates of the billiardmap fn necessary to produce the pair (xkn

n , θknn ). An initial condition of an orbit

of Ω(Fn) will always be referred to as (x0n, θ

0n).

In what follows, we are presupposing an orbit can be formed by iterating thebilliard map forward in time and backwards in time, whenever f−k

n (x0n, θ

0n) is de-

fined.An orbit making finitely many collisions in the boundary is called a closed

orbit. If, in addition, there exists m ∈ Z such that fmD (x0, θ0) = (x0, θ0), then

the resulting orbit is called periodic; the smallest positive integer m such thatfmD (x0, θ0) = (x0, θ0) is called the period of the periodic orbit. In the event that a

basepoint xj of f jD(x0, θ0) is a corner of Ω(D) (that is, a vertex of the polygonal

boundary D) and reflection cannot be determined in a well-defined manner, thenthe resulting orbit is said to be singular. In addition, if there exists a positiveinteger k such that the basepoint x−k of f−k

D (x0, θ0) is a corner of Ω(D) (here,

f−kD denotes the kth inverse iterate of fD), then the resulting orbit is closed and

the path traced out by the billiard ball connecting xj and x−k is called a saddleconnection. Finally, we note that a periodic orbit with period m is a closed orbitfor which reflection is well defined at each basepoint xi of f i

D(x0, θ0), 0 ≤ i ≤ mand fm

D (x0, θ0) = (x0, θ0).We say that an orbit O(x0, θ0) is dense in a rational billiard table Ω(D) if

the path traversed (forward and backward in time) by the billiard ball in Ω(D)


is dense in Ω(D). That is, the closure of the set of points comprising the pathtraversed by the billiard ball is exactly Ω(D). Likewise, the points of incidence(i.e., the footprint) of a dense orbit will be dense in the boundary D, as explainedin Remark 2.2.

Remark 2.2. Consider a rational polygonal billiard Ω(D). The associatedtranslation surface S(D) can be constructed as described in §2.1. As we will showin §2.2, the geodesic flow on a translation surface is dynamically equivalent tothe billiard flow. A dense orbit will have an initial direction preventing the pathfrom being parallel to any side of Ω(D) (except, possibly, for finitely many initialdirections, and hence, for a measure-zero set). The corresponding path on theassociated translation surface4 must also be dense in the surface. Since the path onthe surface is arbitrarily close to every side appropriately identified with anotherside of a copy of Ω(D) and not parallel to any side, the path will be transversalwith respect to each side. Thus, the collection of basepoints of a dense orbit mustbe dense in D.

Definition 2.3 (Footprint of an orbit). Let OD(x0, θ0) be an orbit of a billiardΩ(D) with an initial condition (x0, θ0) ∈ D×S1. Then the trace of an orbit on theboundary D,

(1) OD(x0, θ0) ∩D,

is called the footprint of the orbit OD(x0, θ0) and is denoted by FD(x0, θ0). Whenwe are only interested in a prefractal billiard Ω(Fn), we denote the footprint of anorbit by Fn(x0

n, θ0n).

For the remainder of the article, when discussing polygonal billiards, we will fo-cus our attention on what are called rational polygonal billiards, or, more succinctly,rational billiards.

Definition 2.4 (Rational polygon and rational billiard). If D is a nontrivialconnected polygon such that for each interior angle θj of D there are relativelyprime integers pj ≥ 1 and qj ≥ 1 such that θj =

pj

qjπ, then we call D a rational

polygon and Ω(D) a rational billiard.

2.1. Translation surfaces and properties of the flow. In this subsection,we will discuss what constitutes a translation surface and how to construct a trans-lation surface from a rational billiard. Then, in §2.2, we will see how to relate thecontinuous billiard flow on (Ω(D)×S1)/ ∼ with the geodesic flow on the associatedtranslation surface.

Definition 2.5 (Translation structure and translation surface). Let M be acompact, connected, orientable surface. A translation structure on M is an atlasω, consisting of charts of the form (Uα, ϕα)α∈A , where Uα is a domain (i.e., aconnected open set) in M and ϕα is a homeomorphism from Uα to a domain in R2,such that the following conditions hold:

(1) The collection {Uα}α∈A covers the whole surface M except for finitelymany points z1, z2, ..., zk, called singular points ;

(2) all coordinate changing functions are translations in R2;(3) the atlas ω is maximal with respect to properties (1) and (2);

4See §2.1 for an explanation of what constitutes a translation surface.


Figure 2. The equilateral triangle billiard Ω(Δ) can be actedon by a particular group of symmetries to produce a translationsurface that is topologically equivalent to the flat torus. In thisfigure, we see that opposite and parallel sides are identified in sucha way that the orientation is preserved. This allows us to examinethe geodesic flow on the surface. We will see in §2.2 that thegeodesic flow on the translation surface is dynamically equivalentto the continuous billiard flow.

(4) for each singular point zj , there is a positive integer mj , a punctured

neighborhood Uj of zj not containing other singular points, and a map ψj

from this neighborhood to a punctured neighborhood Vj of a point in R2

that is a shift in the local coordinates from ω, and is such that each pointin Vj has exactly mj preimages under ψj .

We say that a connected, compact surface equipped with a translation structure isa translation surface.

Remark 2.6. Note that in the literature on billiards and dynamical systems,the terminology and definitions pertaining to this topic are not completely uniform;see, for example, [VoGaSt,Gut1,GutJu1,GutJu2,HuSc,Mas,MasTa,Ve1,Ve2,Vo,Zo]. (We note that in [MasTa] and [Zo], ‘translation surfaces’ are referredto as ‘flat surfaces’.) We have adopted the above definition for clarity and thereader’s convenience.

We now discuss how to construct a translation surface from a rational billiard.Consider a rational polygonal billiard Ω(D) with k sides and interior angles

pj

qjπ at

each vertex zj , for 1 ≤ j ≤ k, where the positive integers pj and qj are relativelyprime. The linear parts of the planar symmetries generated by reflection in thesides of the polygonal billiard Ω(D) generate a dihedral group DN , where N :=lcm{qj}kj=1 (the least common multiple of the qj ’s). Next, we consider Ω(D)×DN

(equipped with the product topology). We want to glue ‘sides’ of Ω(D) × DN

together and construct a natural atlas on the resulting surface M so that M becomesa translation surface.

As a result of the identification, the points of M that correspond to the verticesof Ω(D) constitute (removable or nonremovable) conic singularities of the surface.Heuristically, Ω(D) × DN can be represented as {rjΩ(D)}2Nj=1, in which case it iseasy to see what sides are made equivalent under the action of ∼. That is, ∼identifies opposite and parallel sides in a manner which preserves the orientation.See Example 2.7 and Figure 2 for an example of a translation surface constructedfrom the equilateral triangle billiard Ω(Δ).


Figure 3. Partially unfolding an orbit of the square billiard Ω(Q).The ‘R’ is shown so as to provide the reader with a frame of refer-ence.

R R

R R

R

RR

R R

R R

Figure 4. Unfolding an orbit of the square billiard Ω(Q).

Example 2.7. Consider the equilateral triangle Δ. The corresponding billiardis denoted by Ω(Δ). The interior angles are {π

3 ,π3 ,

π3 }. Hence, the group acting on

Ω(Δ) to produce the translation surface is the dihedral group D3. The resultingtranslation surface is topologically equivalent to the flat torus. We will make useof this fact in the sequel.

2.2. Unfolding a billiard orbit and equivalence of flows. Consider arational polygonal billiard Ω(D) and an orbit O(x0, θ0). Reflecting the billiardΩ(D) and the orbit in the side of the billiard table containing the basepoint x1 ofthe orbit (or an element of the footprint of the orbit) partially unfolds the orbitO(x0, θ0); see Figure 3 for the case of the square billiard Ω(Q). Continuing thisprocess until the orbit is a straight line produces as many copies of the billiard tableas there are elements of the footprint; see Figure 4. That is, if the period of anorbit O(x0, θ0) is some positive integer p, then the number of copies of the billiardtable in the unfolding is also p. We refer to such a straight line as the unfolding ofthe billiard orbit.

Given that a rational billiard Ω(D) can be acted on by a dihedral group DN toproduce a translation surface in a way that is similar to unfolding the billiard table,we can quickly see how the billiard flow is dynamically equivalent to the geodesicflow; see Figure 5 and the corresponding caption.

One may modify the notion of “reflecting” so as to determine orbits of billiardtables tiled by a rational polygon D. As an example, we consider the unit-squarebilliard table. An appropriately scaled copy of the unit-square billiard table can betiled by the unit-square billiard table by making successive reflections in the sides ofthe unit square. One may then unfold an orbit of the unit-square billiard table intoa larger square billiard table. When the unfolded orbit of the original unit-squarebilliard intersects the boundary of the appropriately scaled (and larger) square,then one continues unfolding the billiard orbit in the direction determined by the


Figure 5. Rearranging the unfolded copies of the unit square fromFigure 4 and correctly identifying sides so as to recover the flattorus, we see that the unfolded orbit corresponds to a closed geo-desic of the translation surface.

R RR

R RR

R RR

Figure 6. Unfolding the orbit of the unit-square billiard in a(larger) scaled copy of the unit-square billiard. This constitutes anexample of a reflected-unfolding. The edges of the original unit-square billiard table and the segments comprising the orbit havebeen thickened to provide the reader with a frame of reference.

law of reflection (that is, assuming the unfolded orbit is long enough to reach a sideof the larger square). We will refer to such an unfolding as a reflected-unfolding.

We may continue this process in order to form an orbit of a larger scaled squarebilliard table. Suppose that an orbit O(x0, θ0) has period p. The footprint of the

orbit is then FB(x0, θ0) = {f iB(x0, θ0)}p−1

i=0 . If s is a positive integer (i.e., s ∈ N),

then the footprint F sB(x0, θ0) := {f i

B(x0, θ0)}s(p−1)i=0 of an orbit constitutes the

footprint of an orbit that traverses the same path s-many times. For sufficientlylarge s ∈ N, an orbit that traverses the same path as an orbit O(x0, θ0) s-many timescan be reflected-unfolded in an appropriately scaled square billiard table to form anorbit of the larger billiard table; see Figure 6. Such a tool is useful in understandingthe relationship between the billiard flow on a rational polygonal billiard Ω(D) anda billiard table tiled by D, and will be particularly useful in understanding thenature of particular orbits of a self-similar Sierpinski carpet billiard in §5.4.

As one may expect, if D is a rational polygon that tiles a billiard table Ω(R),then an orbit of Ω(R) may be folded up to form an orbit of Ω(D). This is done bymaking successive reflections in D, the result being an orbit of Ω(D); see Figure 7for the case of a square billiard table.

3. The fractals of interest

We are primarily interested in fractals with boundaries either partially or com-pletely comprised of self-similar sets and fractals that are self-similar. So as to makethe material discussed in §4–5 more accessible, we provide a few basic definitionsfrom the subject of fractal geometry.


...

Figure 7. Illustrated in this figure is the process of folding upan orbit of a square billiard table, as discussed at the end of §2.2.In the first image, we see an orbit of unit-square billiard table.Partitioning the unit square into nine equally sized squares, we seethat we can fold up the orbit by making successive reflections inthe sides of the squares comprising the partition. Using sufficientlymany reflections results in an orbit of one of the squares of thepartition.

Definition 3.1. Let (X, d) be a metric space and φ : X → X.

(i) (Contraction). If there exists 0 < c < 1 such that

d(φ(x), φ(y)) ≤ cd(x, y)

for every x, y ∈ X, then φ is called a contraction (or contraction mapping).(ii) (Similarity contraction). If there exists 0 < c < 1 such that

d(φ(x), φ(y)) = cd(x, y),

for every x, y ∈ X, then φ is called a similarity contraction. This uniquevalue c ∈ (0, 1) is called the scaling ratio of φ.

Definition 3.2. Let (X, d) be a complete metric space.

(i) (Iterated function system and attractor). Let {φi}ki=1 be a family of con-tractions defined on X. Then {φi}ki=1 is called an iterated function system(IFS).

An iterated function system is so named because the map Φ : K → K,

given by Φ(·) :=⋃k

i=1 φi(·) and defined on the space K of nonemptycompact subsets of X, can be composed with itself. Indeed, for eachm ∈ N, we have

Φm(·) =

k⋃i1=1

...

k⋃im=1

φi1 ◦ · · · ◦ φim(·).(2)

Furthermore, there exists a unique nonempty compact set F ⊂ X (i.e.,F ∈ K), called the attractor of the IFS, such that

F = Φ(F ) :=

k⋃i=1

φi(F ).(3)

(ii) (Self-similar system and self-similar set). In the special case where eachφi is a contraction similarity, for i = 1, ..., k, then the IFS {φi}ki=1 is saidto be a self-similar system and its attractor F is called a self-similar set(or a self-similar subset of X).


If X is complete, then so is K (equipped with the Hausdorff metric5) andhence, since it can be shown that Φ : K → K is a contraction, it follows from thecontraction mapping theorem that Φ has a unique fixed point (thereby justifyingthe definition of the attractor F above) and that for any E ∈ K, Φm(E) → F , asm → ∞ (where, as in Equation (2), Φm is the mth iterate of Φ). (See [Hut].)

We state the next property in the special case which will be of interest to us,namely, that of an IFS in a Euclidean space.

Theorem 3.3 ([Hut]; see also [Fa, Thm. 9.1]). Consider an iterated functionsystem given by contractions {φi}ki=1, each defined on a compact set D ⊆ Rn, suchthat φi(D) ⊆ D for each i ≤ k, and with attractor F . Then F ⊆ D and in fact,

F =∞⋂

m=0

Φm(E)(4)

for every set E ∈ K such that φi(E) ⊆ E for all i ≤ k. Here, the transformationΦ : K → K is given as in part (i) of Definition 3.2.

Notation 3.4. Suppose F is a fractal set. Then, the nth prefractal approx-imation of F is denoted by Fn. In the case of a self-similar fractal F , the nthprefractal approximation of F is usually defined by

⋂nm=0 Φm(E), where E ∈ K.

Not every fractal is self-similar or embedded in Euclidean space. However,such sets represent an important collection of examples of fractal sets. In the nextsubsection, we will discuss the fractal subsets (self-similar or not) of R or of R2 ofdirect interest to us in this paper.

3.1. Cantor sets. A Cantor set is a set with very rich and counter-intuitiveproperties; topologically, it is a compact and totally disconnected (i.e., perfect)space. In order to illustrate some of the properties that make a Cantor set sointeresting, we refer to the canonical example of a Cantor set: the ternary Cantorset. We focus on three methods for constructing the ternary Cantor set: 1) bytremas, 2) as the unique fixed point attractor of an iterated function system, and3) in terms of an alphabet.

Before we discuss the ternary Cantor set, we mention that this set was first dis-covered by Henry J. S. Smith in 1875. Later, in 1881, Vito Volterra independentlyrediscovered the ternary Cantor set. Smith’s and Volterra’s records being obscuredover the years for one reason or another, it was the German mathematician GeorgCantor whom, in 1883, history credits with the discovery of a bounded, totally dis-connected, perfect and uncountable set with measure zero, that is now commonlyreferred to as “the Cantor set”.

We now proceed to construct the ternary Cantor set, hereafter denoted by C ,by the method known as construction by tremas, which is Latin for ‘cuts’. Beginwith the unit interval I and remove the middle open third ( 13 ,

23 ) from I, leaving

the two closed intervals [0, 13 ] and [ 23 , 1]. Next, remove the middle open ninth from

each closed subinterval. What remains are the closed intervals [0, 19 ], [ 29 ,

13 ], [ 23 ,

79 ],

[ 89 , 1]. Continuing this process ad infinitum, we construct the ternary Cantor set;see Figure 8.

5See [Ba] for details on the Hausdorff metric.


Figure 8. The ternary Cantor set.

One may also construct C by utilizing an appropriately defined iterated func-tion system. Consider the following contraction maps defined on the real line R:

φ1(x) =1

3x, φ2(x) =

1

3x +

2

3.(5)

Then, limn→∞ Φn(I) = C , where Φ is given as in part (i) of Definition 3.2. More-over, since {φi}2i=1 is a family of similarity contractions and C = Φ(C ), we havethat C is a self-similar set.

A third—and equivalent—construction of the ternary Cantor set can be givenin terms of the symbols l, c, and r. Recall that the elements of R can be expressedin terms of a base-3 number system. We focus our attention on elements of the unitinterval I. So-called ternary numbers6 in I have two equivalent expansions: onethat is finite and one that is infinite. For example, 1

3 can be written in base-3 as 0.1

or, equivalently, as 0.02 (where the overbar indicates that the digit 2 is repeatedinfinitely often).

We next discuss a similar addressing system that has the benefit of preventingternary numbers from having a finite representation. The characters l, c and r areto remind the reader of choosing left, center and right. We identify an element ofthe unit interval I by an infinite address that indicates where in I the element islocated. Motivated by the construction of C by tremas, one can identify any point ofI by an infinite address consisting of the characters l, c and r. While elements of Ccan be represented by infinite addresses consisting of c’s, we make the stipulationthat no element of C will be represented by an infinite address containing c’s.7

Moreover, this method of representing elements of I (or C ) provides every elementwith an infinite representation and never a finite representation.

Example 3.5. The values 14 , 1

3 and 12 have the ternary representations lr, lr

and c, respectively.8 While 13 has a finite ternary expansion given by 0.2 = 0.01,

it does not have a finite ternary representation. It should be noted that elementslike 1

4 and 12 will play an important role in our analysis of the Koch snowflake

fractal billiard. The occurrence of infinitely many c’s or infinitely many l’s and r’sis critical to developing some of the theory regarding the Koch snowflake fractalbilliard.

So that some of the results concerning the Koch snowflake fractal billiard canbe more succinctly expressed, we introduce a notation used for describing a value’stype of ternary representation.

6An element x ∈ I is a ternary number if x = p3y

, 0 ≤ p ≤ 3y , p, y ∈ N.7In other words, we do not allow an element of C to be approximated by a sequence {zi}∞i=1

of elements of C c, where C c is the complement of the ternary Cantor set in I.8Equivalently, 1

3has a representation given by cl. Although, we will not consider this as a

representation for 13on account of 1

3∈ C .


Notation 3.6 (The type of ternary representation). The type of ternary rep-resentation can be defined as follows. If x ∈ I, then the first coordinate of [·, ·]describes the characters that occur infinitely often and the second coordinate of[·, ·] describes the characters that occur finitely often. If we want to discuss manydifferent types of ternary representations, then we use ‘or’. That is, the notation[·, ·] ∨ [·, ·] ∨ ... ∨ [·, ·] is to be read as [·, ·] or [·, ·] or ... or [·, ·]. If the collection ofcharacters occurring finitely often is empty, then we denote the corresponding typeof ternary representation by [·, ∅].

Example 3.7. The value 12 has a ternary representation of c. Hence, 1

2 has

a type of ternary representation given by [c, ∅]. Moreover, the value 712 has a

ternary representation given by crl, which means that 712 has a type of ternary

representation given by [lr, c].

We note that “the” type of representation of a point x ∈ I is not unique, ingeneral. For instance, the value 1

3 has a ternary representation of type [r, l] or [l, c].A thorough understanding of the ternary Cantor set is not only important for

understanding many of the results on the Koch snowflake prefractal and fractalbilliard. In general, Cantor sets will be ever-present and instrumental in our anal-ysis of other fractal billiard tables. In each example of a fractal billiard, we willclearly indicate where and how a particular Cantor set is important in analyzing aparticular fractal billiard table.

3.2. The Koch curve and Koch snowflake. The Koch curve KC is con-structed as shown in Figure 9 and is the unique fixed point attractor of the followingiterated function system on the Euclidean plane (here, i =

√−1):

φ1(x) =1

3x, φ2(x) =

1

3ei

π3 x + (

1

3, 0),(6)

φ3(x) =1

3e−iπ

3 x + (2

3,

√3

6), φ4(x) =

1

3x + (

2

3, 0).

Since each contraction map in the iterated function system is a similarity transfor-mation (i.e., {φj}4j=1 is a self-similar system) and KC = Φ(KC), we have that KCis a self-similar set; see part (ii) of Definition 3.2. There are additional propertiesof the Koch curve that are reminiscent of the Cantor set; this is more than just acoincidence and is discussed in more detail below.

If we allow the iterated function system to act on the triangle R = {(x, y)|0 ≤x ≤ 1

2 , 0 ≤ y ≤√36 x} ∪ {(x, y)| 12 ≤ x ≤ 1, 0 ≤ y ≤ −

√36 x +

√36 }, as shown in

Figure 9, sequential iterates of the iterated function system very quickly producea prefractal that is visually indiscernible from the true limiting set. But there is amore common construction that allows us to visualize the curve KC more readily,this being depicted in Figure 10. The technical caveat which we are brushingunder the carpet is that each polygonal approximation shown in Figure 10 does notcontain the Koch curve KC, while each approximation in the sequence shown inFigure 9 does contain KC.9

Notation 3.8. For each integer n ≥ 0, we denote by KCn the nth (inner)polygonal approximation of the Koch curve KC.

9Recall from Definition 3.2 and Theorem 3.3 that for a set F to be the unique fixed pointattractor of an IFS, each Fn must be such that F ⊆ Φ(Fn), so that Φ(Fn) = Φn+1(F0).


Figure 9. The construction of the Koch curve KC. Here, the self-similar set KC is viewed as a limit of the prefractal approximations{Φm(R)}∞m=0, where R is the initial triangle and the map Φ isdefined in terms of the IFS given by Equation (6), as in Definition3.2. (See Theorem 3.3 and the text preceding it.)

Figure 10. One typically sees this construction of the Koch curveKC when learning about fractal sets. Beginning with the unitinterval I, one removes the middle third and replaces it with thetwo other sides of an equilateral triangle, as shown. One thenrepeats this process infinitely often for every remaining interval;the resulting limiting set is KC. Such a sequence {KCn}∞n=0 ofapproximations converges to KC, because it is a subsequence ofthe convergent sequence of prefractal approximations {Φm(R)}∞m=0

shown in Figure 9. (Here, we are using the notion of convergencein the sense of the Hausdorff metric.)

Intuitively, one expects the Koch curve to have finite length, since it is the limitof a sequence of polygonal approximations. On the contrary, the Koch curve KC hasinfinite length, which can be seen by the following calculation given in terms of thenth prefractal KCn, where KCn is one of the polygonal approximations indicatedin Figure 10:

length of KCn =

(4

3

)n

.(7)

Then, limn→∞(43

)n= ∞.

The Koch snowflake KS is a fractal comprised of three abutting copies of theself-similar Koch curve; see Figure 11.

Notation 3.9. For each integer n ≥ 0, we denote by KSn the nth (inner)polygonal approximation of the Koch snowflake KS.

As a closed (simple) curve, the Koch snowflake KS bounds a region of theplane; furthermore, the area of this region can be calculated as follows:

area bounded by KSn = 1 +

n∑i=0

(2

3

)i

.(8)


1 2

3

Figure 11. The Koch snowflake is comprised of three Kochcurves. We have encapsulated each Koch curve in order to high-light how KS is the union of three abutting copies of KC.

Figure 12. The construction of the T -fractal T .

Then, as n increases, the right-hand side of (8) tends to a finite value. The area

bounded by the Koch snowflake is thus given by limn→∞ 1 +∑n

i=0

(23

)i= 3, as-

suming the sides of KS0 have length one.As we noted at the end of §3.1, Cantor sets are ever present in the context of

self-similarity. In the case of the Koch snowflake, KS ∩ KSn is the union of 3 · 4nself-similar ternary Cantor sets, each spanning a distance of 1

3n . Such a fact will beimportant in determining certain sequences of what we will call compatible orbits(see Definitions 4.2–4.5) and certain families of well-defined orbits of Ω(KS).

3.3. The T -fractal. The T -fractal T , discussed in [AcST] in a different con-text, is not a self-similar set. However, T contains, as a proper subset, a topologicalCantor set and is constructed in a way that is reminiscent of an iterated functionsystem acting on a compact set so as to produce a self-similar set.10 As shown inFigure 12, one constructs the T -fractal by appending scaled copies of the initial Tshape T0 to each successive approximation. Specifically, Tn+1 is constructed fromTn by appropriately appending 2n+1 copies of 1

2n+1 T0 to Tn.11

The overall height of T can be calculated and the total area bounded by Tcan be shown to be finite, as shown in the following calculations (we assume herethat the base of T0 is two units in length):

10Recall from Definition 3.2 and Theorem 3.3 that each prefractal approximation Fn mustcontain the unique fixed point attractor F .

11See [AcST, §2.1] and [LapNie6] for a more precise description of the definition of T .


Figure 13. Ω(T0) can be tiled by the unit square Q.

height of Tn = 3 +3

2+

3

4+ ... +

3

2n= 3

n∑i=0

1

2i.(9)

Then, limn→∞ 3∑n

i=0121 = 6, which is the height of T . Furthermore, the area

bounded by Tn is calculated as follows. There are eight squares, each with side-length one, comprising T0; see Figure 13. Hence, the area of T0 is eight square-units. Therefore,

area bounded by Tn = 8 + 2 · 8

4+... + 2n · 8

4n= 8

n∑i=0

1

2i.(10)

Then, limn→∞ 8∑n

i=012i = 16, which is the total area bounded by T .

There is a natural fractal subset of T , but, for each n ≥ 0, no point of Tn

is in this fractal subset, which is unlike what we have seen in the case of theKoch snowflake fractal KS. In fact, the fractal subset in question is given by{(x, 6)|x ∈ R}∩T . We note that this fractal subset is not self-similar, though it isa (topological) Cantor set.

3.4. Self-similar Sierpinski carpets. A Sierpinski carpet can be constructedby systematically removing particular open subsquares from the unit square Q ={(x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. Depending on how one chooses the sizes of the opensubsquares to be removed, one can either construct a self-similar Sierpinski carpetor a non-self-similar Sierpinski carpet, these being defined below. This methodof construction is called construction by tremas and is described in the caption ofFigure 14, using the standard “1/3-Sierpinski carpet” as an example. Such a con-struction process should be very familiar, since “removing middle thirds” is exactlywhat we did to construct the ternary Cantor set in §3.1.

As referred to in the caption of Figure 14, one may also construct the 1/3-Sierpinski carpet by applying an appropriately defined iterated function system tothe unit square Q. Consider the following iterated function system, which is aself-similar system.

φ1(x) =1

3x, φ2(x) =

1

3x +

(0,

1

3

),(11)

φ3(x) =1

3x +

(0,

2

3

), φ4(x) =

1

3x +

(1

3, 0

),

φ5(x) =1

3x +

(1

3,2

3

), φ6(x) =

1

3x +

(2

3, 0

),

φ7(x) =1

3x +

(2

3,1

3

), φ8(x) =

1

3x +

(2

3,2

3

).


Figure 14. The 1/3-Sierpinski carpet is a self-similar carpet con-structed in one of two ways: 1) by tremas and 2) an iterated func-tion system (in fact, a self-similar system). We describe here theconstruction of the 1/3-Sierpinski carpet by tremas, the latter be-ing further discussed in the main text. Beginning with the unitsquare, one then removes the middle open square with side-length13 . From each remaining subsquare of side-length 1

3 , one then re-

moves the middle open square of side-length 19 . One continues this

procedure of removing subsquares of remaining squares until thereis no area left. As one would expect, each step of the constructionprocess can be emulated by applying the correct iterated functionsystem, which is given in Equation (11).

Then, denoting the 1/3-Sierpinski carpet by S3, we have that limn→∞ Φn(Q) = S3.Since each contraction in the iterated function system is a similarity contractionand S3 = Φ(S3), it follows that S3 is a self-similar set.

We discuss here the relevant results and material from [Du-CaTy]. For ourpurposes, the first level approximation of a Sierpinski carpet Sa will always be theunit square Q and denoted by S0. Since every (self-similar and non-self-similar)Sierpinski carpet has the same zeroth level approximation and zero is never a scalingratio, such notation will never cause any confusion.

What follows is a general description on how to construct a Sierpinski carpetby removing appropriately sized middle open squares. Consider the unit squareQ = S0. Let a0 = 2k0 + 1 for some k0 ∈ N. Partition S0 into a0 squares ofside-length a−1

0 . Next, remove the middle open subsquare. Let a1 = 2k1 + 1 forsome k1 ∈ N. Each subsquare may then be partitioned into a21 many squares withside-length (a0 · a1)−1. We then remove each middle open subsquare of side-length(a0 · a1)−1; see Figure 14. Continuing this process, let an−1 = 2kn−1 + 1 wherekn−1 ∈ N and let an = 2kn + 1 for some kn ∈ N . Then we partition a subsquare ofside-length (a0 · a1 · · · an−1)

−1 into a2n many squares. We then remove the middleopen square from each subsquare in the partition. Continuing in this manner adinfinitum, one constructs a Sierpinski carpet denoted by Sa, where a = {a−1

i }∞i=0.

Definition 3.10 (A self-similar Sierpinski carpet). If a = {a−1i }∞i=0, with ai =

2ki + 1 and ki ∈ N, is a periodic sequence of rational values, then the Sierpinskicarpet Sa is called a self-similar Sierpinski carpet.

We have described the construction of a self-similar Sierpinski carpet Sa interms of the removal of particular open squares. As the name would suggest, thereexists a suitably defined iterated function system {φi}ki=1 such that Sa = Φ(Sa).Viewing Sa as the unique fixed point attractor of an appropriately defined iteratedfunction system will be useful in stating some of the results in the subsequent


sections. More precisely, Sa is viewed as the self-similar set associated with a self-similar system, as in part (ii) of Definition 3.2.

While we do not discuss any results concerning non-self-similar Sierpinski carpetbilliards in this paper, we provide the definition for completeness.

Definition 3.11 (A non-self-similar Sierpinski carpet). If a = {a−1i }∞i=0, with

ai = 2ki + 1 and ki ∈ N, is an aperiodic sequence of rational values, then theSierpinski carpet Sa is called a non-self-similar Sierpinski carpet.

Definition 3.12 (A cell of Sa,n). Let a0 = 2k0+1, k0 ∈ N. Consider a partition

of the unit square Q = S0 into a20 many squares of side-length a−10 . A subsquare

of the partition is called a cell of S0 and is denoted by C0,a0. Furthermore, let Sa

be a Sierpinski carpet. Consider a partition of the prefractal approximation Sa,n

into subsquares with side-length (a0 · a1 · · · an)−1. A subsquare of the partitionof Sa,n is called a cell of Sa,n and is denoted by Cn,a0a1···an

and has side-length(a0 · a1 · · · an)−1.

Definition 3.13 (Peripheral square). In accordance with the conventionadopted in [Du-CaTy], the boundary of an open square removed in the construc-tion of Sa is called a peripheral square of Sa. Furthermore, by convention, the unitsquare Q = S0 is not a peripheral square.

Definition 3.14 (Nontrivial line segment of Sa). A nontrivial line segment ofSa is a (straight-line) segment of the plane contained in Sa and which has nonzerolength.

Unless otherwise indicated, in what follows, we assume that Sa is a self-similarSierpinski carpet with a single scaling ratio a; that is, a = {a−1}∞i=0, where a =2k + 1 for some fixed k ∈ N. In addition, when a = {a−1}∞i=0, Sa is denoted by Sa.

We next state the following theorem, due to Durand-Cartagena and Tyson in[Du-CaTy] and which will be very useful to us in this context (see §4.4 and §5.4).

Theorem 3.15 ([Du-CaTy, Thm. 4.1]). Let Sa be a self-similar Sierpinskicarpet. Then the set of slopes Slope(Sa) of nontrivial line segments of Sa is theunion of the following two sets :

A =

{p

q: p + q ≤ a, 0 ≤ p < q ≤ a− 1, p, q ∈ N ∪ {0}, p + q is odd

},(12)

B =

{p

q: p + q ≤ a− 1, 0 ≤ p ≤ q ≤ a− 2, p, q ∈ N, p, q are odd

}.(13)

Moreover, if α ∈ A, then each nontrivial line segment in Sa with slope α touchesvertices of peripheral squares, while if α ∈ B, then each nontrivial line segment inSa with slope α is disjoint from all peripheral squares.

Notation 3.16. Let a, b be odd positive integers such that 3 ≤ b ≤ a and letSlope(Sa) and Slope(Sb) be the set of slopes of nontrivial line segments of Sa andSb, respectively. We denote by Aa (resp., Ab) the subset A ⊆ Slope(Sa) (resp.,A ⊆ Slope(Sb)) given in Equation (12) of Theorem 3.15. Similarly, we denote byBa (resp., Bb) the subset B ⊆ Slope(Sa) (resp., Slope(Sb)) given in Equation (13)of Theorem 3.15.12

12In the case of Ab (resp., Bb), a should of course be replaced by b in Equation (12) (resp.,Equation (13)).


If Sa and Sb are self-similar Sierpinski carpets with b ≤ a, then it is clear thatSlope(Sb) ⊆ Slope(Sa). Moreover, in this case, we also have that Ab ⊆ Aa andBb ⊆ Ba.

Remark 3.17. We note that if α is the slope of a nontrivial line segment inSa, then so is −α, 1

α and − 1α by symmetry of the carpet. However, we restrict our

attention in this paper to the slopes described in the above result of [Du-CaTy].

4. Prefractal (rational) billiards

In the previous sections, we surveyed basic facts and results from mathematicalbilliards and fractal geometry, with most of our attention being focused on thesubject of rational billiards and sets exhibiting self-similarity. We also discussed theimportance of examining the dynamically equivalent geodesic flow on an associatedtranslation surface. In this section, we will examine examples from particular classesof prefractal (rational) billiards. We are interested in tables that can be tiled by asingle polygon which can also tile the (Euclidean) plane. The main examples wewill discuss are the Koch snowflake prefractal billiard table, the T -fractal prefractalbilliard table and a self-similar Sierpinski carpet prefractal billiard table. Eachexample of a prefractal billiard table constitutes a rational billiard table, but isan element of a sequence of rational billiard tables approximating a fractal billiardtable with radically different qualities when compared to the others. That is, theKoch snowflake has an everywhere nondifferentiable boundary; the T -fractal billiardtable is certainly a fractal billiard table, since its boundary T contains a fractalset, but the portion of the boundary that is nondifferentiable has Lebesgue measurezero; a Sierpinski carpet billiard table can possibly have no area, yet yield billiardorbits of finite length.

4.1. A general structure. We restrict our attention to billiard tables withfractal boundary F , where F can be approximated by a suitably chosen sequence ofrational polygons {Fn}∞n=0. More specifically, we are interested in a fractal billiardtable Ω(F ) with the property that, for every n ≥ 0, Ω(Fn) can be tiled by a singlepolygon Dn, where Dn = cnD0. Here, 0 < cn ≤ 1 is a suitably chosen scaling ratioand D0 is a polygon that tiles both the (Euclidean) plane as well as the rationalbilliard Ω(F0).

13

The focus in this subsection is on developing a general framework for discussingbilliards on prefractal approximations. If Ω(Fn) and Ω(Fn+1) are two prefractalbilliard tables approximating a given fractal billiard table Ω(F ), then we wantto have a systematic way of determining how and if two orbits On(x0

n, θ0n) and

On+1(x0n+1, θ

0n+1) of Ω(Fn) and Ω(Fn+1) are related.

Notation 4.1. We will primarily measure angles relative to a fixed coordinatesystem, with the origin being fixed at a corner of a prefractal approximation F0.However, we will sometimes measure an angle relative to a side of Fn on which abilliard ball lies. In such situations, we will write the angle as �(θ) in order toindicate that the inward pointing direction is θ, measured relative to the side onwhich the vector is based.

13In the case of certain prefractal approximations, Ω(F0) is exactly D0. In general, however,D0 does not always equal Ω(F0), but certainly tiles Ω(F0). An example of this situation is Ω(T0);see §4.3. Such a billiard is tiled by the unit square, which is the associated polygon D0.


Figure 15. In the first image, we have the orbit O0(c,π3 ) of

Ω(KS0). In the second image, we see that the orbit O0(c,π3 ), when

embedded in Ω(KS1), is not an orbit of Ω(KS1). In the third image,the given orbit of Ω(KS1) intersects sides of Ω(KS1) and appearsto be related to O0(c,

π3 ) in some way.

To motivate our general discussion, consider the orbit O0(x00,

π3 ) of Ω(KS0),

where x00 = c ∈ I; see the first image in Figure 15 (and recall our earlier discussion

in §3.1). The same orbit, viewed as a continuous curve embedded in Ω(KS1), doesnot constitute an orbit of Ω(KS1); see the second image in Figure 15. Consider theorbit O1(x

01,

π3 ) shown in the third image in Figure 15. Such an orbit does intersect

the boundary of Ω(KS1) and appears to be related to O0(c,π3 ), but in what way

we have not yet explicitly said. Initially, we notice that, as a continuous curveembedded in Ω(KS1), the orbit O0(c,

π3 ) is a subset of O1(x

01,

π3 ). Being eager to

establish a proper notion of “related”, we may be inclined to declare that two orbitsare related if one is a subset of the other, when viewed as continuous curves in theplane. Unfortunately, we quickly see that such a definition is highly restrictive.A more general observation is that x0

1 and x00 are collinear in the direction of

π3 , without any portion of KS1 intersecting the segment x0

1x00. We then say that

(x00,

π3 ) and (x0

1,π3 ) are compatible initial conditions. We state the formal definition

as follows.

Definition 4.2 (Compatible initial conditions). Without loss of generality,suppose that n and m are nonnegative integers such that n > m. Let (x0

n, θ0n) ∈

(Ω(Sa,n) × S1)/ ∼ and (x0m, θ0m) ∈ (Ω(Sa,m) × S1)/ ∼ be two initial conditions of

the orbits On(x0n, θ

0n) and Om(x0

m, θ0m), respectively, where we are assuming thatθ0n and θ0m are both inward pointing. If θ0n = θ0m and if x0

n and x0m lie on a segment

determined from θ0n (or θ0m) that intersects Ω(Sa,n) only at x0n, then we say that

(x0n, θ

0n) and (x0

m, θ0m) are compatible initial conditions.

Remark 4.3. When two initial conditions (x0n, θ

0n) and (x0

m, θ0m) are compati-ble, then we simply write each as (x0

n, θ0) and (x0

m, θ0). If two orbits Om(x0m, θ0m)

and On(x0n, θ

0n) have compatible initial conditions, then we say such orbits are com-

patible.

Depending on the nature of Ω(F ), not every orbit must pass through the regionof Ω(Fn) corresponding to the interior of Ω(F0), let alone pass through the interiorof Ω(Fm), for any m < n. Because of this, it may be the case that an initialcondition (x0

n, θ0) is not compatible with (x0

m, θ0), for any m < n. As such, inDefinitions 4.4 and 4.5, we consider sequences beginning at i = N , for some N ≥ 0.

Definition 4.4 (Sequence of compatible initial conditions). Let {(x0i , θ

0i )}∞i=N

be a sequence of initial conditions, for some integer N ≥ 0. We say that thissequence is a sequence of compatible initial conditions if for every m ≥ N and for


every n > m, we have that (x0n, θ

0n) and (x0

m, θ0m) are compatible initial conditions.In such a case, we then write the sequence as {(x0

i , θ0)}∞i=N .

Definition 4.5 (Sequence of compatible orbits). Consider a sequence of com-patible initial conditions {(x0

n, θ0)}∞n=N . Then the corresponding sequence of orbits

{On(x0n, θ

0)}∞n=N is called a sequence of compatible orbits.

If Om(x0m, θ0m) is an orbit of Ω(Fm), then Om(x0

m, θ0m) is a member of a sequenceof compatible orbits {On(x0

n, θ0)}∞n=N for some N ≥ 0. It is clear from the definition

of a sequence of compatible orbits that such a sequence is uniquely determined bythe first orbit ON (x0

N , θ0). Since the initial condition of an orbit determines theorbit, we can say without any ambiguity that a sequence of compatible orbits isdetermined by an initial condition (x0

N , θ0).

Definition 4.6 (A sequence of compatible P orbits). Let P be a property(resp., P1, ...,Pj a list of properties). If every orbit in a sequence of compatibleorbits has the property P (resp., a list of properties P1, ...,Pj), then we call such asequence a sequence of compatible P (resp., P1, ...,Pj) orbits.

The following theorem can be deduced from Theorem 3 of Gutkin’s paper[Gut2]; see [LapNie3, §3.2].

Theorem 4.7. Consider a prefractal rational billiard Ω(Fn). If Ω(Fn) is tiledby a rational polygon Dn such that Dn tiles the Euclidean plane, then, for a fixeddirection θ0n, every orbit On(x0

n, θ0n) of Ω(Fn) is closed or every orbit On(x0

n, θ0n) is

dense in Ω(Fn),14 regardless of the initial basepoint x0n.

Remark 4.8. When Ω(Fn) is tiled by Dn, where Dn is a rational polygon tilingthe plane, then Ω(Fn) is more generally referred to as an almost integrable billiard,this being the language used in [Gut2].

The following is a generalization to this broader setting of Corollary 16 from[LapNie3]. It is established in the same manner.

Theorem 4.9. Let Ω(F ) be a fractal billiard table approximated by a suitablesequence of rational polygonal billiard tables {Ω(Fn)}∞n=0. If there exists a polygonD0 that tiles the plane and such that for every n ≥ 0 there exists 0 < cn ≤ 1 withDn := cnD0 tiling Ω(Fn), then any sequence of compatible orbits is either entirelycomprised of closed orbits or entirely comprised of orbits that are dense in theirrespective billiard tables.

4.2. The prefractal Koch snowflake billiard. The billiard Ω(KSn) can betiled by equilateral triangles. Specifically, if Δ is the equilateral triangle with sideshaving unit length, then Ω(KSn) is tiled by 1

3n Δ, for every n ≥ 0. Moreover, as iswell known, Δ = KS0 tiles the plane. Therefore, Theorems 4.7 and 4.9 hold for theprefractal billiard Ω(KSn).

Our goal for this subsection and §4.2.1 is to survey some of the main resultsof [LapNie1,LapNie2,LapNie3]. We will focus on pertinent examples that willmotivate a richer discussion in §5.2. Initially, we focus on properties of orbits withan initial direction of π

3 and π6 .15

14Recall that these notions were introduced towards the beginning of §2.15Equivalently, we could focus on orbits with an initial direction of π

3and π

2, since π

2is

the rotation of π6

through the angle π3, the angle π

3being an angle that determines an axis of

symmetry of KSn, for n ≥ 0.


If O0(x00,

π3 ) is an orbit of Ω(KS0), so long as x0

0 is not a corner of KS0, theorbit will be periodic, as expected. However, depending on the nature of the ternaryrepresentation of x0

0, the compatible orbit O1(x01,

π3 ) may be singular in Ω(KS1).

16

Theorem 4.10 ([LapNie3]). Let x00 ∈ I ⊆ KS0. If x0

0 has a ternary represen-tation of type [l, cr] ∨ [r, lc], then there exists N ≥ 0 such that the compatible orbitON (x0

N , π3 ) will be singular in Ω(KSN ). Moreover, for every n ≥ N , On(x0

n,π3 ) will

also be singular in Ω(KSn).

Theorem 4.11 ([LapNie3]). If x00 has a ternary representation of the form

[c, lr] ∨ [lc, r] ∨ [cr, l] ∨ [lcr, ∅] ∨ [lr, c], then the sequence of compatible orbits givenby {On(x0

n,π3 )}∞n=0 is a sequence of compatible periodic orbits.

Theorem 4.12. The length and period of an orbit Om(x0m, π

3 ) ∈ {On(x0n,

π3 )}∞n=0

is dictated by the ternary representation of x00. (See [LapNie2] for the correspond-

ing specific formulas.)

Remark 4.13. See §4.4 of [LapNie2] for a precise statement and proof of thisresult, as well as for additional properties of orbits with an initial direction of π

3 .

Example 4.14 (A sequence of compatible hook orbits). Let x00 ∈ I have a

ternary representation given by rl. Such a representation indicates that, in eachprefractal approximation KSn, x0

0 is an element of an open, connected neighborhoodcontained in KSn. The point x0

0 corresponds to the value 3/4 ∈ I. If we consideran orbit of Ω(KS0) with an initial direction of π

6 , the ternary representation ofthe basepoints at which the billiard ball path forms right angles with the sides ofΩ(KS0) is of the type [c, lr]. This is a degenerate periodic hybrid orbit, meaningthat it doubles back on itself, and the next orbit in the sequence of compatibleperiodic hybrid orbits has the initial condition (x0

1,π6 ) = (x0

0,π6 ). Since the ternary

representation of the basepoint of f0(x00,

π6 ) is rc and θ00 = θ01 = π

6 , it follows

that the basepoint of f1(x01,

π6 ) is a point which, for every prefractal approximation

KSn, is an element of an open, connected neighborhood contained in KSn. Then thebasepoint of f2

1 (x01,

π6 ) (where f2

1 denotes the second iterate of the billiard map f1)has a ternary representation of type [c, lr]. This same pattern is repeated for everysubsequent orbit in the sequence of compatible orbits. It follows that the resultingsequence of compatible orbits forms a sequence of orbits that is converging to aset which is well defined. That is, such a set will be some path in the fractalbilliard table Ω(KS) with finite length which is effectively determined by the lawof reflection in each prefractal approximation of Ω(KS).

Such orbits are introduced in [LapNie3] and referred to as hook orbits, becausethey appear to be “hooking” into the Koch snowflake; see Figure 16.

The hook orbits of Example 4.14 are special cases of a general class of orbitscalled hybrid orbits, which were introduced, as well as studied, in [LapNie3].

Definition 4.15 (Hybrid orbit). Let On(x0n, θ

0n) be an orbit of Ω(KSn). If

all but at most two basepoints xknn ∈ Fn(x0

n, θ0n) have ternary representations

(determined with respect to the side sn,ν on which each point resides) of type[c, lr] ∨ [cl, r] ∨ [cr, l] ∨ [lcr, ∅] ∨ [lr, ∅], then we call On(x0

n, θ0n) a hybrid orbit of

Ω(KSn).

16Recall that the notion of type of a ternary representation was introduced in Notation 3.6of §3.1.


Figure 16. An example of a hook orbit. The same initial condi-tion is used in each prefractal billiard.

A hybrid orbit is so named for the fact that it may have qualities reminiscentof an orbit On+1(x

0n+1, θ

0n+1) that is identical to the compatible orbit On(x0

n, θ0n)

and an orbit On+1(y0n+1, γ

0n+1) that is visually different from the compatible orbit

On(y0n, γ0n); see Figure 17 and its caption.

Definition 4.16 (A P hybrid orbit). If On(x0n, θ

0n) is a hybrid orbit with

property P, then we say that it is a P hybrid orbit.

Proposition 4.17. If On(x0n, θ

0n) is a dense orbit of Ω(KSn), then On(x0

n, θ0n)

is a dense hybrid orbit.

Applying the results in Theorem 4.9 and Proposition 4.17, we state the followingresult.

Theorem 4.18 (A topological dichotomy for sequences of compatible orbits,[LapNie3]). Let {On(x0

n, θ0)}∞n=N be a sequence of compatible orbits. Then we

have that {On(x0n, θ

0)}∞n=N is either entirely comprised of closed orbits or is entirelycomprised of dense hybrid orbits.17

Theorem 4.19 ([LapNie3]). If O0(x00, θ

00) is a periodic hybrid orbit of Ω(KS0)

with no basepoints corresponding to ternary points (i.e., points having ternary rep-resentations of the types [l, cr] ∨ [r, lc]), then for every n ≥ 0, the compatible orbitOn(x0

n, θ0n) is a periodic hybrid orbit of Ω(KSn).

In order to fully understand the following result, we define what it means fora vector to be rational with respect to a basis {u1, u2} of R2. If z = mu1 + nu2,for some m,n ∈ Z, then we say that z is rational with respect to the basis {u1, u2}.Otherwise, we say that z is irrational with respect to {u1, u2}.

Theorem 4.20 (A sequence of compatible periodic hybrid orbits, [LapNie3]).Let x0

0 ∈ I and consider a vector (a, b) that is rational with respect to the basis

{u1, u2} := {(1, 0), (1/2,√

3/2)}. Then, we have the following :

(1) If a and b are both positive integers with b being odd, x00 = r

4s , for some

r, s ∈ N with s ≥ 1, 1 ≤ r < 4s being odd and θ0 := arctan b√3

2a+b , then

the sequence of compatible closed orbits {On(x0n, θ

0)}∞n=0 is a sequence ofcompatible periodic hybrid orbits.

(2) If a = 1/2, b is a positive odd integer, x00 = r

2s , for some r, s ∈ N with

s ≥ 1, 1 ≤ r < 2s being odd and θ0 := arctan b√3

2a+b , then the sequence

of compatible closed orbits {On(x0n, θ

0)}∞n=0 is a sequence of compatibleperiodic hybrid orbits.

17Recall that the notions of “closed orbit” and “dense orbit” were defined in §2, just beforeDefinition 2.3.


Figure 17. Three examples of periodic hybrid orbits. These arethe first three elements of the sequence of compatible periodic hy-brid orbits described in Example 4.22. In order to understandexactly what is discussed in the paragraph immediately followingDefinition 4.15, compare and contrast the hybrid orbits shown herewith the hybrid orbits shown in Figures 15, 16 and 18. Certain seg-ments of the hybrid orbits shown here remain intact and becomesubsets of subsequent compatible periodic hybrid orbits, yet theorbits are visually different from one another.

Remark 4.21. We want to emphasize that the angle θ0 in Part (1) and Part (2)of Theorem 4.20 is not necessarily π

3 , π2 or π

6 , but can assume countably infinitelymany values.

Example 4.22 (A sequence of compatible periodic hybrid orbits). In Figure17, three periodic hybrid orbits are displayed. These three orbits constitute thefirst three terms in a sequence of compatible periodic hybrid orbits.18 If we choosex00 = c ∈ I and θ00 to be an angle such that x0

0 connects with the midpoint ofthe lower one-third interval on the side of Ω(KS0), we can see that O0(x

00, θ

00) is

a periodic hybrid orbit. More importantly, there are elements of the footprintF0(x

00, θ

00) with ternary representations of type [lr, c]. This observation is key for

constructing what we call nontrivial paths of Ω(KS), a topic which is discussed inmore detail in §5.2.

Given a nonnegative integer N , we say that a sequence of compatible orbits{On(x0

n, θ0)}∞n=N is a constant sequence of compatible orbits if the path traversed by

On+1(x0n+1, θ

0n+1) is identical to the path traversed by On(x0

n, θ0n), for every n ≥ N .

Furthermore, we say that a sequence of compatible orbits {On(x0n, θ

0)}∞n=0 is even-tually constant if there exists a nonnegative integer N such that {On(x0

n, θ0)}∞n=N

is constant, in the above sense.

Theorem 4.23 (A constant sequence of compatible periodic hybrid orbits,

[LapNie3]). Let O0(x00, θ

00) be an orbit of Ω(KS0) such that every xk0

0 ∈ F0(x00, θ

00)

has a ternary representation of type [lr, c]. Then {On(x0n, θ

0)}∞n=0 is a sequenceof compatible periodic hybrid orbits. Moreover, there exists N ≥ 0 such that{On(x0

n, θ0)}∞n=N is a constant sequence of compatible periodic hybrid orbits.

Example 4.24 (A constant sequence of compatible periodic hybrid orbits).Consider x0

0 = 7/12 in the base of the equilateral triangle. Such a value has aternary representation of type [lr, c]. Consider the initial condition (x0

0,π3 ). Then

18By Theorem 4.20(2), the angle θ00 determined by the initial segment of the orbit and the ini-

tial basepoint x00 = 1

2= c both guarantee that the sequence of compatible orbits {On(x0

n, θ0)}∞n=0

is a sequence of compatible periodic hybrid orbits.


Figure 18. An eventually constant sequence of compatible peri-odic hybrid orbits. We see that the initial basepoint x0

0 = 7/12 lieson the middle third of the unit interval. The basepoint x0

1 of thecompatible initial condition (x0

1,π3 ) has a ternary representation of

type [lr, ∅].

the sequence of compatible orbits {On(x0n,

π3 )}∞n=1 is a constant sequence. This

follows from the fact that the ternary representation of x01 is rl. Moreover, the

representation of every basepoint of On(x0n,

π3 ) is lr. In Figure 18, we show the first

three orbits in this (eventually) constant sequence of compatible periodic hybridorbits.

As of now, the only examples of constant sequences of compatible periodichybrid orbits are those for which the initial direction is π

3 and π6 (and, equivalently,

π2 ). When the initial angle of an orbit of a constant sequence of compatible periodichybrid orbits is π

6 (or, equivalently, π2 ), then the orbit will be degenerate. For

example, the orbit O1(34 ,

π2 ) traverses a path that is a vertical line. This orbit

has period p = 2. While {On( 34 ,π2 )}∞n=1 is an important example of a constant

sequence of compatible periodic hybrid orbits, it is arguably less interesting thanthe constant sequence of compatible periodic hybrid orbits {On( 34 ,

π3 )}∞n=1.

4.2.1. The corresponding prefractal translation surface S(KSn). In §2.1 we sawhow to construct a translation surface from a rational billiard table. In the case ofthe equilateral triangle billiard table Ω(Δ) = Ω(KS0), there are 2 · lcm{3, 3, 3} = 6copies of Ω(Δ) used in the construction of the associated translation surface S(Δ);see Example 2.7 and the associated Figure 2. In the case of the prefractal billiardtable Ω(KSn), only six copies of Ω(KSn) are needed in the construction of theassociated translation surface S(KSn), for every n ≥ 0; see Figure 19. (We refer to[LapNie1,LapNie2,LapNie3] for further discussion of the topics in the presentsubsection.)

The vertices of Ω(KSn) correspond to conic singularities of the translationsurface. However, only certain singularities are removable. The vertices with anglesmeasuring π

3 (measured from the interior), constitute removable singularities ofthe translation surface. That is, the geodesic flow can be appropriately definedat these points. The vertices with angles measuring 4π

3 constitute nonremovablesingularities. Hence, it is possible to define reflection at certain vertices of theprefractal billiard Ω(KSn), but impossible to define at others. Moreover, definingreflection at acute corners of Ω(KSn) in this way is independent of n. That is, fora given vertex v of Ω(KSn) with an acute angle π

3 , the general rule for reflectionin v states that an incoming trajectory reflect through the angle bisector of v. Abilliard ball entering v along the same path in Ω(KSn+1) as in Ω(KSn) will then


Figure 19. The translation surfaces S(KS1), S(KS2) and S(KS3)associated with the Koch snowflake prefractal approximations KS1,KS2 and KS3, respectively.

reflect in v in Ω(KSn+1) in exactly the same way as it did when considering v as avertex of Ω(KSn).

Such insight is clearly helpful in further understanding the behavior of a billiardball on the Koch snowflake fractal billiard Ω(KS), but we must be careful not toextrapolate more than is possible from this observation. Knowing that we candetermine an orbit of a prefractal billiard Ω(KSn) by unfolding the orbit of Ω(KS0)in Ω(KSn), we are inclined to allow orbits of Ω(KS0) that make collisions withcorners. However, a priori, we cannot conclude that such orbits do not unfold toform saddle connections in Ω(KSn) connecting two nonremovable singularities. Inthe event an orbit Om(x0

m, θ0m) of Ω(KSm) intersects the boundary KSm solely inacute corners, then such an orbit is an element of a sequence of compatible orbits{On(x0

n, θ0)}∞n=N with Oj(x

0j , θ

0j ) = Om(x0

m, θ0m), for every j ≥ m.

4.3. The T -fractal prefractal billiard. We refer to §3.3 for a discussion ofthe T -fractal T and of its prefractal approximations Tn, for n = 0, 1, 2, ...; see,in particular, Figure 12. Recall that the base of T0 has a length of two units.The prefractal billiard Ω(T0) can be tiled by the unit square Q; see Figure 13. Ingeneral, for every n ≥ 0, Ω(Tn) can be tiled by the square 1

2nQ. As such, and sinceQ obviously tiles the plane, we can apply Theorems 4.7 and 4.9.

Much like the case of the prefractal Koch snowflake billiard Ω(KSn), we areinterested in forming sequences of compatible orbits of prefractal billiards exhibitingparticular properties. The results in this subsection appear here for the first timeand will be further discussed in [LapNie6]. It is true that if a periodic orbit has aninitial condition (x0

0, θ00), then there may exist a compatible orbit ON (x0

N , θ0N ) thatforms a saddle connection if x0

0 has a finite binary expansion. This is not to suggest

that ON (x0N , θ0N ) must form a saddle connection. However, if every basepoint xk0

0

of a periodic orbit OQ(x00, θ

00) of the unit square has an infinite binary expansion

(with no equivalent finite binary expansion), then viewing O0(x00, θ

00) in Ω(T0) as

the reflected-unfolding of OQ(x00, θ

00), the corresponding sequence of compatible

orbits {On(x0n, θ

0)}∞n=0 will be a sequence of compatible periodic orbits. We statethis formally in the following theorem.

Theorem 4.25. Let (x00, θ

00) be an initial condition of an orbit OQ(x0

0, θ00) of

Ω(Q). Suppose every element of the footprint FQ(x00, θ

00) has an infinite binary

expansion (and no equivalent finite binary expansion) and (x00, θ

00) is then the initial

condition of an orbit of Ω(T0) that constitutes the reflected-unfolding of OQ(x00, θ

00)


Figure 20. A sequence of compatible periodic orbits of Ω(T0),Ω(T1) and Ω(T2), respectively.

in Ω(T0). Then the sequence of compatible orbits {On(x0n, θ

0)}∞n=0 (where (x00, θ

0) =(x0

0, θ00)) of the prefractal billiards Ω(Tn) is a sequence of compatible periodic orbits.

Example 4.26. Let x00 = 4

3 and θ00 = π4 . Then, {On(x0

n,π4 )}∞n=0 is a noncon-

stant sequence of compatible periodic orbits; see Figure 20.

The following two theorems are ultimately concerning the prefractal billiardΩ(Tn). Determining which intercepts and slopes yield line segments in the planethat avoid lattice points of the form ( a

2c ,b2d

) is equivalent to specifying an initialcondition of an orbit of a square billiard table that avoids corners of the billiardtable. Then, using the fact that an appropriately scaled square billiard table tilesΩ(Tn), we can reflect-unfold such an orbit in Ω(Tn) in order to determine an orbitof Ω(Tn).

Theorem 4.27. Let x00 = t

3kwith k, t ∈ N, t and 3 relatively prime, k �= 0 and

0 < t < 3k. Further, let m ∈ R. If for every p, q, r, s ∈ Z, r, s ≥ 0, we have that

m �= q2r−s3k

p3k − t2r,(14)

then the line y = m(x − x00) does not contain any point of the form ( a

2c ,b2d

),a, b, c, d ∈ Z, with c, d ≥ 0.

Note that the condition (14) above is automatically satisfied if the slope m isirrational.

Theorem 4.28. Let x00 = t

3k, with k, t ∈ N, t and 3 relatively prime, k �= 0

and 0 < t < 3k. If

m =2γ

(2α + 1)β,

with α, β, γ ∈ N, α, β, γ ≥ 0, then, for every p, q, r, s ∈ Z with r, s ≥ 0, the point( p2r ,

q2s ) does not lie on the line y = m(x− x0

0).

Finally, Theorems 4.27 and 4.28 combined with the fact that an initial con-dition of an orbit of Ω(TN ), N ≥ 0, determines a sequence of compatible orbits{On(x0

n, θ0)}∞n=N , allows us to determine a countably infinite family of sequences

of compatible periodic orbits.

4.3.1. The corresponding prefractal translation surface S(Tn). For every n ≥ 0,the interior angles of Tn are π

2 and 3π2 . To form the associated translation surface

S(Tn), we appropriately identify four copies of Ω(Tn); see Figure 21 for a depictionof the first three translation surfaces.

Then, every point of S(Tn) associated with a vertex of Ω(Tn) measuring π2 con-

stitutes a removable singularity of S(Tn). Similarly, every point of S(Tn) associated


1 12 2 2 21 1 2 21 1

3 3 3 34 4 4 4

Figure 21. The translation surfaces S(T0), S(T1) and S(T2) as-sociated with the T -fractal prefractal approximations T0, T1 andT2, respectively.

with a vertex of Ω(Tn) of interior angle measuring 3π2 constitutes a nonremovable

singularity of S(Tn). Therefore, not every vertex of Ω(Tn) will present a problemfor the billiard flow.

Consider an orbit of Ω(Q), where the orbit has basepoints corresponding tovertices of Q, the unit square. Since such vertices correspond to removable singu-larities in the corresponding translation surface (this being the flat torus, see §2.1),we see that the same orbit reflected-unfolded in the billiard Ω(Tn) (if one first scalesthe billiard Ω(Q) and the orbit contained therein by 1

2n , see §2.2) can potentiallyintersect vertices of Tn that are associated with nonremovable singularities in thecorresponding translation surface.

4.4. A prefractal self-similar Sierpinski carpet billiard. Let Sa be aself-similar Sierpinski carpet, as defined in Definition 3.10, and let us denote itsnatural prefractal approximations by Sa,i for i = 0, 1, 2, ... (as in §3.4). The cor-responding billiard is then denoted by Ω(Sa). In this subsection, we examine thebehavior of the billiard flow on the rational polygonal billiard given by the prefrac-tal approximations Ω(Sa,i).

19 In the event a billiard ball collides with a corner of aperipheral square, we must terminate the flow and such a trajectory is then calledsingular. In addition to being singular, such a trajectory will form a saddle con-nection (see the beginning of §2 for a discussion of closed billiard orbits that formsaddle connections). As we have discussed, an examination of the correspondingtranslation surface may prove useful in determining whether or not a billiard ballcan reflect in a vertex.

Definition 4.29 (Obstacle of Ω(D)). Let Ω(D) be a polygonal billiard. ThenΩ(D) can be modified by placing in its interior a piecewise smooth segment thatinhibits the billiard flow and causes a billiard ball to reflect. Such a segment iscalled an obstacle of Ω(D).

Clearly, each prefractal billiard Ω(Sa,i) can be interpreted as a square billiardwith obstacles.

Notation 4.30. Due to the fact that Theorem 3.15 refers to the slope of anontrivial line segment and we make heavy use of this theorem, we will denote theinitial condition (x0

n, θ0n) of an orbit of Ω(Sa,n) by (x0

n, α0n), where α0

n = tan(θ0n).

19We note that the results in this subsection appear here for the first time and will be furtherdiscussed in [CheNie].


Definition 4.31 (An orbit of the cell Ck,ak of Ω(Sa,k)). Consider the boundary

of a cell Ck,ak of Ω(Sa,k) as a barrier.20 Then an orbit with an initial conditioncontained in the cell is called an orbit of the cell Ck,ak of Ω(Sa,k).

Remark 4.32. So as to be clear, the boundary of the cell does not form anobstacle to the billiard flow, as defined in Definition 4.29. Rather, we are treatingthe cell Ck,ak as a billiard table in its own right, embedded in the larger prefractalapproximation Ω(Sa,k).

Recall from §3.4 that a self-similar Sierpinski carpet Sa is the unique fixed

point attractor of a suitably chosen iterated function system {φj}a2−1

j=1 consisting

of similarity contractions. In light of this, an orbit of a cell Ck,ak of Ω(Sa,k) is the

image of an orbit O0(x00, α

00) of the unit-square billiard Ω(S0) under the action of

a composition of contraction mappings φmk◦ · · · ◦ φm1

, with 1 ≤ mi ≤ a2 − 1 and

1 ≤ i ≤ k, determined from the iterated function system {φj}a2−1

j=1 of which Sa isthe unique fixed point attractor.

Lemma 4.33. Consider a self-similar Sierpinski carpet Sa. Let k ≥ 0 and Sa,k

be a prefractal approximation of Sa. If α ∈ Ba,21 then the line segment beginning

at a midpoint of a cell Ck,ak of Sa,k is a nontrivial line segment (in the sense ofDefinition 3.14). Moreover, such a segment avoids the boundary of the peripheralsquares of Sa with side-length a−m, m ≥ k + 1.

The statement in Lemma 4.33 asserts that a segment beginning at a midpointof a cell with slope α ∈ Ba will be a nontrivial line segment in Sa. In addition tothis, any line segment contained in R2 that contains a nontrivial line segment of Sa

must necessarily avoid the peripheral squares in a tiling of R2 by Sa. Otherwise,there exists k ≥ 1 such that scaling the line segment in R2 and the tiling of R2 bya−k results in a segment contained in the nontrivial line segment which intersectsperipheral squares of Sa. This is a contradiction of the fact that the segmentbeginning at (2−1, 0) with slope α ∈ Ba is a nontrivial line segment of Sa. We thendeduce the following result.

Theorem 4.34. Consider a self-similar Sierpinski carpet Sa. Let k ≥ 0 andSa,k be a prefractal approximation of Sa. Furthermore, let α ∈ Ba and x0

k =(p(2ak)−1, 0) with p ≤ ak a positive, odd integer. If Ok(x

0k, α

0k) is an orbit of

Ω(Sa,k), then the initial condition (x0k, α

0k) determines a sequence of compatible

periodic orbits {On(x0n, α

0)}∞n=k of the prefractal approximations Ω(Sa,n).

As one may suspect, there exists N ≥ k ≥ 0 such that a sequence of compati-ble orbits {On(x0

n, α0)}∞n=N is a constant sequence of compatible orbits. Moreover,

x0n = x0

N , for every n ≥ N . This is not any different from the case of a constantsequence of compatible orbits of prefractal billiards Ω(KSn), as discussed in The-orem 4.23 and Example 4.24. However, in the context of a self-similar Sierpinskicarpet billiard table, every sequence of compatible orbits we will examine will bea sequence for which there exists N ≥ 0 such that {On(x0

n, α0)}∞n=N is a constant

sequence of compatible orbits.

20Here, Ck,ak is a cell of the kth prefractal approximation Sa,k, as given in Definition 3.12

with all numbers aj equal to a.21Recall from Notation 3.16 that Ba is the set of slopes given by Equation (13).


Figure 22. Interpreting the translation surface S(Sa,n) as a flattorus with obstacles.

4.4.1. The corresponding prefractal translation surface S(Sa,i). In much thesame way the billiard Ω(Sa,i) can be interpreted as a square billiard with obstacles,the corresponding translation surface can be interpreted as a “torus with obstacles”;see Figure 22.

In light of the fact that S(Sa,n) can be interpreted as a torus with obstacles andthe presence of a dynamical equivalence between the billiard flow and the geodesicflow on the corresponding translation surface (see §2.2), we see that reflection inthe vertices with angles measuring π

2 (relative to the interior) can be defined. Morespecifically, the geodesic flow can be defined at points corresponding to verticeswith angles measuring π

2 , because these points constitute removable singularities ofthe geodesic flow.

This fact is crucial in determining orbits of Ω(Sa) for which the slope α is anelement of Aa and not Ba (see Notation 3.16), and the orbit avoids all peripheralsquares of Ω(Sa). While one may say that this contradicts part of Theorem 3.15(and he/she would be correct), in [CheNie] a more precise formulation of Theorem3.15 is given that clarifies which slopes are permissible and which ones are not. Thatis, if α0

n ∈ Aa, it may be possible for an orbit On(x0n, α

0n) to begin at the origin

and avoid the peripheral squares of each billiard Ω(Sa,m), for every m ≥ n. Wedo not give here an explicit reformulation of Theorem 3.15, but Example 5.12 in§5.4 exhibits a situation showing that the latter half of Theorem 3.15 is not statedprecisely enough.

5. Fractal billiards

The theme that will tie together all of the examples in §4 is that suitable limitsof sequences of compatible orbits may constitute billiard orbits of each respectivefractal billiard table. We have shown that in the case of Ω(KS), Ω(T ) and Ω(Sa),we can determine a sequence of compatible periodic orbits. We will see that ineach case of a fractal billiard, under certain conditions, a sequence of compatibleperiodic orbits (or a proper subset of points from each footprint Fn(x0

n, θ0n)) will

converge to a set which can be thought of as a true orbit of a fractal billiard table(or such a sequence will yield a subsequence of basepoints converging to what weare calling an elusive point in [LapNie2,LapNie3]).

5.1. A general framework for Ω(KS), Ω(T ) and Ω(Sa). We restrict ourattention to the family of fractal billiard tables Ω(F ) where F is a fractal approxi-mated by a suitable sequence of rational polygons {Fn}∞n=0, with each Fn tiled byDn = cnD0 for suitably chosen cn ∈ (0, 1] and D0 a polygon that tiles the plane.


Specifically, we are interested in developing a general framework for dealing with afractal billiard table Ω(F ) that is constructed in a way which is similar to that ofΩ(KS), Ω(T ) and Ω(Sa).

Before we begin our discussion of the fractal billiard tables Ω(KS), Ω(T ) andΩ(Sa), we define certain terms. The following definitions were initially motivatedby the work in [LapNie3], but later generalized for this paper in order to accountfor a larger class of fractal billiard tables. (From now on, we assume that Ω(F )is a fractal billiard table with prefractal billiard approximations {Ω(Fn)}∞n=0 asdescribed just above.)

Definition 5.1 (A corner). Let z ∈ F . If there exists n ≥ 0 such that z ∈ Fn

and z is a vertex of Fn, then z is called a corner of F .

Definition 5.2 (A Cantor point). Let z ∈ F be such that z is not a corner ofF . If there exists N ≥ 0 such that for every n ≥ N , z ∈ Fn and every connectedneighborhood of z contained in Fn becomes totally disconnected when intersectedwith F , then z is called a Cantor point of F .

In the Koch snowflake KS, every Cantor point is a smooth point of infinitelymany prefractals KSn approximating KS. That is, if z is a Cantor point in KS,then there exists N ≥ 0 such that for every n ≥ N , there exists a well-definedtangent at z ∈ KSn.22 We deduce from this that the law of reflection holds atz ∈ KSn, for every n ≥ N . Moreover, since the billiard ball reflects at z ∈ KSn atthe same angle for every n ≥ N , we deduce that the tangent at z is the same foreach KSn, n ≥ N . This observation then prompts us to generalize the definition ofa Cantor point in order to account (for example) for points of the T -fractal whichare not Cantor points, but are points for which a well-defined tangent can be foundin infinitely many prefractal approximations.

Definition 5.3 (Smooth fractal point). Let z ∈ F and N ≥ 0 be such thatz ∈ Fn for every n ≥ N . If there exists a well-defined tangent at z ∈ Fn for everyn ≥ N , then z is called a smooth fractal point.

To be clear, a Cantor point of F is an example of a smooth fractal point ofF . The special nature of a Cantor point warrants a formal definition. In the T -fractal billiard, there are certainly corners and elusive points. There are also smoothconnected segments contained in the boundary of Ω(T ). Points contained in suchsegments that do not correspond to corners are then called smooth fractal points.

Definition 5.4 (An elusive point). Let z ∈ F . If z /∈⋃∞

n=0 Fn, then z is calledan elusive point of F .

Consider a piecewise linear path in Ω(F ), such that every linear segment of thepath is joined at the endpoint of another segment with the coincidental endpointsintersecting the boundary F at a smooth fractal point of F (in the sense of Definition5.3). In the following definition, we define a particular type of piecewise linear curvein a fractal billiard Ω(F ).

Definition 5.5 (A nontrivial path). Suppose that there exists a piecewiselinear curve in Ω(F ) as described immediately above. If at each point z for whichthe piecewise linear path intersects the boundary F , the angle formed by the first

22Here and Definition 5.3 below, z is viewed as a point of the smooth subarc of Fn to whichit belongs.


segment is equal to the angle formed by the second segment, relative to the side ofFn on which z lies,23 then the piecewise linear path is called a nontrivial path ofΩ(F ).

Remark 5.6. In [LapNie3], a nontrivial path was called a nontrivial polygonalpath. The change in name is purely based on aesthetics.

Definition 5.7 (A Cantor orbit). Suppose ON (x0N , θ0N ) is an orbit of Ω(FN ),

for some N ≥ 0, such that every point of the footprint FN (x0N , θ0N ) corresponds

to a smooth fractal point of F . This then readily implies that On(x0n, θ

0n) is the

same as ON (x0N , θ0N ) for every n ≥ N .24 Then On(x0

n, θ0n) is called a Cantor orbit

of Ω(F ) and is denoted by O(x0, θ0).

If Ω(F ) is a fractal billiard table, then it may or may not be possible to constructCantor orbits or nontrivial paths of Ω(F ). We will next discuss three examples offractal billiard tables with different dynamical properties that lend themselves well(or not) to determining well-defined billiard orbits.

Remark 5.8. We note that applying Definitions 5.1, 5.2 and 5.4 to Ω(KS) andthe sequence of rational polygon prefractal approximations Ω(KSn) which we havediscussed in §3.2 yields exactly the sets of points we are considering as corners,Cantor points and elusive points of Ω(KS), respectively. Moreover, applying Defi-nitions 5.1, 5.3 and 5.4 to Ω(T ) and the prefractal approximations Ω(Tn) which wediscussed in §3.3 yields exactly the sets of points that we are considering as corners,smooth fractal points and elusive points of Ω(T ). Finally, applying Definitions 5.1and 5.3 to Ω(Sa) and the prefractal approximations Ω(Sa,n) which we discussedin §3.4 yields exactly the set of points we are considering as corners and smoothfractal points of Ω(Sa).

5.2. The Koch snowflake fractal billiard. As we have noted before at theend of §3.2, for each n ≥ 0, KSn ∩ KS can be realized as the union of 3 · 4n self-similar ternary Cantor sets, each spanning a distance of 1

3n . Within each Cantorset, we find Cantor points and corners of the Koch snowflake.

We begin our discussion of orbits of Ω(KS) by examining the limiting behaviorof a particular sequence of compatible orbits with the initial condition (x0

N , π3 ),

where x0N is a Cantor point of KS (i.e., x0

N is a point of KSN with a well-definedtangent in KSn for every n ≥ N). For the sake of simplicity, we let N = 0 andx0N = 1

4 be on the base of the equilateral triangle KS0 (recall that we are assumingthat the left corner of KS0 is at the origin and the length of each side is oneunit). Then, O0(x

00,

π3 ) is an orbit that remains fixed as one constructs Ω(KS1)

from Ω(KS0). More correctly, {On(x0n,

π3 )}∞n=0 is a sequence of compatible orbits

with Fn(x0n,

π3 ) = F0(x

00,

π3 ) for every n ≥ 0 (that is, with the same footprint in

each prefractal approximation).In general, if (x0

N , π3 ) is an initial condition of an orbit of Ω(KSN ) and x0

N is aCantor point, then the sequence of compatible orbits is such that for every n ≥ N ,the footprints Fn(x0

n,π3 ) and FN (x0

N , π3 ) are the same.

23Recall from Definition 5.3 that a smooth fractal point z of F is necessarily a point ofinfinitely many prefractal approximations Fm. Hence, there is a least nonnegative integer n suchthat z ∈ Fn.

24In other words, {On(x0n, θ

0)}∞n=N is a constant sequence of compatible orbits, where a

sequence of compatible orbits was defined in Definition 4.5.


...

Figure 23. A nontrivial path of the Koch snowflake fractal bil-liard table Ω(KS) beginning at x = 1

2 .

Theorem 5.9. If x ∈ KS is a Cantor point, then there exists a well-definedorbit of Ω(KS) with an initial condition (x,�(π3 )), where the angle �(π3 ) is de-termined with respect to the side on which x lies in a prefractal approximationΩ(KSn).

There are many more properties of {On(x0n,

π3 )}∞n=N which we could discuss

here. These properties largely rely on the nature of the ternary representation ofx0N , and are elaborated upon in [LapNie2,LapNie3]. We now proceed to illustrate

how we can connect two elusive points of Ω(KS). Such a result has already beenpresented in greater detail in [LapNie3], so we will be brief. In §5.3, we will showthat an identical construction holds for the billiard table Ω(T ).

Recall from Example 4.22 that we were able to construct a sequence of com-patible periodic hybrid orbits. From such a sequence we can derive a sequence ofbasepoints that is converging to an elusive point of Ω(KS). The latter sequence ofbasepoints constitutes the vertices of a nontrivial path; see Figure 23. One may con-sider a direction γ0

0 that is the reflection of θ00 through the normal at x00. Then, the

resulting sequence of compatible periodic hybrid orbits {On(x0n, γ

00)}∞n=N yields a

sequence of basepoints converging to another elusive point. Again, such a sequenceof basepoints constitute the vertices of a nontrivial path of Ω(KS); see Figure 24.Together, these two nontrivial paths constitute a single nontrivial path connectingtwo elusive points of Ω(KS).

In conjunction with Theorem 4.20, we can determine countably infinitely manyinitial conditions (x0

n, θ0n), each of which determines a sequence of compatible peri-

odic hybrid orbits yielding a sequence of basepoints converging to an elusive pointof Ω(KS).

5.3. The T -fractal billiard. The results in this subsection appear here forthe first time and will be further discussed in [LapNie6]. We begin our discussionof the billiard Ω(T ) by recalling (and referring the reader back to) Example 4.26from §4.3. The sequence of compatible periodic orbits provided by Example 4.26gives rise to a nontrivial path that connects 4

3 with an elusive point of Ω(T ). Fur-

thermore, considering the sequence of compatible periodic orbits {On( 43 ,3π4 )}∞n=N ,

we determine another nontrivial path that connects 43 with another elusive point

of Ω(T ); see Figure 25. This behavior is analogous to the one which we observedfor the Koch snowflake billiard in §5.2.

As was the case with Ω(KS), we can analogously build upon Theorems 4.27 and4.28 in order to determine a sequence of basepoints converging to an elusive point.That is, Theorems 4.27 and 4.28 guide our search for a sequence of compatible


Figure 24. Two nontrivial paths connecting two elusive pointsof Ω(KS). (As is explained in the text, these two paths can beconcatenated to obtain a single nontrivial path connecting the twoelusive points.) In the first figure, we only show the relevant por-tions of the Koch snowflake. In the second figure, we magnify theregions containing the nontrivial paths so as to highlight the factthat such paths are converging to elusive points. Actually, there isan obvious geometric similarity one can take advantage of in orderto produce more segments of the nontrivial path.

Figure 25. Two nontrivial paths connecting two elusive points of Ω(T ).

periodic orbits which yields a sequence of basepoints converging to an elusive pointof Ω(T ).

Theorem 5.10. Let {On(x0n, θ

0)}∞n=N be a sequence of compatible orbits. Then,there are countably infinitely many directions and countably infinitely many pointsfrom which to choose so that {On(x0

n, θ0)}∞n=N is a sequence of compatible peri-

odic orbits yielding a sequence of basepoints {xknn }∞n=N that converges to an elusive

point of Ω(T ). The collection of basepoints {xknn }∞n=N constitutes the vertices of

a nontrivial path of Ω(T ). Moreover, once such a nontrivial path is constructed,

letting x′0N = x0

N , an additional nontrivial path can be determined from a sequence

of compatible periodic orbits {On(x′0n, π − θ0)}∞n=N in exactly the same fashion.

5.4. A self-similar Sierpinski carpet billiard. In [Du-CaTy], nontrivialline segments of Sierpinski carpets are constructed. Building on the main resultsof [Du-CaTy], the second author and Joe P. Chen have been able to construct afamily of Cantor periodic orbits of a self-similar Sierpinski carpet, in the sense of[LapNie2,LapNie3] recalled in Definition 5.7.25 Such orbits constitute Cantororbits of the self-similar Sierpinski carpet. As of yet, we have not attempted toconstruct a nontrivial path of a Sierpinski carpet.

25The results described in this subsection appear here for the first time and will be furtherdiscussed in [CheNie].


Figure 26. An orbit with an initial condition beginning at (0, 0)and with an initial direction constituting a slope of α = 2/3 ∈ A7,where A7 is defined as in Notation 3.16. While it would appearthat this orbit intersects corners of peripheral squares, it in factremains away from all peripheral squares. The same is true forfiner approximations.

In light of Theorem 4.34, we say that the trivial limit of a constant sequenceof compatible periodic orbits constitutes a periodic orbit of a self-similar Sierpinskicarpet billiard Ω(Sa). In the event an orbit has an initial direction α0

0, we maystill be able to determine a constant sequence of compatible periodic orbits. Thetrivial limit of such a sequence then constitutes a periodic orbit of Ω(Sa). Using thefact that reflection can be defined in the vertices with interior angles measuring π

2 ,we can state the following result. (Recall from §4.4 that Sa,n is the nth prefractalapproximation of Sa.)

Theorem 5.11. Recall from Notation 4.30 that if θ is the initial direction of abilliard orbit, then α = tan θ. Let x0 = (0, 0), α ∈ Q and let O(x0, α) be an orbitof Ω(S0). If O(x0, α), as an orbit of Ω(Sa,1), avoids the middle peripheral square,then the initial condition (x0, α) will determine an orbit of Ω(Sa). Specifically, thepath traversed by the orbit O(x0, α) of Ω(Sa,1) is exactly the path traversed by theorbit of Ω(Sa) determined by (x0, α).

Example 5.12. Let x0 = (0, 0), α = 2/3 ∈ Slope(S5). Consider an orbitof Ω(S7,2) with an initial condition (x0, α); see Figure 26. We see that the orbitavoids the peripheral square of Ω(S7,1). By Theorem 5.11, the initial condition(x0, α) determines an orbit of Ω(S7). The path traversed by the orbit of Ω(S7) isexactly the path traversed by the orbit O(x0, α).

6. Concluding remarks

It is clear from the preceding sections that much work remains to be developedin order to determine a well-defined phase space (Ω(F ) × S1)/ ∼ for the yet to bedefined fractal billiard flow. We have discussed several examples of what clearlyconstitute periodic orbits of Ω(KS) and Ω(Sa). Furthermore, for both Ω(KS) andΩ(T ), we were able to connect two elusive points of each billiard table via suitablychosen nontrivial paths. These nontrivial paths were determined from suitablychosen sequences of compatible periodic orbits.

Question 6.1. Let F be either KS or T . Suppose that two suitably chosennontrivial paths converge to two distinct elusive points of Ω(F ). For each of the


two elusive points, is it possible to determine another nontrivial path converging toa different elusive point?

If we can answer Question 6.1 in the affirmative (or answer it in the affirmativeunder specific conditions), will this help us gain insight into how to determine a well-defined phase space for the billiard flow on Ω(F )? An alternate approach, discussedin the concluding remarks of [LapNie3], entails determining a well-defined fractaltranslation surface.

Following this line of thought to its logical end, for certain fractal billiard tables(e.g., Ω(T )), is it possible to determine which directions produce recurrent orbits?More generally, can one prove that, in almost every direction, the billiard flow isergodic in Ω(F )?

Question 6.2. Regarding a self-similar Sierpinski carpet billiard Ω(Sa), wehave determined a countable set of points from which a periodic billiard orbit canbegin. Can we show that the set of points from which a periodic orbit can begin isin fact uncountable and, furthermore, a set of full (Lebesgue) measure in the baseof the unit square S0?

It is possible to construct a nontrivial line segment of Sa beginning from( 12 , 0) with slope α ∈ Slope(Sa), that, when translated to (0, 0), no longer liesentirely in Sa. However, if we consider the sequence of compatible periodic or-bits {On((0, 0), α0)}∞n=0, is it possible to determine a well-defined limit? The workof [HuLeTr] may prove useful in further exploring the behavior of a sequence ofcompatible periodic orbits. Building on the work of [HuLeTr], the author of [De]has examined the behavior of nonperiodic orbits in what is an example of whatis called a wind-tree billiard, and what is also strongly suggestive of a Sierpinskicarpet. The work of [CoGut] discusses recurrence and ergodicity for more generalinfinite periodic billiard tables. Such works may provide insight into examining thebehavior of a sequence of compatible dense orbits.

Question 6.3. In analogy with the prefractal billiard and associated transla-tion surface, can a thorough understanding of the geodesic flow on the limiting(and still to be mathematically defined) ‘fractal translation surface’ S(F ) aid us indetermining a well-defined billiard flow on Ω(F )?

The work in progress in [LapNie4] draws upon the work of Gabriela Weitze-Schmithusen [We-Sc] and attempts to answer Question 6.3 from an algebraic per-spective.

Approaching the problem of determining a well-defined billiard flow on a fractalbilliard table from many different points of view may prove useful. The theoriesof translation surfaces and rational billiards are intimately tied together and moredeeply understood by knowing the structure of what is called the Veech group (thisbeing the group studied in, for example, [HuSc,Ve3,Ve4,Vo,We-Sc]). In short,the Veech group of a translation surface S(D) determined from a rational polygonD is the stabilizer of S(D).

Question 6.4. Let Ω(F ) be a fractal billiard table, with F being approximatedby a suitably chosen sequence of rational polygons {Fn}∞n=0. Is it then possible toconstruct a Veech group for Ω(F ) (or rather, of S(F )), presumably in terms ofthe Veech groups for the prefractal approximations Ω(Fn) (or rather, of the asso-ciated translation surfaces S(Fn))? Will the knowledge of such a group aid us indetermining a well-defined billiard flow on Ω(F )?


References

[AcST] Yves Achdou, Christophe Sabot, and Nicoletta Tchou, Diffusion and propagationproblems in some ramified domains with a fractal boundary, M2AN Math. Model.Numer. Anal. 40 (2006), no. 4, 623–652, DOI 10.1051/m2an:2006027. MR2274772(2007k:35061)

[Ba] Michael F. Barnsley, Superfractals: Patterns of nature, Cambridge University Press,Cambridge, 2006. MR2254477 (2008c:28006)

[CheNie] Joe P. Chen and Robert G. Niemeyer, Periodic billiard orbits of self-similar Sierpinskicarpets, 29 pages, e-print, arXiv:1303.4032v1,2013.

[CoGut] Jean-Pierre Conze and Eugene Gutkin, On recurrence and ergodicity for geodesicflows on non-compact periodic polygonal surfaces, Ergodic Theory Dynam. Systems32 (2012), no. 2, 491–515, DOI 10.1017/S0143385711001003. MR2901357

[De] Vincent Delecroix, Divergent directions in some periodic wind-tree models, J of Mod.Dyn. 7 (2013), no. 1, 1–29.

[Du-CaTy] Estibalitz Durand-Cartagena and Jeremy T. Tyson, Rectifiable curves in

Sierpinski carpets, Indiana Univ. Math. J. 60 (2011), no. 1, 285–309, DOI10.1512/iumj.2011.60.4382. MR2952419

[Fa] Kenneth J. Falconer, Fractal Geometry: Mathematical foundations and applications,2nd ed., John Wiley & Sons Inc., Hoboken, NJ, 2003. MR2118797 (2006b:28001)

[Gut1] Eugene Gutkin, Billiards in polygons: survey of recent results, J. Statist. Phys. 83(1996), no. 1-2, 7–26, DOI 10.1007/BF02183637. MR1382759 (97a:58099)

[Gut2] Eugene Gutkin, Billiards on almost integrable polyhedral surfaces, Ergodic Theory Dy-nam. Systems 4 (1984), no. 4, 569–584, DOI 10.1017/S0143385700002650. MR779714(86m:58123)

[GutJu1] Eugene Gutkin and Chris Judge, The geometry and arithmetic of translation surfaceswith applications to polygonal billiards, Math. Res. Lett. 3 (1996), no. 3, 391–403.MR1397686 (97c:58116)

[GutJu2] Eugene Gutkin and Chris Judge, Affine mappings of translation surfaces: geometryand arithmetic, Duke Math. J. 103 (2000), no. 2, 191–213, DOI 10.1215/S0012-7094-00-10321-3. MR1760625 (2001h:37071)

[HuLeTr] Pascal Hubert, Samuel Lelievre, and Serge Troubetzkoy, The Ehrenfest wind-treemodel: periodic directions, recurrence, diffusion, J. Reine Angew. Math. 656 (2011),223–244, DOI 10.1515/CRELLE.2011.052. MR2818861 (2012f:37079)

[HuSc] Pascal Hubert and Thomas A. Schmidt, An introduction to Veech surfaces, Handbookof Dynamical Systems. vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 501–526, DOI10.1016/S1874-575X(06)80031-7. MR2186246 (2006i:37099)

[Hut] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981),no. 5, 713–747, DOI 10.1512/iumj.1981.30.30055. MR625600 (82h:49026)

[KaZe] A. N. Zemljakov and Anatole B. Katok, Topological transitivity of billiards in polygons,Mat. Zametki 18 (1975), no. 2, 291–300 (Russian). MR0399423 (53 #3267)

[LapNie1] Michel L. Lapidus and Robert G. Niemeyer, Towards the Koch snowflake fractal bil-liard: computer experiments and mathematical conjectures, Gems in ExperimentalMathematics, Contemp. Math., vol. 517, Amer. Math. Soc., Providence, RI, 2010,pp. 231–263, DOI 10.1090/conm/517/10144. MR2731085 (2012b:37101)

[LapNie2] Michel L. Lapidus and Robert G. Niemeyer, Families of periodic orbits of the Kochsnowflake fractal billiard, 63 pages, e-print, arXiv:1105.0737v1, 2011.

[LapNie3] Michel L. Lapidus and Robert G. Niemeyer, Sequences of compatible periodic hy-brid orbits of prefractal Koch snowflake billiards, Discrete and Continuous Dynam-ical Systems–Ser. A 33 (2013), no. 8, 3719–3740. [E-print: IHES/M/12/16, 2012;arXiv:1204.3133v1 [math.DS], 2012.]

[LapNie4] Michel L. Lapidus and Robert G. Niemeyer, Experimental evidence in support of afractal law of reflection, in progress, 2013.

[LapNie5] Michel L. Lapidus and Robert G. Niemeyer, Veech groups Γn of the Koch snowflakeprefractal translation surfaces S(KSn), in progress, 2012.

[LapNie6] Michel L. Lapidus and Robert G. Niemeyer, Sequences of compatible periodic orbitsof the T -fractal billiard, in progress, 2012.


[Mas] Howard Masur, Closed trajectories for quadratic differentials with an application tobilliards, Duke Math. J. 53 (1986), no. 2, 307–314, DOI 10.1215/S0012-7094-86-05319-6. MR850537 (87j:30107)

[MasTa] Howard Masur and Sergei Tabachnikov, Rational billiards and flat structures, Hand-book of Dynamical Systems, vol. 1A, Elsevier, Amsterdam, 2002, pp. 1015–1089, DOI10.1016/S1874-575X(02)80015-7. MR1928530 (2003j:37002)

[Sm] John Smillie, Dynamics of billiard flow in rational polygons, in: Dynamical Systems,

Encyclopedia of Math. Sciences, vol. 100, Math. Physics 1 (Ya. G. Sinai, ed.), Springer-Verlag, New York, 2000, pp. 360–382.

[Ta1] Sergei Tabachnikov, Billiards, Panor. Synth. 1 (1995), vi+142 (English, with Englishand French summaries). MR1328336 (96c:58134)

[Ta2] Sergei Tabachnikov, Geometry and Billiards, Student Mathematical Library, vol. 30,American Mathematical Society, Providence, RI, 2005. MR2168892 (2006h:51001)

[Ve1] William A. Veech, The billiard in a regular polygon, Geom. Funct. Anal. 2 (1992),no. 3, 341–379, DOI 10.1007/BF01896876. MR1177316 (94a:11074)

[Ve2] William A. Veech, Flat surfaces, Amer. J. Math. 115 (1993), no. 3, 589–689, DOI10.2307/2375075. MR1221838 (94g:30043)

[Ve3] William A. Veech, The Teichmuller geodesic flow, Ann. of Math. (2) 124 (1986), no. 3,441–530, DOI 10.2307/2007091. MR866707 (88g:58153)

[Ve4] William A. Veech, Teichmuller curves in moduli space, Eisenstein series and anapplication to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583, DOI10.1007/BF01388890. MR1005006 (91h:58083a)

[Vo] Yaroslov B. Vorobets, Plane structures and billiards in rational polygons: TheVeech alternative, Uspekhi Mat. Nauk 51 (1996), no. 5(311), 3–42, DOI10.1070/RM1996v051n05ABEH002993 (Russian); English transl., Russian Math. Sur-veys 51 (1996), no. 5, 779–817. MR1436653 (97j:58092)

[VoGaSt] Yaraslov B. Vorobets, Gregory A. Gal′perin, and Anatolii M. Stepin, Periodic bil-liard trajectories in polygons: generation mechanisms, Uspekhi Mat. Nauk 47 (1992),no. 3(285), 9–74, 207, DOI 10.1070/RM1992v047n03ABEH000893 (Russian, withRussian summary); English transl., Russian Math. Surveys 47 (1992), no. 3, 5–80.

MR1185299 (93h:58088)[We-Sc] Gabriela Schmithusen, An algorithm for finding the Veech group of an origami, Ex-

periment. Math. 13 (2004), no. 4, 459–472. MR2118271 (2006b:30080)[Zo] Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I,

Springer, Berlin, 2006, pp. 437–583, DOI 10.1007/978-3-540-31347-2 13. MR2261104(2007i:37070)

University of California, Department of Mathematics, 900 Big Springs Rd., River-

side, California 92521-0135


University of New Mexico, Department of Mathematics & Statistics, 311 Terrace

NE, Albuquerque, New Mexico 87131-0001



Long-Range Dependence and the Rank of Decompositions

Celine Levy-Leduc and Murad S. Taqqu

Abstract. We review and compare different methodologies for studying theasymptotic behavior of partial sums of nonlinear functionals of the following

type∑N

i=1 h(Xi) in the long-range dependence setting. Here (Xi)i≥1 is eithera stationary mean-zero Gaussian process or a linear process. The method-ologies, we consider, are based on different decompositions of the function h.This includes the decomposition of [Sur82] and of [HH97] in the case of lin-ear processes. The so-called “rank” of these decompositions plays an essentialrole. We show that all ranks coincide when the function h is a polynomial.

1. Introduction

We focus here on long-range dependence and on its impact on central, or moreprecisely, non-central limit theorems. Long-range dependence, also called “long-memory” or “strong dependence”, occurs in a stationary time series when the co-variances of that series tend to zero like a power function but so slowly that theirsums diverge. Such a behavior is often observed in economics, telecommunicationsand hydrology and was of great interest to Benoıt Mandelbrot. Many of his articleson the subject have been collected in his book [Man02].

The notion of long-range dependence is closely related to self-similarity. Self-similarity refers to invariance in distribution under a suitable change of scale. Moreprecisely, the process (Z(t), t ≥ 0) is self-similar with parameter H if (Z(at), t ≥ 0)has the same finite-dimensional distributions as (aHZ(t), t ≥ 0), for all non negativea. For instance, Brownian motion is self-similar with parameter H = 1/2. In suchan example the increments Z(t + 1) − Z(t) are stationary and independent overdisjoint intervals. But now consider standard fractional Brownian motion. It isself-similar with parameter 0 < H < 1, satisfies E[Z(t)] = 0, E[Z(t)2] = 1 and hasstationary increments. This last fact, together with self-similarity, implies that itscovariance function equals

(1.1) E[Z(t1)Z(t2)] = {|t1|2H + |t2|2H − |t1 − t2|2H}/2, t1, t2 ≥ 0.

Observe that if H = 1/2, then (1.1) reduces to E[Z(t1)Z(t2)] = min(t1, t2), fort1, t2 ≥ 0, which is the covariance of Brownian motion. If H �= 1/2, however,the increments of standard fractional Brownian motion, while stationary, are notindependent anymore. In fact, when 1/2 < H < 1, they have the long-range

2010 Mathematics Subject Classification. Primary 60G18, 62M10.Key words and phrases. Long-range dependence, Gaussian processes, linear processes.


289

290 CELINE LEVY-LEDUC AND MURAD S. TAQQU

dependence property. Indeed,

E[{Z(t + 1) − Z(t)}{Z(s + t + 1) − Z(s + t)}] ∼ H(2H − 1)s2H−2 ,

as s tends to infinity. Since H > 1/2, the sum of these covariances diverges.We will consider Gaussian processes converging to fractional Brownian motion

as well as linear processes which may be non-Gaussian. A linear process (Xi)i∈Z isdefined as

(1.2) Xi =∑j≥1

ajεi−j , i ∈ Z ,

Here the εi’s are “innovations”. These are zero-mean independent and identically(i.i.d.) random variables with at least finite second moments. The coefficients aj in(1.2) are such that

∑j≥1 a

2j < ∞, which ensures that E(X2

i ) < ∞. By choosing the

aj ’s judiciously, we can construct linear processes (Xi) with long-range dependence

such that their partial sums∑[Nt]

i=1 Xi, 0 ≤ t ≤ 1, suitably normalized, converge tofractional Brownian motion as N tends to infinity.

In this paper, we focus on processes (h(Xi))i≥1 which are non-linear functionalsof Gaussian or linear processes with long-range dependence and are interested in the

asymptotic behavior of their partial sums∑N

i=1 h(Xi). We will study this behaviorin various settings. Each setting involves a decomposition and a notion of “rank”.Our goal is to highlight the connections between the different methodologies.

Because h is in general a non-linear function, the limits are typically non-

Gaussian. The limits are called Hermite process {Z(m)D (t) , t ≥ 0} which are defined

in terms of multiple Wiener-Ito integrals as follows:(1.3)

Z(m)D (t) = am,D

∫x1<···<xm

⎧⎨⎩∫ t

0

m∏j=1

(s− xj)−(1+D)/2+ ds

⎫⎬⎭dB(x1) . . .dB(xm) ,

where 0 < D < 1/m, am,D is a constant, {B(x)}x∈R denotes the standard Brownianmotion, and

(u)+ =

{u, if u ≥ 0;

0, if u < 0,

is the ”positive part” function. Physically, Z(m)D (t) is an aggregation of products

of independent Gaussian noises with power weights. The multiple integrals arewell-defined because∫

x1<···<xm

∣∣∣∣∣∣∫ t

0

m∏j=1

(s− xj)−(1+D)/2+ ds

∣∣∣∣∣∣2

dx1 . . .dxm < ∞ .

The representation (1.3) is called a time-domain representation.

There are other equivalent ones. There is the spectral representation of Z(m)D (t),

namely

bm,D

∫λ1<···<λm

ei(λ1+···+λm)t − 1

i(λ1 + · · · + λm)

m∏j=1

|λj |−(1−D)/2dB(λ1) . . .dB(λm) , t ≥ 0 ,

LONG-RANGE DEPENDENCE AND THE RANK OF DECOMPOSITIONS 291

where bm,D is a constant and {B(λ)}λ∈R denotes a complex Brownian motion.There is the positive half-time representation

cm,D

∫0<x1<···<xm<t

⎧⎨⎩∫ t

0

m∏j=1

x−(1−D)/2j (1 − sxj)

−(1+D)/2

⎫⎬⎭dB(x1) . . .dB(xm) ,

t ≥ 0 ,

where cm,D is a constant and {B(x)}x∈R is a standard Brownian motion. There isfinally the finite interval representation

dm,D

∫0<x1<···<xm<t

⎧⎨⎩

m∏j=1

x−(1−D)/2j

∫ t

0sm(1−D)/2

m∏j=1

(s− xj)−(1+D)/2+ ds

⎫⎬⎭ dB(x1) . . . dB(xm) ,

where dm,D is a constant and {B(x)}x∈R is a standard Brownian motion. For moredetails see [PT10, Theorem 1.1].

The Hermite processes have interesting properties: they have mean zero, finitemoments of all order, have stationary increments and are self-similar in the sensethat for all positive a,

{Z(m)D (at)}t≥0

d= {aHZ

(m)D (t)}t≥0 ,

where

H = 1 − mD

2,

and hence depends on D and m.The paper is organized as follows. In Section 2, we state conditions for∑[Nt]

i=1 h(Xi), suitably normalized, to converge to Z(m)D (t) when (Xi)i≥1 is a station-

ary mean-zero long-range dependent Gaussian process. In Section 3, we considerthe case where (Xi)i≥1 is not a Gaussian process anymore but is a long-range de-pendent linear process and h is an entire function as done in [Sur82]. In Section4, (Xi)i≥1 is still assumed to be a long-range dependent linear process but the reg-ularity assumptions on h are somewhat alleviated as done in [HH97]. In Section5, we consider the particular case where h is a polynomial and show that all rankscoincide. Finally, in Section 6, we illustrate the methods by providing sketches ofproofs.

2. The Gaussian case

We suppose here that, the underlying process (Xi)i≥1 satisfies the followingassumption:

(A1) (Xi)i≥1 is a stationary mean-zero Gaussian process with covariances ρ(k)= E(X1Xk+1) satisfying:

ρ(0) = 1 and ρ(k) = k−DL(k), 0 < D < 1 ,

where L is slowly varying at infinity and is positive for large k.

Recall that a slowly varying function L(x), x > 0 is such that L(xt)/L(x) → 1, asx → ∞ for any t > 0. Constants and logarithms are examples of slowly varyingfunctions.


If h is such that E[h(X1)2] < ∞, the idea is to use the expansion of h in the

basis of Hermite polynomials, that is:

h(x) =∑k≥0

(J(k)/k!)Hk(x) ,

where Hk is the kth Hermite polynomial with leading coefficient equal to 1, that isH0(x) = 1, H1(x) = x, H2(x) = x2 − 1, H3(x) = x3 − 3x, . . . and

J(k) = E[h(X1)Hk(X1)].

Definition 2.1. We shall say that h is of Hermite rank m ≥ 1 if m is thesmallest positive integer such that

(2.1) J(m) = E[h(X)Hm(X)] �= 0 ,

where X is a standard Gaussian random variable. The corresponding rank coeffi-cient is J(m).

Suppose that not only (Xi)i≥1 but also h(Xi)i≥1 is long-range dependent whichhappens if 0 < D < 1/m. Then by the reduction theorem of [Taq75], the leading

term of∑N

i=1 h(Xi) properly normalized is the first term of the decomposition ofh in the Hermite polynomials basis, namely

J(m)

m!

N∑i=1

Hm(Xi) .

Moreover, this leading term, properly normalized, converges in distribution to

Z(m)D (1), where {Z(m)

D (t) , t ≥ 0} is the Hermite process of order m, evaluatedat time t = 1. For this last step, one needs to show that

(2.2)

N∑i=1

Hm(Xi)

m!/Var

( N∑i=1

Hm(Xi)/m!)1/2

converges in distribution to Z(m)D (1), as N → ∞. To gain some insight, note

that not only the limit Z(m)D (1) is represented by a multiple Wiener-Ito integral of

order m (see (1.3)), but also the summands Hm(Xi). This is because the Gaussiansequence Xi, i ≥ 1, can be expressed as

Xi =

∫R

ψi(x)dB(x),

∫R

ψi(x)ψj(x)dx = E(XiXj) , i, j ≥ 1 ,

with E(X2i ) = 1, i ≥ 1. Then one has (see Proposition 8.1.2 of [PT11] or Theorem

9.6.9 in [Kuo06]),

(2.3)Hm(Xi)

m!=

∫x1<x2<···<xm

ψi(x1) . . . ψi(xm)dB(x1) . . .dB(xm) ,

sometimes written

Hm(Xi) =

∫ ′

Rm

ψi(x1) . . . ψi(xm)dB(x1) . . .dB(xm) ,

where the prime indicates that one does not integrate over the diagonals.Here is the precise result (see [Taq75], [DM79] and [Taq79].)


Theorem 2.2. Assume that h is such that E[h(X1)2] < ∞ and that m is the

smallest integer greater than 1 such that J(m) = E[h(X1)Hm(X1)] �= 0, where Hm

denotes the mth Hermite polynomial. Assume also that Assumption (A1) holds withD in (0, 1/m). Then

(2.4)1

σN,m

N∑i=1

{h(Xi) − E[h(Xi)]

}d−→ J(m)Z

(m)D (1) , N → ∞ ,

where

(2.5) σ2N,m = Var

( N∑i=1

Hm(Xi)/m!)∼ 2N2−mDLm(N)

m!(1 −mD)(2 −mD), N → ∞ ,

and {Z(m)D (t)}t∈R is the Hermite process of order m and parameter D defined by

( 1.3) where

am,D =

[m!(1 −mD/2)(1 −mD)

{∫ ∞

0

(x + x2)−(1+D)/2dx

}−m]1/2

,

ensures that E[Z(m)D (1)2] = 1.

A sketch of proof of Theorem 2.2 is given in Section 6.

3. The linear case: Surgailis approach

Suppose now that (Xi) is a linear process. Thus, replace Assumption (A1) bythe following assumption.

(A2) (Xi) is defined by

(3.1) Xi =∑j≥1

ajεi−j , i ∈ Z ,

where the innovations εi’s are zero-mean i.i.d. random variables havingat least finite second moments and the aj ’s are such that

∑j≥1 a

2j < ∞.

The aj ’s are assumed to be such that

(3.2) aj = j−βL(j) ,

where β ∈ (1/2, 1) and L is a slowly varying function at infinity.

Note that under Assumption (A2),

E(X1Xk+1) ∼ CL(k)2k1−2β , as k → ∞ ,

where C is a positive constant.In this situation, [Sur82] proposes a methodology for studying the limiting

behavior of∑N

i=1 h(Xi) in the case where h is an entire function, that is, h(z) =∑k≥0 ckz

k, z ∈ C. The idea is to prove that, this time, the leading term is

E[h(m)(X1)]

m!

N∑i=1

Xmi ,

where the rank m is the exponent rank of h.


Definition 3.1. We shall say that h is of exponent rank m ≥ 1 if m is thesmallest integer such that

(3.3) E[h(m)(X1)] �= 0 ,

where h(m) denotes the mth derivative of h. The corresponding rank coefficient isE[h(m)(X1)].

In view of the definition (3.1) of a linear process, the idea is to show that∑Ni=1 h(Xi) has the same asymptotic behavior as

E[h(m)(X1)]

m!

N∑i=1

Xmi =

E[h(m)(X1)]

m!

N∑i=1

(∑j≥1

ajεi−j)m ,

which in turns has the same asymptotic behavior as E[h(m)(X1)]YN,m, where

(3.4) YN,m =N∑

n=1

∑1≤j1<j2<···<jm

m∏s=1

ajsεn−js .

One has then finally to prove that YN,m/[Var(YN,m)]1/2 converges to the Wiener-Ito multiple integral defined in (1.3). In view of (2.3), this is not too different fromfocusing on (2.2). Observe, however, that in (3.4) we are dealing with a discreteconvolution and that the ε’s are not assumed normal.

Here is the precise result of [Sur82].

Theorem 3.2. Let h be an entire function defined by h(z) =∑

k≥0 ckzk, z ∈ C,

such that ∑k,j≥0

|ck||cj |(k!j!)222(k+j)μk+j < ∞ ,

where μk = E(|ε1|k), k ≥ 0 and let m be the smallest integer larger than 1 such that

E[h(m)(X1)] �= 0 .

Then, under Assumption (A2), with 0 < D = 2β − 1 < 1/m, β being defined in( 3.2),

s−1N,m

N∑n=1

{h(Xn) − E[h(Xn)]}

has the same limit in distribution, as N tends to infinity, as

E[h(m)(X1)]YN,m

sN,m,

where YN,m is defined in ( 3.4) and s2N,m = Var(YN,m). Moreover, that limit is

E[h(m)(X1)]Z(m)D (1) ,

where {Z(m)D (t)}t∈R is the Hermite process of order m and parameter D defined in

( 1.3).

A sketch of proof of Theorem 3.2 is given in Section 6. A related approach,focusing on Appell polynomials, can be found in [AT87].


4. The linear case: Ho and Hsing approach

We need to introduce first some notation. Let F be the distribution of thelinear process Xn =

∑i≥1 aiεn−i and Fj the distribution of

Xn,j =∑

1≤i≤j

aiεn−i,

for j ≥ 1, with the convention: Xn,0 = 0. Let

(4.1) hj(x) =

∫R

h(x + y)dFj(y) , h∞(x) =

∫R

h(x + y)dF (y) ,

and

(4.2) h(r)j (x) =

dr

dxr

∫R

h(x + y)dFj(y) , h(r)∞ (x) =

dr

dxr

∫R

h(x + y)dF (y) .

If the rth derivative h(r)j of hj exists, define

h(r)j,λ(x) = sup

|y|≤λ

|h(r)j (x + y)| , λ ≥ 0 .

We shall say that h satisfies the Condition C(r, j, λ) if

(1) h(r)j (x) exists for all x and h

(r)j is continuous.

(2) For all x ∈ R,

supI⊂{1,2,... }

E[{h(r)j,λ(x +

∑i∈I

aiεi)}4] < ∞ ,

where the supremum is taken over all subsets I of {1, 2, . . . }.Let us comment on Condition C(r, j, λ). It is satisfied if the rth derivative of h

is bounded and continuous, in which case one can take any j. Moreover, if h is anypolynomial, then C(r, j, λ) holds provided that εi has finite moments of sufficientlyhigh order.

The novelty here is that C(r, j, λ) can hold without h being smooth. An impor-tant example is the indicator function. If h(x) = �{x≤u}, for some fixed u, let usprove that h satisfies C(r, 1, λ) for all positive λ as soon as the probability densityfunction g of ε1 has a continuous and integrable rth derivative.

Since Xn,1 = a1εn−1, we have

h1(x) =

∫R

h(x + y)dF1(y) =

∫R

h(x + a1y1)g(y1)dy1 = a−11

∫R

h(z)g

(z − x

a1

)dz .

Note that,

∂r

∂xr

{h(z)g

(z − x

a1

)}=

(−1)r

ar1h(z)g(r)

(z − x

a1

).

Since, by assumption,∫R|g(r)(y)|dy < ∞, we get

h(r)1 (x) =

(−1)r

ar+11

∫R

h(z)g(r)(z − x

a1

)dz =

(−1)r

ar1

∫R

h(x + a1y1)g(r)(y1)dy1 .

Moreover, h(r)1 is a continuous function since g(r) is assumed to be a continuous

function. This gives (1) of C(r, 1, λ). Let us now check (2) of C(r, 1, λ). For all


subset I of {1, 2, . . . }, we have

E[{

sup|y|≤λ

∣∣∣h(r)1

(x +

∑i∈I

aiεi + y)∣∣∣}4]

=1

a4r1E[{

sup|y|≤λ

∣∣∣ ∫R

h(x +

∑i∈I

aiεi + y + a1y1

)g(r)(y1)dy1

∣∣∣}4],

which is bounded by a−4r1 (

∫R|g(r)(y1)|dy1)4, which is finite since

∫R|g(r)(y)|dy < ∞

and thus ensures that h(x) = �{x≤u} satisfies Condition C(r, 1, λ).Observe that the indicator function h(x) = �{x≤u} is allowed in the Gaussian

case but not in the situation considered by [Sur82]. As we have just seen, it isallowed in the methodology proposed by [HH97].

The idea here is to use a mixingale decomposition as explained in Section 6and to prove that the leading term is once again YN,m, defined in (3.4), where herem is the power rank of h defined as follows.

Definition 4.1. We shall say that h is of power rank m ≥ 1 if it is the smallestinteger such that

(4.3) h(m)∞ (0) �= 0 .

The corresponding rank coefficient is h(m)∞ (0).

The idea is once again to prove that Yn,m properly normalized converges tothe Wiener-Ito multiple integral defined in (1.3). Here is the precise result due to[HH97].

Theorem 4.2. Assume that Assumption (A2) holds and that h is of power

rank m, that is, m is the smallest positive integer such that h(m)∞ (0) �= 0. Assume

also that for some j and λ, condition C(r, j, λ) holds for r = 0, . . . ,m + 2 andE[h(X1)

2] < ∞. If

0 < D = (2β − 1) < 1/m,

β being defined in ( 3.2), and E(|ε1|2m∨8) < ∞ then

s−1N,m

N∑i=1

{h(Xi) − E[h(Xi)]}

has the same limit in distribution, as N tends to infinity, as

h(m)∞ (0)

YN,m

sN,m,

where YN,m is defined in ( 3.4) and s2N,m = Var(YN,m). Moreover, that limit is

h(m)∞ (0)Z

(m)D (1), where (Z

(m)D (t))t∈R is defined in ( 1.3).

A sketch of proof of Theorem 4.2 is given in Section 6.

Remark 4.3. The second parts of Theorems 3.2 and 4.2 are similar. This

shows that the key is to reduce the original∑N

n=1 h(Xn) to YN,m. It is YN,m whichconverges to the limit after suitable normalization.


5. Application to the polynomial case

In the case where h is a polynomial, we prove in the following proposition thatthe three definitions of ranks introduced previously coincide.

Proposition 5.1. If h is a polynomial defined by h(x) =∑K

k=0 ckxk then the

three rank coefficients

J(m), E[h(m)(X1)] and h(m)∞ (0)

defined in ( 2.1), ( 3.3) and ( 4.3) respectively, are identical and equal to

(5.1)K∑

k=m

ck k(k − 1) . . . (k −m + 1)E[Xk−m1 ] ,

where m is the corresponding rank.

Proof of Proposition 5.1. Let

h(X) =

K∑k=0

ckXk

denote the polynomial.Suppose first that X is a standard Gaussian random variable. The mth coeffi-

cient of the expansion in Hermite polynomials of h is given by

J(m) =

K∑k=0

ckE[XkHm(X)].

We first show that J(m) is equal to (5.1). Using the relation between powers andHermite polynomials [Kuo06, p. 159], we have, for all x ∈ R and k ∈ N,

(5.2) xk =

[k/2]∑�=0

k!

(k − 2�)!�!2�Hk−2�(x) ,

where [y] denotes the integer part of y. Thus, by orthogonality of the Hk’s in L2

equipped with the N (0, 1) Gaussian measure,

E[XkHm(X)] =

[k/2]∑�=0

k!

(k − 2�)!�!2�E[Hk−2�(X)Hm(X)]

=k!

m!{(k −m)/2}!2(k−m)/2m!

=k!

(k −m)!

(k −m)!

{(k −m)/2}!2(k−m)/2

= k(k − 1) . . . (k −m + 1)E[Xk−m] ,

when k ≥ m and k−m is even, otherwise, E[XkHm(X)] = 0. We used here the factthat when X is a standard Gaussian random variable, E[X2p+1] = 0 and E[X2p] =(2p)!/(p!2p). Now ckE[XkHm(X)] vanishes if either ck = 0 or if E[Xk−m] = 0.The Hermite rank m is thus equal to the smallest k such that both ck �= 0 andE[Xk−m] �= 0. The corresponding coefficient J(m) is thus given by (5.1).


We now suppose that (Xi)i≥1 is a linear process and consider E[h(m)(X1)]defined in (3.3). Observe that

h(m)(x) =

K∑k=m

ck k(k − 1) . . . (k −m + 1)xk−m,

for all x in R. Thus,

(5.3) E[h(m)(X1)] =K∑

k=m

ck k(k − 1) . . . (k −m + 1)E[Xk−m1 ] ,

which is the same as (5.1).

We finally consider h(m)∞ (0) defined in (4.3). Since

h∞(x) =K∑

k=0

ckE[(x + X1)k] =

K∑k=0

ck

k∑j=0

(k

j

)xj E[Xk−j

1 ] ,

we get

(5.4) h(m)∞ (0) =

K∑k=0

ck

(k

m

)m!E[Xk−m

1 ] =

K∑k=m

ck k(k−1) . . . (k−m+1)E[Xk−m1 ] ,

which is the same as (5.1) and (5.3). �

To understand the significance of the proposition, let h(X) =∑K

k=1 ckXk be a

polynomial and set for convenience

C(k, j) = k(k − 1) . . . (k − j + 1) , k ≥ j .

As noted, the rank of h(X) is m if

(5.5)

K∑k=j

ck C(k, j)E[Xk−j] = 0 , j = 1, . . . ,m− 1 ,

and

em =

K∑k=m

ck C(k,m)E[Xk−m] �= 0 .

In this case,∑N

i=1 h(Xi) behaves asymptotically like emYN,m, where YN,m is definedin (3.4) and where, here, em stands for any of the three rank coefficients J(m),

E[h(m)(X1)] and h(m)∞ (0).

Example 5.2. Suppose h(X) = ckXk, ck �= 0, k ≥ 1. If k is odd, then

E[Xk−1] �= 0 and the rank is m = 1. The corresponding rank coefficient isck C(k, 1)E[Xk−1]. If k ≥ 2 is even, then E[Xk−1] = 0, E[Xk−1] �= 0 and thusthe corresponding rank is m = 2 and the rank coefficient is ck C(k, 2)E[Xk−2].

Example 5.3. Suppose h(X) = ckXk + ck+1X

k+1, ck �= 0, ck+1 �= 0, k ≥ 1.Since

ck C(k, 1)E[Xk−1] + ck+1 C(k + 1, 1)E[Xk+1−1] �= 0 ,

the rank is m = 1 and the rank coefficient is ck+1 C(k+ 1, 1)E[Xk] if k is even andck C(k, 1)E[Xk−1] if k is odd.


Example 5.4. Suppose h(X) = X3 − 3X. For j = 1, (5.5) equals

C(3, 1)E[X2] − 3C(1, 1) = C(3, 1) − 3C(1, 1) = 3 − 3 = 0 ,

and for j = 2, it equals C(3, 2)E[X3−2] = 0. For j = 3, it equals

C(3, 3)E[X3−3] = C(3, 3) = 3 × 2 = 6 .

Hence the rank is m = 3 and the rank coefficient is 6. One can arrive to thisconclusion immediately by supposing X ∼ N (0, 1) and noting that

J(j) = E[h(X)Hj(X)] = E[(X3 − 3X)Hj(X)] = E[H3(X)Hj(X)]

equals 0 if j �= 3 and equals 3! = 6 if j = 3.

6. Sketches of proofs of Theorems 2.2, 3.2 and 4.2

The proof for each theorem involves a decomposition of the function h and hastwo parts. The first involves showing that the only contribution to the limit is dueto the term of the decomposition with index m, where m is the rank. The secondpart consists in showing that the term with index m converges in distribution to

Z(m)D (1). In fact convergence also holds for the finite-dimensional distributions as

well as in function space. We will focus here on the first part of each proof.

Sketch of proof of Theorem 2.2. The first part of the proof consists in

showing that the remainder h�(x) = h(x) − J(m)m! Hm(x) is negligible, namely that

N−2+mDL−m(N) Var

(N∑i=1

h�(Xi)

)→ 0 , as N → ∞ .

This implies that N−1+mD/2L−m/2(N)∑N

i=1 h�(Xi) = oP (1) and that the conver-

gence in distribution of

N−1+mD/2L−m/2(N)N∑i=1

h(Xi)

reduces to the convergence in distribution of

N−1+mD/2L−m/2(N)

N∑i=1

(J(m)/m!)Hm(Xi).

Using the Mehler formula, namely

E[Hp(Xi)Hq(Xj)] = δp,qJ(p)2

p!(ρ(i− j))p ,

where δp,q is 1 if p = q and 0 if p �= q, we get that

Var

(N∑i=1

h�(Xi)

)≤

N∑i=1

N∑j=1

⎛⎝∑p≥1

J(p)2

p!

⎞⎠ |ρ(i− j)|m+1

= E[h2(X1)]

N∑i=1

N∑j=1

|ρ(i− j)|m+1

≤ NE[h2(X1)]∑

|k|<N

|ρ(k)|m+1 .


Taking into account the normalization, we have(6.1)

N−2+mDL−m(N) Var

(N∑i=1

h�(Xi)

)≤ N−1+mDL−m(N)E[h2(X1)]

∑|k|<N

|ρ(k)|m+1 .

There are two possibilities: either the series∑

k∈Z|ρ(k)|m+1 is convergent and in

this case the expression (6.1) tends to zero since D < 1/m or the series∑k∈Z

|ρ(k)|m+1 is divergent. In the latter case,∑

|k|<N |ρ(k)|m+1 is of order

N−(m+1)D+1 and thus again, the expression of (6.1) tends to zero since D > 0.To conclude the proof, it remains to show that, as N → ∞,

J(m)

σN,m

N∑i=1

Hm(Xi)/m!d−→ J(m)Z

(m)D (1) ,

where σN,m is defined in (2.5). This is done in [DM79], [Taq79] and [Maj81]. �

Sketch of proof of Theorem 3.2. Let (Xj) be defined in (3.1). The ideaof the proof consists of decomposing

(6.2) (Xj)k =

∑p1,p2,...,pk≥1

ap1ap2

. . . apkεj−p1

εj−p2. . . εj−pk

in terms of the cardinality |{p1, . . . , pk}| of the set {p1, . . . , pk}. When |{p1, . . . , pk}|= k, the term ap1

ap2. . . apk

εj−p1εj−p2

. . . εj−pkis not modified. When |{p1, . . . , pk}|

< k and for instance equal to k − 1 with p1 = p2, it is split in two parts:

ap1ap2

. . . apk(εj−p1

εj−p2. . . εj−pk

) = a2p1ap3

. . . apk(ε2p1−jεj−p3

. . . εj−pk)(6.3a)

= a2p1ap3

. . . apk(μ2 εj−p3

. . . εj−pk) + a2p1

ap3. . . apk

(ηp1(2) εj−p3

. . . εj−pk) ,

(6.3b)

where μ� = E(ε�1) and ηp(�) = ε�p − μ�. One then shows that the term with ηp1(2)

is negligible.Let us focus on the first term in 6.3b and consider the general case. The idea

is to replace (Xj)k by

(6.4) (Zj)∗k =

k∑�=0

(k

�

)∑(p)�

ap1εj−p1

. . . ap�εj−p�

∑(V )(k−�)

∑(q)r

av1q1μv1 . . . avrqrμvr ,

where the summation over (p)� corresponds to the summation over the sets {p1, . . . , p�}of cardinality |{p1, . . . , p�}| = �, that is, over p1, . . . , p� which take different values.In the perspective of (6.3b), one should take the sets {q1, . . . , qr} and {p1, . . . , p�}disjoint, but we shall not impose this restriction in (6.4). The sum

∑(V )(k−�) is

taken over all partitions of the set {1, . . . , k− �} of cardinality v1, . . . , vr such thatvi ≥ 2, for all i. By convention, this sum equals 1 for (V )(0). In contrast to (Xj)

k,the notation (Zj)

∗k is a shorthand for the r.h.s of (6.4) and does not mean Zj tothe power k. The difference between (Xj)

k and (Zj)∗k is that when there is an

ε�p−j with � > 1 in (6.2), it is replaced by μ� = E(ε�1) in (6.4). Observe also that

the summands of (Zj)∗k with � > 0 have zero mean. This ensures that

E[(Xj)k] = E[(Zj)

∗k] =∑

(V )(k)

∑(q)r

av1q1μv1 . . . avrqrμvr ,


for k ≥ 0 and hence for 0 ≤ s ≤ k,

(6.5) E[(Xj)k−s] = E[(Zj)

∗(k−s)] =∑

(V )(k−s)

∑(q)r

av1q1μv1 . . . avrqrμvr .

Let us now define formally h∗(Zj) as∑

s≥0 cs(Zj)∗s, where (Zj)

∗s is given in (6.4)and prove that

(6.6) h∗(Zj) =∑s≥0

cs(Zj)∗s =

∑s≥0

(es/s!)∑(p)s

ap1εj−p1

. . . apsεj−ps

,

where here

es = E[h(s)(Xj)] .

Observe that by (5.3) and (6.5),

ess!

=E[h(s)(Xj)]

s!=

1

s!

∑k≥s

ck k(k − 1) . . . (k − s + 1)E[(Xj)k−s]

=∑k≥s

ck

(k

s

)E[(Xj)

k−s] =∑k≥s

ck

(k

s

)E[(Zj)

∗(k−s)] .

Using this and again (6.5), we note that the last term of (6.6) can be expressed as∑s≥0

ess!

∑(p)s

ap1εj−p1

. . . apsεj−ps

=∑s≥0

⎛⎝∑k≥s

ck

(k

s

)E[(Zj)

∗(k−s)]

⎞⎠∑(p)s

ap1εj−p1

. . . apsεj−ps

=∑s≥0

⎛⎝∑k≥s

ck

(k

s

) ∑(V )(k−s)

∑(q)r

av1q1μv1 . . . avrqrμvr

⎞⎠∑(p)s

ap1εj−p1

. . . apsεj−ps

=∑k≥0

ck

⎛⎝ k∑s=0

(k

s

)∑(p)s

ap1εj−p1

. . . apsεj−ps

∑(V )(k−s)

∑(q)r

av1q1μv1 . . . avrqrμvr

⎞⎠=∑k≥0

ck(Zj)∗k ,

by (6.4), hence proving (6.6).

The proof of [Sur82] consists in showing that∑N

j=1 h(Xj) can be replaced by∑Nj=1 h

∗(Zj) and that the leading term in∑N

j=1 h∗(Zj) is the term corresponding

to s = m in∑N

j=1 h(Zj), that is by (6.6) and (3.4),

emYN,m = (em/m!)N∑j=1

∑(p)m

ap1εj−p1

. . . apmεj−pm

.

More precisely, it is proved in Lemma 2 and 3 of [Sur82] that

E[{ N∑

j=1

(h(Xj) − h∗(Zj)

)}2]≤ CN ,


where C is a positive constant and that

E[{ N∑

j=1

∑�≥m+1

∑(p)�

(e�/�!)ap1εj−p1

. . . ap�εj−p�

}2]= E

[{ ∑�≥m+1

e�YN,�

}2]= o(L(N)2mN2−mD) ,

as N tends to infinity.To conclude, it remains to show that, as N → ∞,

emYN,m

sN,m

d−→ emZ(m)D (1) ,

where s2N,m = Var(YN,m). This is done in Lemma 5 of [Sur82]. The basic ideais to express YN,m as a discrete multiple stochastic integral and use the fact that

N−1/2∑N

k=1 εk converges to a normal distribution. �

Sketch of proof of Theorem 4.2. The idea is to condition on the σ-fields

Fk = σ(εi; i < k),

using the telescoping expression

(6.7) h(Xn) − E[h(Xn)] =∑j≥1

{E[h(Xn)|Fn−j+1] − E[h(Xn)|Fn−j ]} ,

since the extreme summands are such that E[h(Xn)|Fn] = h(Xn) and E[h(Xn)|F−∞]= E[h(Xn)]. Now write

E[h(Xn)|Fn−j ] = E[h(Xn,j + Xn,j)|Fn−j ] ,

with

Xn,j =

j∑i=1

aiεn−i and Xn,j =∑i>j

aiεn−i.

Since Xn,j is Fn−j-measurable and Xn,j is independent of Fn−j ,

(6.8) E[h(Xn)|Fn−j ] =

∫R

h(x + Xn,j)dFj(x) = hj(Xn,j) ,


where Fj is the distribution of Xn,j and hj(y) = E[h(Xn,j + y)]. Using (6.7) and(6.8), we get that

N∑n=1

{h(Xn) − E[h(Xn)]} =N∑

n=1

∑j1≥1

[hj1−1(Xn,j1−1) − hj1(Xn,j1)]

=N∑

n=1

∑j1≥1

(Xn,j1−1 − Xn,j1)h(1)j1

(Xn,j1)

+[hj1−1(Xn,j1−1) − hj1(Xn,j1) − (Xn,j1−1 − Xn,j1)h

(1)j1

(Xn,j1)]

=

N∑n=1

∑j1≥1

aj1εn−j1h(1)j1

(Xn,j1)

+[hj1−1(Xn,j1−1) − hj1(Xn,j1) − aj1εn−j1h

(1)j1

(Xn,j1)]

(6.9)

≈N∑

n=1

∑j1≥1

aj1εn−j1h(1)j1

(Xn,j1) ,(6.10)

after proving that the terms in brackets can be neglected. We have introduced

the ε’s. We need now to introduce h(1)∞ (0), . . . , h

(m)∞ (0). To do so, we express the

summands in the remaining term (6.10) as

(6.11) aj1εn−j1h(1)j1

(Xn,j1) = aj1εn−j1h(1)∞ (0) + aj1εn−j1 [h

(1)j1

(Xn,j1) − h(1)∞ (0)]

= aj1εn−j1h(1)∞ (0) + aj1εn−j1

∑j2≥j1+1

[h(1)j2−1(Xn,j2−1) − h

(1)j2

(Xn,j2)] .

Focusing on the term in brackets, we write as before (see (6.9)),

(6.12) h(1)j2−1(Xn,j2−1) − h

(1)j2

(Xn,j2)

= aj2εn−j2h(2)j2

(Xn,j2) +[h(1)j2−1(Xn,j2−1) − h

(1)j2

(Xn,j2) − aj2εn−j2h(2)j2

(Xn,j2)]

≈ aj2εn−j2h(2)j2

(Xn,j2) = aj2εn−j2h(2)∞ (0) + aj2εn−j2 [h

(2)j2

(Xn,j2) − h(2)∞ (0)] ,

where in that last equality, we proceeded as in (6.11). Relations (6.11) and (6.12)yield

aj1εn−j1h(1)j1

(Xn,j1) = aj1εn−j1h(1)∞ (0)

+ aj1εn−j1

∑j2≥j1+1

{aj2εn−j2h

(2)∞ (0) + aj2εn−j2 [h

(2)j2

(Xn,j2) − h(2)∞ (0)]

}.

Thus, we get

(6.13)

N∑n=1

{h(Xn) − E[h(Xn)]} = YN,1h(1)∞ (0) + YN,2h

(2)∞ (0)

+N∑

n=1

∑j2>j1≥1

aj1εn−j1aj2εn−j2 [h(2)j2

(Xn,j2) − h(2)∞ (0)] ,


where

YN,r =N∑

n=1

∑1≤j1<j2<···<jr

r∏s=1

ajsεn−js .

Iterating, we get

(6.14)N∑

n=1

{h(Xn)−E[h(Xn)]} = YN,1h(1)∞ (0)+YN,2h

(2)∞ (0)+ · · ·+YN,mh(m)

∞ (0)+RN ,

where RN can be shown to be a negligible remainder term. Hence, the first term of

the expansion (6.14) is given by YN,mh(m)∞ (0), where m is the power rank, namely

the first k such that h(k)∞ (0) �= 0. This is the same YN,m as in (3.4). One concludes

by applying the last part of Theorem 3.2. �

7. Conclusion

We considered a stationary sequence (Xi) which is either

• Gaussian with long-range dependence• a linear process with long-range dependence,

and focused on the convergence of

N∑i=1

(h(Xi) − E[h(Xi)]

)properly normalized to that of a Hermite process Z(t), at t = 1, when (h(Xi))i≥1

itself is long-range dependent. We

• described the type of functions h considered in the literature,• showed that their notions of rank coincide for h polynomial,• and indicated heuristically why we expect the limit to be a Hermite pro-

cess.

Acknowledgments

The authors would like to thank Eric Moulines for interesting discussions andVladas Pipiras for his comments. Murad S. Taqqu was supported by the NationalScience Foundation grant DMS-1007616 at Boston University. He would like tothank Telecom ParisTech in Paris for their hospitality.

References

[AT87] Florin Avram and Murad S. Taqqu, Noncentral limit theorems and Appell polynomials,Ann. Probab. 15 (1987), no. 2, 767–775. MR885142 (88i:60058)

[DM79] R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear function-als of Gaussian fields, Z. Wahrsch. Verw. Gebiete 50 (1979), no. 1, 27–52, DOI10.1007/BF00535673. MR550122 (81i:60019)

[HH97] Hwai-Chung Ho and Tailen Hsing, Limit theorems for functionals of moving averages,Ann. Probab. 25 (1997), no. 4, 1636–1669, DOI 10.1214/aop/1023481106. MR1487431(98m:60027)

[Kuo06] Hui-Hsiung Kuo, Introduction to stochastic integration, Universitext, Springer, NewYork, 2006. MR2180429 (2006e:60001)

[Maj81] Peter Major, Multiple Wiener-Ito integrals, Lecture Notes in Mathematics, vol. 849,Springer, Berlin, 1981. With applications to limit theorems. MR611334 (82i:60099)


[Man02] Benoit B. Mandelbrot, Gaussian self-affinity and fractals, Selected Works of Benoit B.Mandelbrot, Springer-Verlag, New York, 2002. Globality, the earth, 1/f noise, and R/S;Selecta (old or new) Volume H; Includes contributions by F. J. Damerau, M. Frame, K.McCamy, J. W. Van Ness, J. R. Wallis, and others. MR1878884 (2003a:01026)

[PT10] Vladas Pipiras and Murad S. Taqqu, Regularization and integral representationsof Hermite processes, Statist. Probab. Lett. 80 (2010), no. 23-24, 2014–2023, DOI10.1016/j.spl.2010.09.008. MR2734275 (2012f:60134)

[PT11] Giovanni Peccati and Murad S. Taqqu, Wiener chaos: moments, cumulants and dia-grams, Bocconi & Springer Series, vol. 1, Springer, Milan, 2011. A survey with computerimplementation; Supplementary material available online. MR2791919 (2012d:60157)

[Sur82] D. Surgailis, Domains of attraction of self-similar multiple integrals, Litovsk. Mat. Sb.22 (1982), no. 3, 185–201 (Russian, with English and Lithuanian summaries). MR684472(84h:60055)

[Taq75] Murad S. Taqqu, Weak convergence to fractional Brownian motion and to the Rosen-blatt process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 287–302.MR0400329 (53 #4164)

[Taq79] Murad S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z.Wahrsch. Verw. Gebiete 50 (1979), no. 1, 53–83, DOI 10.1007/BF00535674. MR550123(81i:60020)

AgroParisTech 16, Rue Claude Bernard F-75231 Paris Cedex 05 France


Department of Mathematics and Statistics, Boston University, 111 Cummington

Street, Boston, Massachusetts 02215



Hitting Probabilities of the Random Covering Sets

Bing Li, Narn-Rueih Shieh, and Yimin Xiao

Abstract. Let E be the Dvoretzky random covering sets on the circle. Byapplying the method of limsup type random fractals, as illustrated in Khosh-nevisan, Peres and Xiao [24], we determine the hitting probability P(E∩G �= ∅)and the packing dimension of the intersection E ∩G, where G is an arbitraryBorel set on the circle.

Introduction

We begin with a brief review on random coverings. Let {ωn}n≥1 be a sequenceof independent random variables on (Ω,B,P) which are uniformly distributed overthe unit interval I = [0, 1). Let {ln}n≥1 be a sequence of positive real numberswhich is decreasing to zero. For every n ≥ 1, denote by In := (ωn, ωn + ln)(mod 1)the random interval whose starting point and lengthe are determined by ωn and lnrespectively. Define the random covering set as

E := lim supn→∞

In = {t ∈ I : t ∈ In for infinitely many n ≥ 1}.

The set E consists of the points which are covered by {In} infinitely often (i.o. forshort). The Borel-Cantelli Lemma implies that the Lebesgue measure of the randomset E is either 1 or 0 almost surely according to the divergence or convergence ofthe series

∑∞n=1 ln.

It was Dvoretzky [5] who called the attention on study of such random sets; heraised the question that under what condition on {ln} one can have

(0.1) [0, 1) = lim supn→∞

In a.s.

In the literature this is referred to as the Dvoretzky covering problem and hadattracted the attention of P. Billard, J.-P. Kahane, B. Mandelbrot, among others,before it was completely solved by L. A. Shepp in 1972. Shepp [31] provided anecessary and sufficient condition for (0.1) to hold, namely

(0.2)∞∑

n=1

1

n2exp

(l1 + · · · + ln

)= ∞.

2010 Mathematics Subject Classification. Primary 60D05, 52A22, 28A78, 28A80.Key words and phrases. Random covering sets, Dvoretzky random covering, hitting proba-

bility, packing dimension, Hausdorff dimension, limsup random fractals.Research supported in part by NSF of China Grant #11201155 and Fundamental Research

Funds for the Central Universities 2011ZM0083.Research supported in part by NSF grant DMS-1006903.

c©2013 American Mathematical Society307

308 BING LI, NARN-RUEIH SHIEH, AND YIMIN XIAO

Since then the topic has been under active development and there have been manyextensions and refinements. We refer to [20, Chapter 11] for a systematic accounton the Dvoretzky covering problem and its higher dimensional extensions, to thesurvey articles [8,22,23] for historical accounts and connections to multiplicativeprocesses, and to [1,4,7–10,14,21] and the references therein for further informa-tion. It should also be mentioned that Jonasson and Steif [16] (see also [15]) haverecently extended the Dvoretzky covering model by including time dynamics (Inthe first variant, they identify I = [0, 1) with the unit circle C and allow the centersof In (n ≥ 1) perform independent Brownian motions on C, each with variance 1.In the second variant, they associate independent Poisson processes with the differ-ent intervals.) The work of Jonasson and Steif [16] has revealed rich structures indynamical random coverings and raised more interesting questions about proper-ties of the dynamical random covering sets, including their fractal dimensions andhitting probabilities.

This paper is concerned with the geometric and potential-theoretic properties ofthe Dvoretzky covering set E = lim sup

n→∞In. It is known that the set E is a.s. dense

in I and is of second category ([20, Chapter 5, Proposition 11]). Thus, the upperbox dimension of the set E is 1 almost surely. Several authors have investigatedthe Hausdorff dimension and other fractal properties of E and/or its complementF∞ = I\E (which is called the uncovered set). For example, Fan and Wu [10]considered the Hausdorff dimension of the set E for the special case ln = a

nγ , where

a > 0 and γ > 1 are constants, they proved that dimH(E) = 1γ a.s., where dimH

denotes Hausdorff dimension. Durand [4] considered a general sequence {ln} with∑∞n=1 ln < ∞ and proved, among other things,

(0.3) dimHE = α and dimPE = 1 a.s.,

where α is defined by

(0.4) α := inf

{s > 0 :

∞∑n=1

lsn < ∞}

= sup

{s > 0 :

∞∑n=1

lsn = ∞}

with the convention that sup ∅ = 0 and inf ∅ = 1.The index α defined in (0.4) is known as the exponent of convergence of the

sequence {ln} and can be calculated by using the following formula

(0.5) α = lim supn→∞

log n

− log ln

(see [27] p.285 or [30] p.26). Besicovitch and Taylor [3] applied the index α (theyalso introduced another index–the lower index for {ln}) to characterize the Haus-dorff measure and Hausdorff dimension of a linear compact set K whose complementforms a sequence of open intervals of lengths {ln}. Hawkes [13] showed that α is theupper box dimension of K, and the lower index of {ln} is the lower box dimensionof K. Kahane [18,19] called α the upper Besicovitch-Taylor index of {ln}. Somerelated indices for {ln} were also discussed in [1] for studying the Carleson problemand covering numbers for the Dvoretzky covering set E.

The following intersection problem is of intrinsic importance in the study ofrandom coverings and other random fractals. For any given set A ⊂ [0, 1), wecan ask whether or not it is a.s. covered infinitely often by {In}. That is, when

HITTING PROBABILITIES OF THE RANDOM COVERING SETS 309

does P(A ⊂ E) = 1 hold? In the case∞∑

n=1ln = ∞, which is opposite to what we

are considering in this paper, the analogous problem for the uncovered set F∞ hasbeen investigated by several authors. For example, Kahane [20] considered the case

ln = βn , 0 < β < 1 and showed that P(A ⊂ E) = 1 (equivalently, P

(A∩F∞ �= ∅

)= 0)

if dimH(A) < β, whilst P(A ⊂ E) = 0 (equivalently, P

(A ∩ F∞ �= ∅

)= 1) if

dimH(A) > β. For a more general case, Hawkes [12] proved that, if the set A

satisfies a regularity condition (which in particular requires dimH(A) = dimP(A)),then P(A ⊂ E) = 1 or 0 according as dimH(A) < τ and dimH(A) > τ , where τ isthe index of {ln} defined by

τ = lim supn→∞

∑ni=1 li

log n.

More precise hitting probability results for the Poisson covering (see Mandelbrot[26]) have been established by using the connection between F∞ and the range of asubordinator; see Fitzsimmons, et al. [11]. However, the problems for determiningthe hitting probabilities of the Dvoretzky random covering set E had never beenstudied.

The purpose of this paper is to study the hitting probabilities of the Dvoretzkycovering set E, as well as fractal dimensions of the intersection E ∩ G, when it isnonempty. Our main result (Theorem 2.1 below) shows that the hitting probabilityP(E ∩ G �= ∅) is determined by dimP(G), the packing dimension of G (see (0.7)below). This is in contrast with the hitting probability results for the random setF∞, where Hausdorff dimension plays the natural role. Theorem 2.1 will allow usto determine the packing dimension of E ∩ G for any analytic set G ⊆ [0, 1) andprovide a refinement (under an extra condition) of (0.3) obtained by Durand [4].

Recall that packing dimension was introduced in the early 1980s by Tricot [33]as follows. For any ε > 0 and any bounded set G ⊂ R, let N(G, ε) be the smallestnumber of balls of radius ε needed to cover G. The upper box dimension of G isdefined as

(0.6) dimM(G) = lim supε→0

logN(G, ε)

− log ε

and the packing dimension of G is defined as

(0.7) dimP(G) = inf

{supn

dimMGn : G ⊂∞⋃

n=1

Gn

},

where the infimum is taken over all countable coverings {Gn} of G. It is well knownthat 0 ≤ dimH(G) ≤ dimP(G) ≤ dimM(G) ≤ 1 for every set G ⊂ R. Similarlyto Hausdorff dimension, packing dimension has been shown to be a useful tool forcharacterizing fractal sets and for studying “roughness” of stochastic processes. Werefer to Falconer [6] and Mattila [28] for further properties of packing dimensionand to Taylor [32] and Xiao [34] for extensive surveys on its applications to randomfractals.

The rest of this paper is organized as follows. In Section 2 we state the mainresults and provide some discussions and examples. The proofs of the theorems aregiven in Section 3 and they rely on the general method on limsup random fractalsin Khoshnevisan, Peres and Xiao [24]. We remark that our argument extends thatin [24] and shows that their Theorems 3.1, 3.3 and corollaries still hold if their


Condition 4 is replaced by a weaker condition. Finally Section 4 contains sometechnical results on the upper Besicovitch-Taylor index. In particular we apply theresults in Lapidus and van Frankenhuysen [25] to show that every sequence {ln}associated to a self-similar string has its upper Besicovitch-Taylor index equal tothe self-similarity dimension and satisfies the condition (C) in this paper.

1. Main results and examples

Throughout this paper we assume∑∞

n=1 ln < ∞. Thus the Lebesgue measureof the Dvoretzky covering set E is 0 almost surely.

Let α be the upper Besicovitch-Taylor index of {ln}. Then by Proposition 3.1below, we have

(1.1) α = lim supk→∞

log2 nk

k,

where log2 is the logarithm in base 2 and nk is defined as

nk = #{n ∈ N : ln ∈ [2−k+1, 2−k+2)

}(k ≥ 2).

Here #A denotes the cardinality of the set A.To state the main results of this paper we will make use of the following con-

dition (C):

(C) There exists an increasing sequence of positive integers {ki} such that

(1.2) limi→∞

ki+1

ki= 1

and

(1.3) limi→∞

log2 nki

ki= α < 1.

Theorem 1.1. Let E be the Dvoretzky covering set associated with the sequence{ln} whose upper Besicovitch-Taylor index is α. If the condition (C) holds, thenfor every analytic set G ⊂ [0, 1), we have

P(E ∩G �= ∅

)=

{0 if dim

P(G) < 1 − α,

1 if dimP(G) > 1 − α.

Remark 1.2. Some remarks are in order.

(i) It is clear that if

(1.4) limk→∞

log2 nk

k= α < 1,

then condition (C) holds. We will give several interesting examples ofsequences {ln} that satisfy (1.4).

(ii) If G is regular in the sense that dimM

(G) = dimM

(G), where dimM

(G)is the lower box dimension of G, which is defined by replacing limsup in(0.6) by liminf, then condition (C) is surplus. This follows from the proofof Theorem 1.1 below, in which we can take N = {ki0 , ki0+1, . . . } for somei0 ≥ 1.

(iii) From the first part of the proof of Theorem 1.1, we see that the conclusiondimP(G) < 1 − α implies P(E ∩ G �= ∅) = 0, even without the condition(C).


(iv) By Proposition 3.3 in Section 4, we see that (1.4) can be replaced by thefollowing: there exists a constant b ∈ (1, 2] such that

(1.5) limk→∞

logb mk

k= α < 1,

where mk = #{n ∈ N : ln ∈ [b−k+1, b−k+2)

}.

(v) When∑∞

n=1 ln = ∞, but Shepp’s condition (0.2) is not satisfied, thenE �= [0, 1). One can consider the random set F∞ = [0, 1)\E of the un-covered points. The fractal dimension and hitting probabilities have beenstudied by Hawkes [12] (see also Kahane [20, Chapter 11]) and have beenshown to be very different from Theorem 1.1.

We can extend Theorem 1.1 to the following, which describes the intersectionof two independent random covering sets of indices α, α′ < 1.

Theorem 1.3. Let E and E′ be two independent Dvoretzky covering sets onthe same probability space, associated to the sequences {ln} and {l′n} respectively.Suppose both {ln} and {l′n} satisfy the condition (C) with the corresponding upperBesicovitch-Taylor indices α, α′ < 1 and possibly different subsequences {ki} and{k′i}. Then for any analytic set G ⊂ [0, 1) satisfying dimP(G) > 1−min{α, α′}, wehave

P(E ∩ E′ ∩G �= ∅

)= 1.

In particular, if dimP(G) > 1 − α, then

dimP(E ∩G) = dim

P(G) a.s.

In the following we provide an estimate on the Hausdorff dimension of theintersection E ∩G for a given set G.

Theorem 1.4. Let E be the Dvoretzky covering set associated with the sequence{ln} which satisfies the condition (C). Then for any analytic set G ⊂ [0, 1), we have

(1.6) dimH(G) − (1 − α) ≤ dimH(E ∩G) ≤ dimP(G) − (1 − α) a.s.

By taking G = [0, 1) in Theorems 1.3 and 1.4, we obtain dimH(E) = α anddim

P(E) = 1 almost surely. This recovers the result (0.3) of Durand [4], under the

extra condition (C). We remark that our method is different from that of Durand[4].

Corollary 1.5. Assume the conditions of Theorem 1.4 hold. For any analyticset G ⊂ [0, 1) satisfying dimH(G) = dimP(G), we have

dimH(E ∩G) = dim

H(G) − (1 − α) a.s.

In particular dimH(E) = α almost surely.

We end this section with some examples.

Example 1.6. 1. If ln ∼ c n−γ , where c > 0 and γ > 1 are constants andln ∼ jn means lim

n→∞lnjn

= 1, then {ln} satisfies (1.4) with α = 1γ . Hence Theorem 2.1

provides results on hitting probabilities for the associated Dvoretzky covering set E.In particular, we have dim

P(E) = 1. This complements the results in Fan and Wu

(2004) on the Hausdorff dimension of E. More generally, we can take ln ∼ c1 n−γ

for even integers n, while ln ∼ c2 n−γ′

for odd integers n, where both constants γand γ′ are larger than 1. Such sequence satisfies (1.4) with α = max{γ−1, γ′−1}.


Thus, by Durand [4] or by our Corollary 1.5, dimH(E) = α < 1. On the other

hand, by [4] or by our Theorem 1.3, dimP(E) = 1.

2. Let {ln} be the sequence corresponding to the complimentary open intervals

of the tertiary Cantor set. That is, ln = 3−m whenm−1∑i=0

2i < n ≤m∑i=0

2i (m =

1, 2, . . .). Then it can be verified directly that the upper Besicovitch-Taylor indexα = log 2/ log 3 and, moreover, (1.4) holds. Hence our results are applicable tothe corresponding random covering set E. Consequently, dim

H(E) = log 2/ log 3

and dimP(E) = 1 almost surely. Similar results hold for the random coveringsets associated with more general self-similar sets (or self-similar strings, in theterminology of Lapidus and van Frankenhuysen [25]). See Proposition 3.2.

3. If ln = a−n, where a > 1 is a constant, then {ln} satisfies the condition (1.5)with α = 0, by Remark 1.2 (iv), more generally, Proposition 3.3. Hence Theorem1.1 holds for such {ln}. In particular, we have dimHE = 0 and dimP(E) = 1 almostsurely.

4. Finally we provide a simple example of {ln} that satisfies condition (C), butnot (1.4). Let β > log 3/ log 2 be a constant. We define

ln =

⎧⎪⎪⎨⎪⎪⎩3−m if

m−1∑i=0

2i < n ≤m−1∑i=0

2i + 2m−1,

n−β ifm−1∑i=0

2i + 2m−1 < n ≤m∑i=0

2i.

Then we can verify that condition (C) is satisfied with α = log 2/ log 3 and the sub-sequence ki = *(log2 3)i+, where *x+ denotes the largest integer ≤ x. However, (1.4)fails. Nevertheless, the theorems in this section are applicable to the correspondingDvoretzky covering set E.

2. Proofs of the theorems

In this section we prove Theorems 1.1, 1.3 and Theorem 1.4. It will be clearthat the method for studying limsup random fractals in Khoshnevisan, Peres andXiao [24] plays an essential role in our proofs. We remark that, even though thesecond half of the proof of Theorem 1.1 is a modification of the proof of Theorem3.1 in [24], our argument is more general and proves that Theorems 3.1 and 3.2and their corollaries in [24] still hold if their Condition 4 is replaced by

Condition 4′: For some constant γ > 0,

lim supk→∞

log2 pkk

= −γ

and there exists an increasing sequence of positive integers {ki} satisfying (1.2)such that

limi→∞

log2 pki

ki= −γ.

For proving Theorem 1.1 (and for extending the results in [24]) we will use thefollowing elementary lemma on upper box dimension.

Lemma 2.1. Let {ki} be an increasing sequence of positive integers which sat-isfies ( 1.2). Then for any bounded set G ⊂ R,

(2.1) dimM(G) = lim supi→∞

log2 N(G, 2−ki)

ki.


Proof. For any ε > 0, there is an integer i such that 2−ki+1 < ε ≤ 2−ki . Thusfor any bounded set G ⊂ R we have

N(G, 2−ki) ≤ N(G, ε) ≤ N(G, 2−ki+1).

This implies that

logN(G, 2−ki)

ki log 2

kiki+1

≤ logN(G, ε)

− log ε≤ logN(G, 2−ki+1)

ki+1 log 2

ki+1

ki.

It is clear that (2.1) follows from the above and (1.2). �

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. Firstly we show that dimP(G) < 1−α implies P

(E∩

G �= ∅)

= 0. By (0.7), it suffices to show that whenever dimM

(G) < 1 − α, thenE ∩G = ∅, a.s.

Fix an arbitrary but small η > 0 such that dimM(G) < 1 − α − η. For anyr > 0, denote by Cr = Cr(G) a collection of the smallest number of the intervalswith length r that cover the set G. Let Nr(G) = #Cr. Since

lim supn→∞

logNln(G)

− log ln≤ lim sup

r→0

logNr(G)

− log r= dim

M(G) < 1 − α− η,

there exists an integer n0 ∈ N such that

(2.2) Nln(G) < l−(1−α−η)n

for all n ≥ n0. For any interval J in [0, 1) with length ln, since ωn is uniformlydistributed on [0, 1), we have

P{In ∩ J �= ∅

}≤ 3ln.

Note that {In ∩G �= ∅

}⊂

⋃J∈Cln

{In ∩ J �= ∅

},

we derive from this and (2.2) that

P{In ∩G �= ∅

}≤∑

J∈Cln

P{In ∩ J �= ∅

}≤ Nln(G) · 3ln < 3lα+η

n

for all n ≥ n0. Hence the series∑∞

n=1 P{In ∩G �= ∅} converges by the definition ofα and η > 0. By the Borel-Cantelli Lemma, we have

P{In ∩G �= ∅ i.o.

}= 0.

That is, P{∃n0, s.t. ∀n ≥ n0, In ∩G = ∅

}= 1. Therefore, E ∩G = ∅ a.s.

In the following, we prove that if dimP(G) > 1 − α, then

P(E ∩G �= ∅

)= 1.

For this purpose, we construct a random subset E∗ ⊂ E and show that P(E∗∩G �=

∅)

= 1. The random subset E∗ is a discrete limsup random fractal as in [24]. Ourproof below is a modification and extension of the method in their Section 3 and isdivided into two steps.

(i) Construction. For any k ≥ 2, let Dk be the collection of dyadic intervals of theform ( i

2k, i+1

2k), i = 3, 4, . . . , 2k − 1. Denote by Tk = {n ∈ N : ln ∈ [2−k+1, 2−k+2)}

and let nk = #Tk.


For every J ∈ Dk, define

Zk(J) =

{1 if ∃ n ∈ Tk such that J ⊂ In = (ωn, ωn + ln),

0 otherwise.

Let

A(k) =⋃

J∈DkZk(J)=1

J

be the union of open dyadic intervals of order k that are contained in some In withlength ln ∈ [2−k+1, 2−k+2). Observe that

A(k) ⊂⋃

n∈Tk

In.

We define E∗ := lim supk→∞ A(k). From the above, we have E∗ ⊂ E.

(ii) Hitting probability. Now let G ⊂ [0, 1) be an analytic set such that dimP(G) >

1 − α. Then by Joyce and Preiss [17], we can find a closed set G∗ ⊂ G, such thatfor all open set V , we have dim

M(G∗ ∩ V ) > 1 − α, whenever G∗ ∩ V �= ∅.

In the following, we show P(E∗ ∩ G∗ �= ∅

)= 1. Our method is a modification

and extension of that in Section 3 of [24].For every J ∈ Dk, the probability

P(Zk(J) = 1

)= P

{∃ n ∈ Tk such that J ⊂ (ωn, ωn + ln)

}does not depend on J due to our assumption on {ωn} and our definition of Dk.Denote the above probability by Pk. Then

(2.3) Pk ≤ nk(ln − 2−k) ≤ 3nk2−k.

On the other hand,

Pk = P

( ⋃n∈Tk

{J ⊂ In})

≥∑n∈Tk

P(In ⊃ J) −∑

m∈Tk

∑n∈Tkn�=m

P(Im ⊃ J, In ⊃ J

)≥ nk

(ln − 2−k

)− 9n2

k 2−2k

≥ nk 2−k(1 − 9nk2−k).

(2.4)

In the above, we have used the independence of Im and In (m �= n) to derive thesecond inequality. Combining (2.3) and (2.4), together with (1.1) and Condition(C), we derive that

(2.5) lim supk→∞

log2 Pk

k= −(1 − α)

and there is an increasing sequence of integers {ki} that satisfies (1.2) such that

(2.6) limi→∞

log2 Pki

ki= −(1 − α).

Hence E∗ is a limsup random fractals which satisfies Condition 4′ with γ = 1−α(which is weaker than Condition 4 in Khoshnevisan, Peres and Xiao [24, p.11]).Still using their terminology, we call E∗ a limsup random fractal of index 1 − α.


Next we verify their Condition 5 regarding the correlation of Zk(J) and Zk(J′)

in [24]. Given J and J ′ ∈ Dk such that the distance d(J, J ′) ≥ 2−k+2. Since

Cov(Zk(J), Zk(J

′))

= E(Zk(J)Zk(J

′))− E

(Zk(J)

)E(Zk(J

′))

= E(Zk(J)Zk(J

′))− P 2

k ,(2.7)

we estimate E(Zk(J)Zk(J′)) first,

E(Zk(J)Zk(J′)) = P

(Zk(J) = 1, Zk(J

′) = 1)

= P{∃ m,n ∈ Tk such that Im ⊃ J and In ⊃ J ′}

≤∑

m∈Tk

∑n∈Tk,

n�=m

P(Im ⊃ J, In ⊃ J ′)

=

( ∑m∈Tk

P(Im ⊃ J

))( ∑n∈Tk,

n�=m

P(In ⊃ J ′)).

(2.8)

By (2.7), (2.8) and the first inequality in (2.4) we derive

Cov(Zk(J), Zk(J′)) ≤ 2

( ∑m∈Tk

P(Im ⊃ J

))·∑

m∈Tk

∑n∈Tkn�=m

P(Im ⊃ J, In ⊃ J ′)

≤ C(nk 2−k

)E(Zk(J)

)E(Zk(J

′)),

(2.9)

where the last inequality follows from (2.3) and C > 0 is a finite constant. It followsfrom (2.9) and (1.1) that for any ε > 0

Cov(Zk(J), Zk(J

′))< εE

(Zk(J)

)E(Zk(J

′))

for all k large enough. This implies that f(k, ε) ≤ 8, where

f(k, ε) = maxJ∈Dk

#{J ′ ∈ Dk : Cov(Zk(J), Zk(J

′)) ≥ εE(Zk(J))E(Zk(J′))}.

In particular,

limk→∞

log f(k, ε)

k= 0.

Thus we have shown that Condition 5 in [24] is satisfied with δ = 0.The rest of the proof follows a similar line as in the proof of Theorem 3.1 in

[24]. For convenience of the reader, we give it below. Notice that our set N isdetermined by Condition (C) and may be different from that in [24].

Fix an open set V ⊂ [0, 1) such that G∗ ∩ V �= ∅. Let Nk be the number ofdyadic intervals J ∈ Dk such that

(2.10) J ∩G∗ ∩ V �= ∅.Since dimM(G∗ ∩ V ) > 1 − α, we use Lemma 2.1 to derive that, for any β ∈(1 − α, dimM(G∗ ∩ V )

), Nki

≥ 2kiβ for infinitely many integers i. This implies theset N defined as

(2.11) N := {i ≥ 1 : Nki≥ 2kiβ}

satisfies #N = ∞. Similarly to [24], we define

Si =∑

J∈DkiJ∩G∗∩V �=∅

Zki(J).


Namely, Si is the total number of intervals J ∈ Dkisuch that

J ∩G∗ ∩ V ∩A(ki) �= ∅.We now show P

(Si > 0 i.o.

)= 1.

To this end, we estimate

Var(Si) =∑

J∈DkiJ∩G∗∩V �=∅

∑J′∈Dki

J′∩G∗∩V �=∅

Cov(Zki

(J), Zki(J ′)

).

Fix ε > 0, for each J ∈ Dkiwhich satisfies (2.10), let Gki

(J) be the collection of allJ ′ ∈ Dki

such that(i) J ′ ∩G∗ ∩ V �= ∅, and(ii) Cov

(Zki

(J), Zki(J ′)

)≤ εP 2

ki.

If J ′ ∈ Dkisatisfies (i), but not (ii), then we say J ′ ∈ Bki

(J). Thus

Var(Si) ≤ εN 2kiP 2ki

+∑

J∈DkiJ′∈Bki

(J)

Cov(Zki(J), Zki

(J ′))

≤ εN 2kiP 2ki

+ NkimaxJ∈Dki

#Bki(J)Pki

,

where the last term comes from the fact that Cov(Zk(J), Zk(J′)) ≤ E

(Zk(J)

)= Pk.

Since we have shown maxJ∈Dk#Bk(J) ≤ 8 for all k large enough, the above implies

lim supi→∞i∈N

Var(Si)

[E(Si)]2≤ ε + lim sup

i→∞i∈N

maxJ∈Dki#Bki

(J)

NkiPki

= ε.

In the above, we have used that facts that E(Si) = NkiPki

and NkiPki

→ ∞ ifi ∈ N and i → ∞ (recall (2.6) and (2.11)). Since ε > 0 is arbitrary, we have

(2.12) lim supi→∞i∈N

Var(Si)

[E(Si)]2= 0.

It follows from the Paley-Zygmund inequality ([20, p.8]) that

P(Si > 0

)≥ (E(Si))

2

E(S2i )

= 1 − Var(Si)

E(S2i )

≥ 1 − Var(Si)[E(Si)

]2 .Combining the above inequality, (2.12) and Fatou’s Lemma, we derive

(2.13) P(Si > 0 i.o.

)≥ lim sup

i→∞P(Si > 0

)= 1.

It follows from (2.13) that

P

{( ∞⋃k=n

A(k)

)∩G∗ ∩ V �= ∅, ∀n ≥ 1

}= 1

for every open set V with G∗ ∩ V �= ∅. Letting V run over all open interval withrational endpoints, we obtain that (∪∞

k=nA(k))∩G∗ is a.s. dense in G∗ for all n ≥ 1.Since ( ∞⋃

k=n

A(k)

)∩G∗


is an open set in G∗ and G∗ is a complete metric space, by Baire’s category theorem(see Munkres [29]), we know ∩∞

n=1(∪∞k=nA(k)) ∩ G∗ is a.s. dense in G∗, that is,

E∗ ∩G∗ is a.s. dense in G∗. In particular, E∗ ∩G∗ �= ∅ a.s. This finishes the proofof Theorem 1.1. �

Proof of Theorem 1.3. We use the same method as in the proof of Theorem3.2 in [24]. Let G∗ be the closed subset of G described in the proof of Theorem1.1. Suppose dim

P(G) > 1 − min{α, α′}, the proof of Theorem 1.1 shows that for

any open set V such that V ∩G∗ �= ∅ we have

P

(( ∞⋃k=n

Ik

)∩ V ∩G∗ �= ∅, ∀n ≥ 1

)= P

(( ∞⋃k=n

I ′k

)∩ V ∩G∗ �= ∅, ∀n ≥ 1

)= 1.

By independence, there exists a single null probability event outside which for allopen intervals V with rational endpoints satisfying V ∩G∗ �= ∅, we have( ∞⋃

k=n

Ik

)∩ V ∩G∗ �= ∅ and

( ∞⋃k=n

I ′k

)∩ V ∩G∗ �= ∅ for all n ≥ 1.

That is,{(

∪∞k=n Ik

)∩ G∗}n≥1 ∪

{(∪∞k=n I ′k) ∩ G∗

}n≥1

is a countable collection

of open, dense subsets of the complete metric space G∗. Again, Baire’s theoremimplies that

P(E ∩E′ ∩G∗ is dense in G∗

)= 1.

In particular, E ∩ E′ ∩ G∗ �= ∅ a.s. That is, P(E ∩ E′ ∩ G �= ∅) = 1. This provesthe first part of Theorem 1.3.

In order to prove the second half, we regard the set E∩G as the target set withrespect to the random covering set E′. By Theorem 1.1, we know that P(E′∩E∩G �=∅) = 1 implies dim

P(E ∩ G) ≥ 1 − α′ a.s. Therefore, from the above we see that

dimP(G) > 1 − min{α, α′} implies dimP(E ∩G) ≥ 1 − α′ a.s.Now we assume dimP(G) > 1−α. For any α′ with 1−dimP(G) < α′ < α, that

is, dimP(G) > 1 − min{α, α′}, we have dim

P(E ∩G) ≥ 1 − α′ a.s. Letting α′ tend

to 1 − dimP(G) along rational numbers, we obtain

dimP(E ∩G) ≥ dimP(G) a.s.

Therefore, dimP(E ∩G) = dim

P(G) a.s. �

Proof of Theorem 1.4. Firstly, we prove the right-hand inequality in (1.6).By (0.7), it suffices to prove that

(2.14) dimH(E ∩G) ≤ dim

M(G) − (1 − α) a.s.

Denote by Cln a collection of the smallest number of the intervals with length ln,the union of such intervals covers the set G. Let Nln(G) = #Cln . Since ξ :=

dimM(G) ≥ lim supn→∞

logNln (G)− log ln

, we have

Nln(G) < l−(ξ+ε)n

as n large enough, say n ≥ n1(ε), where ε > 0 is an arbitrary small real number.Let Gn be the collection of the intervals J ∈ Cln such that J ∩ In �= ∅ and denoteTn = #Gn. For any J ∈ Gn, P(In ∩ J �= ∅) ≤ 3ln. Thus

E(Tn) ≤∑J∈Gn

P(In ∩ J �= ∅) ≤ 3Nln(G)ln ≤ 3l1−ξ−εn .


For any θ > ξ − (1 − α), we choose ε > 0 such that 2ε < θ − ξ + (1 − α), then

E

( ∞∑n=n1(ε)

Tn lθn

)< 3

∞∑n=n1(ε)

l1−ξ−εn lξ−(1−α)+2ε

n = 3

∞∑n=n1(ε)

lα+εn < ∞.

Thus E(∑∞

n=1 Tn lθn) < ∞. It follows that

∑∞n=1 Tn l

θn < ∞ a.s.

For any m ≥ 1, the collection {J ∈ Gn}n≥m is a covering of the set E ∩G, then

Hθ(E ∩G

)≤

∞∑n=m

Tn lθn < ∞ a.s.,

which implies dimH(E ∩G) ≤ θ a.s. Since θ > dimM(G) − (1 − α) is arbitrary, thisproves that (2.14) holds.

The left-hand inequality in (1.6) can be derived from Theorem 1.1 and thefollowing Lemma, due to Khoshnevisan, Peres, and Xiao [24] (Lemma 3.4 withN = 1 and γ = 1 − α). The proof of Theorem 1.3 completed. �

Lemma 2.2. Equip [0, 1] with the Borel σ-field. Suppose E = E(ω) is a randomset in [0, 1] (i.e., the indicator function χE(ω)(x) is jointly measurable) such thatfor any compact set F ⊂ [0, 1] with dim

H(F ) > γ, we have P(E ∩ F �= ∅) = 1.

Then, for any analytic set F ⊂ [0, 1],

dimH(F ) − γ ≤ dimH(E ∩ F ) a.s.

3. Technical results

The upper Besicovitch-Taylor index (or the convergence exponent) of {ln} playsan essential role in this paper. In this section we provide some equivalent charac-terizations for this index and elaborate more on the condition (C) and (1.4).

First we show that

Proposition 3.1. For any constant a > 1, let nk = #{n ∈ N : ln ∈ [a−k+1,a−k+2)}. Then


loga nk

k.

Proof. For any γ > α, we have∑∞

n=1 lγn < ∞ or

∑∞k=1 nka

−γ(k−1) < ∞.Thus

nka−γ(k−1) ≤ 1

for all k large, which implies

lim supk→∞

loga nk

k≤ γ.

Hence we have

lim supk→∞

loga nk

k≤ α.

On the other hand, if γ > lim supk→∞loga nk

k , we choose γ′ such that

lim supk→∞

loga nk

k< γ′ < γ.


This implies nk ≤ akγ′for all k large enough, say k ≥ k0. Hence∞∑

k=k0

nka−kγ ≤

∞∑k=k0

a−k(γ−γ′) < ∞.

This implies α ≤ γ, which proves α ≤ lim supk→∞loga nk

k . Therefore (3.1) holds. �

For any decreasing sequence {ln} of positive numbers such that∞∑

n=1ln < ∞,

one can associate the following Dirichlet series

ζ(s) =

∞∑n=1

lsn =

∞∑n=1

e−s ln(l−1n ),

which is called the geometric zeta function in Lapidus and van Franhenhuysen[25]. Then the upper Besicovitch-Taylor index α defined by (0.4) is the abscissa ofconvergence of the above Dirichlet series. On the other hand, denote by N(x) thecounting function defined by

N(x) = #{n : l−1

n ≤ x},

see [25, p.8]. Then nk in Proposition 3.1 can be written as nk = N(ak−1)−N(ak−2)for a > 1, hence the index α can also be determined by N(x) (we take a = 2):


log2(N(2k−1) −N(2k−2)

)k

.

Thanks to the above we can also apply the results in [25] to calculate the upperBesicovitch-Taylor index of a sequence {ln}. In the following we focus on sequenceswhich are associated to self-similar sets (or self-similar strings in [25]).

Given an integer M ≥ 2 and constants r1, . . . , rM ∈ (0, 1) such that

1 ≥ r1 ≥ r2 ≥ · · · ≥ rM > 0 and R =

M∑i=1

ri < 1,

one can construct self-similar sets in [0, 1] with scaling ratios r1, . . . , rM (cf. [6,25,28]). Similarly to the tertiary Cantor set in Section 2, we denote the correspondingsequence by {ln}, where each ln is of the form

rk11 · · · rkM

M , where k1, . . . , kM ∈ N.

It can be verified that the multiplicity of the length rk11 · · · rkM

M in {ln} is the multi-

nomial coefficient(

qk1 ··· kM

), where q =

M∑i=1

ki; see [25, p.24].

By the proof of Theorem 2.3 in [25] we see that the geometric zeta function of{ln} is

(3.3) ζ(s) =

∞∑q=0

( M∑i=1

rsi

)q

, ∀s ∈ C.

This, together with (0.4), implies the first assertion of Proposition 3.2 below.The asymptotic behavior of the counting function N(x) for a sequence {ln}

associated to a self-similar set has been studied in [25] (see also the referencestherein for further information). We notice that the zeta function ζ(s) in (3.3)


satisfies conditions (H1) and (H′2) in [25, p.80] with κ = 0 and A = rM (see

[25, pp.121–122]). Hence we can apply Theorem 4.8 in [25, p.88] to obtain that

(3.4) N(x) =∑

ω∈D(C)

res

(xsζ(s)

s; ω

)+ constant

for all x > rM . In the above D(C) denotes the set of complex dimensions of {ln}(i.e., the set of poles of ζ(s) or equivalently the set of solutions of the equation∑M

i=1 rωi = 1) and res(g(s);ω) denotes the residue of a meromorphic function g(s)

at s = ω.To obtain more explicit information about the terms on the right hand side of

(3.4), we distinguish two cases:

Nonlattice case: The additive group generated by log r1, . . . , log rM isdense in R.Lattice case: There exists some number δ > 0 such that log r1, . . . , log rM ∈δZ. The largest such δ is called the additive generator and is denoted byr [25, p.34]. The positive constant p = (2π)/(log r−1) is called the oscil-latory period.

In the nonlattice case, it follows from (5.44) in [25, p.126] that

(3.5) N(x) = res(ζ;α)xα

α+ o(xα), as x → ∞.

The lattice case is much simpler since the complex dimension of {ln} are locatedon finitely many vertical lines [25, Theorem 2.13]. It follows from (5.33) and (5.34)in [25, pp.122-123] that

(3.6) N(x) = res(ζ;α)b1−{u}

b− 1

2π

pxα + o(xα), as x → ∞,

where log b = 2πα/p, u = p log x/2π, {x} = x− *x+ is the fractional part of x.By (3.5), (3.6) and (3.2) we derive

(3.7) limk→∞

log2(N(2k−1) −N(2k−2)

)k

= α.

In other words, (1.4) always holds for a self-similar sequence {ln}.Hence we have proved the following proposition.

Proposition 3.2. Let {ln} be the sequence associated to a self-similar set withscaling ratios r1, . . . , rM . Then the upper Besicovitch-Taylor index α of {ln} coin-cides with the self-similarity dimension D, which is the unique constant satisfying

M∑i=1

rDi = 1.

Moreover, (1.4) holds.

As an example, we mention the Fibonacci sequence, which is obtained by takingM = 2, r1 = 1/2 and r2 = 1/4. Then it can be verified directly that α = log2 φ,

where φ = 1+√5

2 is the golden ratio, and its geometric counting function is givenby

NFib(x) =3 + 4φ

5φ−{log2 x}xα − 1 +

7 − 4φ

5φ{log2 x}x−α(−1)�log2 x�,

see [25, p.124]. It can be verified directly that (1.4) holds.


Finally we show that condition (1.4) can be replaced by (1.5), as stated inRemark 1.2 (iv).

Proposition 3.3. For any constants a > b > 1, let mk = #{n : ln ∈

[b−k+1, b−k+2)}

and let nk = #{n : ln ∈ [a−k+1, a−k+2)

}. If lim

k→∞logb mk

k = α,

then

limk→∞

loga nk

k= α.

Proof. We state the elementary fact that if limk→∞

logb mk

k = α, then for any

fixed integer τ0 ≥ 1, we have

(3.8) limk→∞

logb(mk + mk+1 + · · · + mk+τ0)

k= α.

This can be verified by the fact that b(α−ε)k < mk < b(α+ε)k for all k large implies

b(α−ε)k < mk + mk+1 + · · · + mk+τ0 < (τ0 + 1)b(α+ε)(k+τ0)

for all k large.To prove the lemma, observe that

ln ∈ [a−k+1, a−k+2) ⇐⇒ ln ∈ [b−(logb a)(k−1), b−(logb a)(k−2)).

Hence

nk ≤ #{n : ln ∈ [b−�(logb a)(k−1)�−1, b−�(logb a)(k−2)�)

}= m�(logb a)(k−1)�+2 + · · · + m�(logb a)(k−2)�+2,(3.9)

where *x+ denotes the largest integer ≤ x, and note that a > b, we have

nk ≥ #{n : ln ∈ [b−�(logb a)(k−1)�, b−�(logb a)(k−2)�−1)

}= m�(logb a)(k−1)�+1 + · · · + m�(logb a)(k−2)�+3.(3.10)

Since limk→∞logb m�(logb a)k�

(logb a)k = α, we derive from (3.8), (3.9) and (3.10) that

limk→∞

loga nk

k= lim

k→∞

logb nk

(logb a)k= α.

This proves the lemma. �

Acknowledgement. This paper was developed and finished when Bing Lidid his post-doc research at National Taiwan University, under a grant from NCTSTaipei Office, and during his visit to Michigan State University. The hospitalityof the hosts is appreciated. The authors thank Prof. Ai Hua Fan for his helpfulcomments.

References

[1] Julien Barral and Ai-Hua Fan, Covering numbers of different points in Dvoretzky covering,Bull. Sci. Math. 129 (2005), no. 4, 275–317, DOI 10.1016/j.bulsci.2004.05.007. MR2134123(2007f:28006)

[2] Victor Beresnevich and Sanju Velani, A mass transference principle and the Duffin-Schaefferconjecture for Hausdorff measures, Ann. of Math. (2) 164 (2006), no. 3, 971–992, DOI10.4007/annals.2006.164.971. MR2259250 (2008a:11090)

[3] A. S. Besicovitch and S. J. Taylor, On the complementary intervals of a linear closed set ofzero Lebesgue measure, J. London Math. Soc. 29 (1954), 449–459. MR0064849 (16,344d)


[4] A. Durand, On randomly placed arcs on the circle, in: Recent Developments in Fractals andRelated Fields (Applied and Numerical Harmonic Analysis), 343–352 (edited by J. Barraland S. Seuret), Birkhauser, Boston, 2010.

[5] Aryeh Dvoretzky, On covering a circle by randomly placed arcs, Proc. Nat. Acad. Sci. U.S.A.42 (1956), 199–203. MR0079365 (18,75c)

[6] Kenneth Falconer, Fractal geometry—Mathematical foundations and applications, John Wi-ley & Sons Ltd., Chichester, 1990. MR1102677 (92j:28008)

[7] Aihua Fan, How many intervals cover a point in Dvoretzky covering?, Israel J. Math. 131(2002), 157–184, DOI 10.1007/BF02785856. MR1942307 (2003k:60026)

[8] Ai Hua Fan, Some topics in the theory of multiplicative chaos, Fractal geometry andstochastics III, Progr. Probab., vol. 57, Birkhauser, Basel, 2004, pp. 119–134. MR2087136(2005f:60116)

[9] Ai Hua Fan and Jean-Pierre Kahane, Rarete des intervalles recouvrant un point dans unrecouvrement aleatoire, Ann. Inst. H. Poincare Probab. Statist. 29 (1993), no. 3, 453–466(French, with English and French summaries). MR1246642 (95c:60015)

[10] Ai-Hua Fan and Jun Wu, On the covering by small random intervals, Ann. Inst. H. PoincareProbab. Statist. 40 (2004), no. 1, 125–131, DOI 10.1016/S0246-0203(03)00056-6 (English,with English and French summaries). MR2037476 (2005c:60012)

[11] P. J. Fitzsimmons, Bert Fristedt, and L. A. Shepp, The set of real numbers left uncoveredby random covering intervals, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 2, 175–189, DOI10.1007/BF02451427. MR799145 (86k:60017)

[12] John Hawkes, On the covering of small sets by random intervals, Quart. J. Math. OxfordSer. (2) 24 (1973), 427–432. MR0324748 (48 #3097)

[13] John Hawkes, Random re-orderings of intervals complementary to a linear set, Quart. J.Math. Oxford Ser. (2) 35 (1984), no. 138, 165–172, DOI 10.1093/qmath/35.2.165. MR745418(86b:28004)

[14] Svante Janson, Random coverings in several dimensions, Acta Math. 156 (1986), no. 1-2,83–118, DOI 10.1007/BF02399201. MR822331 (87i:60018)

[15] Johan Jonasson, Dynamical circle covering with homogeneous Poisson updating, Statist.Probab. Lett. 78 (2008), no. 15, 2400–2403, DOI 10.1016/j.spl.2008.03.001. MR2462675

(2010a:60361)[16] Johan Jonasson and Jeffrey E. Steif, Dynamical models for circle covering: Brownian motion

and Poisson updating, Ann. Probab. 36 (2008), no. 2, 739–764, DOI 10.1214/07-AOP340.MR2393996 (2009d:60023)

[17] H. Joyce and D. Preiss, On the existence of subsets of finite positive packing measure, Math-ematika 42 (1995), no. 1, 15–24, DOI 10.1112/S002557930001130X. MR1346667 (96g:28010)

[18] J.-P. Kahane, The technique of using random measures and random sets in harmonic analy-sis, Advances in probability and related topics, Vol. 1, Dekker, New York, 1971, pp. 65–101.MR0298321 (45 #7373)

[19] J.-P. Kahane, Ensembles parfaits et processus de Levy, Period. Math. Hungar. 2 (1972), 49–59 (French). Collection of articles dedicated to the memory of Alfred Renyi, I. MR0329050(48 #7392)

[20] Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Ad-vanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR833073(87m:60119)

[21] Jean-Pierre Kahane, Recouvrements aleatoires et theorie du potentiel, Colloq. Math. 60/61(1990), no. 2, 387–411 (French). MR1096386 (92g:60015)

[22] Jean-Pierre Kahane, Recouvrements aleatoires, Gaz. Math. 53 (1992), 115–129 (French).MR1175539 (93k:60030)

[23] Jean-Pierre Kahane, Random coverings and multiplicative processes, Fractal geometry andstochastics, II (Greifswald/Koserow, 1998), Progr. Probab., vol. 46, Birkhauser, Basel, 2000,pp. 125–146. MR1785624 (2001j:60093)

[24] Davar Khoshnevisan, Yuval Peres, and Yimin Xiao, Limsup random fractals, Electron. J.Probab. 5 (2000), no. 5, 24 pp., DOI 10.1214/EJP.v5-60. MR1743726 (2001a:60095)

[25] Michel L. Lapidus and Machiel van Frankenhuysen, Fractal geometry and number theory,Complex dimensions of fractal strings and zeros of zeta functions, Birkhauser Boston Inc.,Boston, MA, 2000. MR1726744 (2001b:11079)


[26] Benoit B. Mandelbrot, Renewal sets and random cutouts, Z. Wahrscheinlichkeitstheorie undVerw. Gebiete 22 (1972), 145–157. MR0309162 (46 #8272)

[27] A. I. Markushevich, Theory of functions of a complex variable. Vol. II, Revised Englishedition translated and edited by Richard A. Silverman, Prentice-Hall Inc., Englewood Cliffs,N.J., 1965. MR0181738 (31 #5965)

[28] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Fractals and rectifia-bility, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press,

Cambridge, 1995. MR1333890 (96h:28006)[29] James R. Munkres, Topology: a first course, Prentice-Hall Inc., Englewood Cliffs, N.J., 1975.

MR0464128 (57 #4063)[30] George Polya and Gabor Szego, Problems and theorems in analysis. I, Classics in Mathemat-

ics, Springer-Verlag, Berlin, 1998. Series, integral calculus, theory of functions; Translatedfrom the German by Dorothee Aeppli; Reprint of the 1978 English translation. MR1492447

[31] L. A. Shepp, Covering the circle with random arcs, Israel J. Math. 11 (1972), 328–345.MR0295402 (45 #4468)

[32] S. James Taylor, The measure theory of random fractals, Math. Proc. Cambridge Philos. Soc.100 (1986), no. 3, 383–406, DOI 10.1017/S0305004100066160. MR857718 (87k:60189)

[33] Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos.Soc. 91 (1982), no. 1, 57–74, DOI 10.1017/S0305004100059119. MR633256 (84d:28013)

[34] Yimin Xiao, Random fractals and Markov processes, Fractal geometry and applications: ajubilee of Benoıt Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc.,Providence, RI, 2004, pp. 261–338. MR2112126 (2006a:60065)

Department of Mathematics, South China University of Technology, 510640,

Guangzhou, P. R. China and Department of Mathematical Sciences, University of Oulu,

P.O. Box 3000 FI-90014, Finland


Department of Mathematics, Honorary Faculty, National Taiwan University,

Taipei 10617, Taiwan.


Department of Statistics and Probability, 619 Red Cedar Road, Michigan State

University, East Lansing, Michigan 48824



Fractal Oscillations Near the Domain Boundary of RadiallySymmetric Solutions of p-Laplace Equations

Yuki Naito, Mervan Pasic, Satoshi Tanaka, and Darko Zubrinic

Abstract. We consider radially symmetric solutions of p-Laplace differentialequation (1): −Δpu = f(|x|)|u|p−2u in an annular domain Ωa,b. Motivatedby [7] and [12], we introduce and study the fractal oscillations near |x| = bof all radially symmetric solutions of equation (1). Precisely, for a given realnumber s ∈ [N,N + 1) we find some sufficient conditions on the coefficientf(r) such that every radially symmetric nontrivial solution u(x) of equation (1)oscillates near |x| = b and the box-dimension dimB Γ(u) of the graph Γ(u) andcorresponding lower and upper s-dimensional Minkowski contents Ms

∗(Γ(u))and M∗s(Γ(u)) satisfy: dimB Γ(u) = s and 0 < Ms

∗(Γ(u)) ≤ M∗s(Γ(u)) <∞. Our argument is based on the study of the fractal dimension of radiallysymmetric functions, and on the analysis of solutions to the correspondingordinary differential equations.

1. Introduction

Let p > 1 and Ωa,b = {x ∈ RN : 0 < a < |x| < b}. We consider the quasilinearelliptic equation in Ωa,b associated to the classic p-Laplacian operator:

(1.1) −div(|∇u|p−2∇u

)= f(|x|)|u|p−2u, x ∈ Ωa,b.

It is said that a function u = u(x) is a solution of equation (1.1) if u(x) satisfiesthe equality in (1.1) for all x ∈ Ωa,b in the classic sense and

u ∈ C(Ωa,b) ∩ C1(Ωa,b) and |∇u|p−2∇u ∈ C1(Ωa,b).

A function u(x) is said to be a radially symmetric function if there is a real functiony = y(r) such that u(x) = y(|x|). A solution u(x) of equation (1.1) is said to be aradially symmetric solution of (1.1) if u(x) is a radially symmetric function.

Definition 1.1. A radially symmetric function u(x) = y(|x|), u ∈ C(Ωa,b), issaid to be oscillatory near |x| = b, if y(r) oscillates near r = b, see Figure 1.

2010 Mathematics Subject Classification. 26A06, 34A26, 34B05, 35J.Key words and phrases. Graph, box dimension, Minkowski content, p-Laplacian, radial so-

lutions, oscillations.The research of Yuki Naito was supported by Grant-in-Aid for Scientific Research (C)

(No.23540244) from JSPS.

The research of Mervan Pasic and Darko Zubrinic was supported by the Ministry of Scienceof the Republic of Croatia under grant no. 036-0361621-1291.

The research of Satoshi Tanaka was supported by Grant-in-Aid for Young Scientists (B)(No.23740113) from JSPS.


325

326 Y. NAITO, M. PASIC, S. TANAKA, AND D. ZUBRINIC

Figure 1: the graph Γ(u) of an oscillating function u : Ωa,b → R near |x| = b.

The oscillations of radially symmetric solutions of equation (1.1) near |x| = bare closely related with the singular behaviour of the coefficient f(r) near r = b, asshown in the following result, which will be proved in Section 6.

Lemma 1.1 (necessary condition for oscillations). Let equation (1.1) have atleast one radially symmetric solution which oscillates near x = |b|.

(i) If f ∈ C([a, b)) and f(r) > 0 in [a, b), then lim supr→b f(r) = ∞.(ii) If f ∈ C1([a, b)), f(r) > 0 and f ′(r) > 0 in [a, b), then limr→b f(r) = ∞.

According to Lemma 1.1, we impose the following basic assumptions on thecoefficient f(r) of equation (1.1):

(1.2) f ∈ C2([a, b)), f(r) > 0 in (a, b) and limr→b

f(r) = ∞.

Very recently in [7], the so-called rectifiable and unrectifiable oscillations ofequation (1.1) have been studied. Precisely, in [7] some sufficient conditions on thecoefficient f(r) have been given such that the surface area of graph Γ(u) of everyradially symmetric solution u(x) of (1.1) is finite and infinite respectively, where

Γ(u) = {(x, xN+1) ∈ RN × R : xN+1 = u(x)}.In this paper, we study a refinement of these results in terms of fractal geometry.Before we state the main result of the paper, we introduce the next definition.

Definition 1.2. A radially symmetric function u ∈ C(Ωa,b) is said to be fractaloscillatory near |x| = b with the fractal dimension s ∈ [N,N + 1) if u(x) oscillatesnear |x| = b and

(1.3) dimB Γ(u) = s and 0 < Ms∗(Γ(u)) ≤ M∗s(Γ(u)) < ∞,

where dimB Γ(u) denotes the box-dimension of the graph Γ(u), and Ms∗(Γ(u))

and M∗s(Γ(u)) denote, respectively, the lower and upper s-dimensional Minkowskicontents of Γ(u). When (1.3) is satisfied, then it is said that Γ(u) is a Minkowskinondegenerate set in RN+1, see [16].

Let us remark that if there is an s ≥ 0 such that 0 < Ms∗(Γ(u)) = M∗s(Γ(u)) <

∞, then Γ(u) is said to be a Minkowski measurable set in RN+1.In general, a bounded set A in RN is said to be Minkowski nondegenerate if

there is d ≥ 0 such that 0 < Md∗(A) ≤ M∗d(A) < ∞, see [16] where this notion was

introduced. Note that this immediately implies that d = dimBA = dimBA, and the

FRACTAL OSCILLATIONS IN PDES 327

common value is denoted by dimB A for short. Note that Minkowski nondegeneracyextends the notion of Minkowski measurability of A, which by definition means that0 < Md

∗(A) = M∗d(A) < ∞. According to our Definition 1.2, fractal oscillatoryfunctions have Minkowski nondegenerate graphs.

In [3] and [8]-[12] the case when A is equal to the graph of a chirp like functiony : (a, b) → R is treated. As can be seen from the proofs in these papers, thecondition of Minkowski nondegeneracy is satisfied for all the results obtained theredealing with fractal oscillations. In particular, all dimension results stated therehold not only for the upper box dimension, but also for the lower box dimension,that is, both of them are equal.

Let us mention by the way that the condition of Minkowski nondegeneracy offractal sets is encountered in numerous results due to Professor Michel L. Lapidusand his collaborators, dealing with fractal strings, see [5] and the related referencestherein. Minkowski nondegeneracy is encountered in the study of spiral trajectoriesof dynamical systems, see e.g. [17] and in the study of some classes of singular inte-grals, see [16]. We can address the interested reader also to the survey paper [13].The property of Minkowski nondegeneracy of sets is preserved under bilipshitz map-pings: if A is Minkowski nondegenerate subset of RN and Φ is a bilipschitz mappingfrom a neighbourhood of A to RN , then Φ(A) is also Minkowski nondegenerate.See [17] for more details and the proof of this fact.

The basic way to explore some elementary classes of real functions u(x) whichare fractal oscillatory near |x| = b is to use the radially symmetric functions u(x) =y(|x|), where y = y(r), y ∈ C([a, b]) is a known class of real functions which isfractal oscillatory near r = b. It is possible because of the following essential result,which also plays an important role in the proof of the main results.

Proposition 1.1. Let u : Ωa,b → R be a bounded radially symmetric function,i.e., u(x) = y(|x|) for all x ∈ Ωa,b, where y = y(r), y : (a, b) → R is a given boundedfunction. Then

(1.4) dimBΓ(u) = dimBΓ(y) + N − 1.

Moreover, if there exists dimB Γ(y), then (1.4) holds when dimB is replaced bydimB. Also, if Γ(y) is Minkowski nondegenerate, then so is Γ(u).

The application of this proposition is given in the next examples.

Example 1.1. Let 0 < α < β and S(t) = sin t or cos t. Then the function

u(x) = (b− |x|)αS((b− |x|)−β

), x ∈ Ωa,b,

is fractal oscillatory near |x| = b with the fractal dimension

s = N + 1 − α + 1

β + 1.

Indeed, for y = y(r) = (b − r)αS((b− r)−β

)we know that dimB Γ(y) = 2 −

(α + 1)/(β + 1), see for instance [10], [11], [14]. This dimension result has beenpreviously obtained using Lapidus zeta functions of fractal sets, see [4]. Now fromProposition 1.1 applied to u(x) = y(|x|), it follows that:

dimB Γ(u) = dimB Γ(y) + N − 1 = N + 1 − α + 1

β + 1.


Example 1.2. Let ρ �= 0 and S(t) = sin t or cos t. Then the function

u(x) =√b− |x| S(ρ ln(b− |x|)), x ∈ Ωa,b,

is fractal oscillatory near |x| = b with the fractal dimension s = N . In fact, fory = y(r) =

√b− r S(ρ ln(b − r)) we know that dimB Γ(y) = 1, see [10, Example

1.2]. Now from Proposition 1.1 applied to u(x) = y(|x|), it follows that

dimB Γ(u) = dimB Γ(y) + N − 1 = N.

Example 1.3. Let ν ∈ R and S(t) = Jν(t) or Yν(t), where Jν(t) and Yν(t)denote the Bessel functions of the first and second kind respectively. Then thefunction

u(x) = S((b− |x|)−1

), x ∈ Ωa,b,

is fractal oscillatory near |x| = b with the fractal dimension s = N + 1/4. In fact,for y = y(r) = S(b−r) we know that dimB Γ(y) = 5/4, see [10, Example 1.7]. Nowfrom Proposition 1.1 applied to u(x) = y(|x|), it follows that

dimB Γ(u) = dimB Γ(y) + N − 1 = N +1

4.

Example 1.4. Let λ > 1/4, ρ =√λ− 1/4 and let y = y(r) be a function

defined by

(1.5) y(r) =

√ln

b

r

[c1 cos

(ρ ln ln

b

r

)+ c2 sin

(ρ ln ln

b

r

)], r ∈ (a, b),

where c1, c2 ∈ R. Then the function u(x) = y(|x|) is fractal oscillatory near |x| = bwith the fractal dimension s = N . Moreover, if N = 2 then all radially symmetricsolutions of the next linear elliptic pde:

(1.6)

⎧⎨⎩ −Δu =λ

|x|2 ln2(b/|x|)u in Ωa,b,

u = 0 on |x| = b,

are given by u(x) = y(|x|), where y(r) is defined by (1.5).

Now we can state the following two main results of the paper. We recall a knownnotation for the same order of asymptotic behaviour of two given functions. Namely,for two functions g(r) and h(r) defined on the interval [a, b), we say that g(r) ∼ h(r)near r = b if there are two positive constants c1, c2 such that c1h(r) ≤ g(r) ≤ c2h(r)near r = b.

Theorem 1.1. Let f(r) satisfy (1.2) and the following Hartman-Wintner typecondition

(1.7) f−θ(f−η

)′′ ∈ L1(a, b),

where η and θ are arbitrary positive constants satisfying θ + η = 1/p. If there is aσ ∈ (p, p2) ∪ (p2,∞) such that

(1.8) f(r) ∼ (b− r)−σ near r = b,

then every radially symmetric nontrivial solution u(x) of equation (1.1) is fractaloscillatory near |x| = b with the fractal dimension s determined by:

(1.9) s =

{N if p < σ < p2,

N + 1 − 1/q − p/σ if σ > p2,


where q = p/(p− 1) is the exponent conjugate to p.

Two most frequent choices for the parameters θ and η appearing in (1.7) are:

θ =1

p− 1

p2, η =

1

p2and θ =

1

2p, η =

1

2p.

In [12, Lemma 1] it is shown that if (1.7) is satisfied for a pair (θ, η) such thatθ + η = 1/p, then (1.7) is satisfied for all such pairs (θ, η).

Example 1.5. We consider the linear elliptic PDE:⎧⎨⎩ −Δu =λ

|x|2(ln(b/|x|))σ u in Ωa,b

u = 0 on |x| = b,

where λ > 0, σ ∈ R. The coefficient function f(r) = λ/(r2(ln(b/r))σ) satisfies theHartman-Wintner type condition (1.7) with p = 2 if and only if σ > 2. In fact, forp = 2, we have

f− 12p

(f− 1

2p

)′′= −λ2 4(ln(b/r))2 + σ(σ − 4)

16r(ln(b/r))−σ2 +2

.

Since ln(b/r) ∼ (b − r) as r → b, we have f− 12p

(f− 1

2p

)′′∈ L1(a, b) if and only if

σ > 2.

Remark 1.1. The coefficient f(r) = λ/[r2 ln2(b/r)] of equation (1.6) does notsatisfy the Hartman-Wintner type condition (1.7) for p = 2. In fact,

f− 12p

(f− 1

2p

)′′= c

ln2 r + 1

r ln(b/r)�∈ L1(a, b),

where c is a real constant.

In the proof of [7, Theorem 1.2], the necessary and sufficient conditions of theHartman-Wintner type for the following large class of functions f(r) were shown.

Proposition 1.2. Let f(r) satisfy (1.2) and let there be a σ ∈ R such that

(1.10) f ′′(r) ∼ (b− r)−σ−2 near r = b.

Then f(r) satisfies the Hartman-Wintner type condition (1.7) if and only if σ > p.

In the case when the condition (1.7) is difficult to be checked, then we sug-gest the following asymptotic result which is a consequence of Theorem 1.1 andProposition 1.2.

Corollary 1.1. Let f(r) satisfy (1.2) and (1.10) for some σ ∈ (p, p2)∪(p2,∞).Then every radially symmetric nontrivial solution u(x) of equation (1.1) is fractaloscillatory near |x| = b with the fractal dimension s determined by (1.9).

As a consqeunce of previous results we consider the next example.

Example 1.6. We consider the p-Laplace elliptic PDE:

(1.11)

⎧⎨⎩ −div(|∇u|p−2∇u) =1

|x|α(ln(b/|x|))σ |u|p−2u in Ωa,b

u = 0 on |x| = b,


where α ∈ R and σ ∈ (p, p2) ∪ (p2,∞). Then every radially symmetric nontrivialsolution u(x) of (1.11) is fractal oscillatory near |x| = b with the fractal dimensions determined by (1.9). In fact, we see that (1.10) holds for the coefficient functionf(r) = 1/(rα(ln(b/r))σ).

The analysis of all radially symmetric solution u(x) of equation (1.1) is relatedto the analysis of all solutions of the corresponding one-dimensional equation:

(1.12)

⎧⎨⎩(rN−1|y′|p−2y′

)′+ rN−1f(r)|y|p−2y = 0, r ∈ (a, b),

y ∈ C([a, b]) ∩ C2([a, b)).

Proposition 1.3. The function u(x) = y(|x|) is a radially symmetric solutionof equation (1.1) if and only if the function y = y(r) is a solution of the onedimensional equation (1.12).

Using Proposition 1.1 and Proposition 1.3, the crucial role in the proof ofTheorem 1.1 plays the following result.

Theorem 1.2. Let f(r) satisfy (1.2), (1.7) and (1.8). Then every nontrivialsolution y(r) of equation (1.12) oscillates near r = b and satisfies:

(1.13) dimB Γ(y) = s0 and 0 < Ms0∗ (Γ(y)) ≤ M∗s0(Γ(y)) < ∞,

where the dimensional number s0 satisfies:

(1.14) s0 =

{1 if p < σ < p2,

2 − 1/q − p/σ if σ > p2,

where q = p/(p− 1).

Since equation (1.12) is more complicated than the basic half-linear differentialequation (HL): (|y′|p−2y′)′+f(r)|y|p−2y = 0, we are not able to prove Theorem 1.2by using the main results on the fractal oscillations of equation (HL) recentlypublished in [12], see Section 4.

The proof of Theorem 1.1 is an immediate consequence of Proposition 1.1,Proposition 1.3 and Theorem 1.2 and hence it is omitted.

With the help of Proposition 1.2, we can derive the next important particularcase of Theorem 1.2.

Corollary 1.2. Let f(r) satisfy (1.2) and (1.10) for some σ ∈ (p, p2)∪(p2,∞).Then every nontrivial solution y(r) of equation (1.12) oscillates near r = b andsatisfy (1.13) with respect to dimensional number s0 satisfying (1.14).

2. A bi-Lipschitz transformation of equation (1.12)

For dimensional number N ≥ 1, let IN be an open bounded interval definedby:

(2.1) IN =

⎧⎪⎪⎪⎨⎪⎪⎪⎩(t0, 0) with t0 = ln a

b ∈ (−∞, 0) if p = N,

(t0, 1) with t0 =(ab

) p−Np−1 ∈ (0, 1) if p > N,

(1, t0) with t0 =(ab

) p−Np−1 ∈ (1,∞) if p < N.


The essential boundary point of IN (which is independent of a and b) is denotedby ∂IN and defined by

(2.2) ∂IN =

{0 if p = N,

1 if p �= N.

Let ϕN : IN → R be a transformation of variable t ∈ IN into r ∈ (a, b) defined by

(2.3) ϕN (t) =

⎧⎨⎩b et if p = N,

b tp−1p−N if p �= N.

It is clear that ϕN (∂IN ) = b, ϕN is increasing if p ≥ N and decreasing if p < Nand

(2.4) ϕ−1N (r) =

⎧⎨⎩ln rb if p = N,(

rb

) p−Np−1 if p �= N.

Proposition 2.1. The transformation ϕN is a bijection from interval IN into(a, b). Moreover, ϕN : IN → (a, b) is a bi-Lipschitz mapping.

Proof. From (2.1) and (2.3) immediately follows that ϕN (IN ) = (a, b) as wellas that ϕN is a bijective transformation. Also, we have:

(2.5)dϕN

dt=

⎧⎨⎩b et if p = N,

b(p−1)p−N t

N−1p−N if p �= N,

which provides two positive constants m,M such that

(2.6) 0 < m ≤∣∣∣∣dϕN

dt

∣∣∣∣ ≤ M for all t ∈ IN .

By the Lagrangue mean value theorem, for all t1, t2 ∈ IN , t1 < t2, there is aξ ∈ (t1, t2) such that:

|ϕN (t1) − ϕN (t2)| =

∣∣∣∣dϕN

dt(ξ)

∣∣∣∣ (t2 − t1).

Now, because of previous inequality and (2.6) we obtain:

m|t2 − t1| ≤ |ϕN (t1) − ϕN (t2)| ≤ M |t2 − t1| for all t1, t2 ∈ IN .

Thus, it is shown that ϕN is a bi-Lipschitz mapping. �

Next, we define the following change of variables:

(2.7) z(t) = y(r) with r = ϕN (t).

Let CN be a positive real number defined by:

(2.8) CN =

⎧⎨⎩1 if p = N,∣∣ p−1p−N

∣∣p−2 p−1p−N bp−N if p �= N.

It is easy to check that from (2.5) and (2.8) it follows:

(2.9)

∣∣∣∣dϕN

dt

∣∣∣∣p−2dϕN

dt= CNrN−1.


Now, with the help of (2.7) and (2.9), we see that equation (1.12) is transformedinto the equivalent one (see for instance Naito [6] and Wang [15]):

(2.10) (|z′|p−2z′)′ + F (t)|z|p−2z = 0, t ∈ IN ,

where the coefficient F (t) satisfies:(2.11)

F (t) = CNϕN−1N (t)

dϕN

dtf(ϕN (t)) =

⎧⎪⎨⎪⎩bNeNtf(b et) if p = N,

bp∣∣∣ p−1p−N

∣∣∣p tpN−1p−N f(b t

p−1p−N ) if p �= N.

Exactly in the same way as in [7, Lemmas 2.3 and 2.4], we can prove the nextauxiliary result about the equivalence between the Hartman-Wintner asymptotictype conditions for the main coefficients f(r) and F (t) of equations (1.1) and (2.10)respectively.

Proposition 2.2. Let F and IN be defined by (2.11) and (2.1), respectively.

(i) The condition (1.2) is equivalent with

F ∈ C2(IN ), F (t) > 0 in IN and limt→∂IN

F (t) = ∞;

(ii) Let η and θ be two arbitrary positive constants satisfying θ + η = 1/p.Then the Hartman-Wintner condition (1.7) is equivalent with

F−θ(F−η)′′ ∈ L1(IN ).

3. Qualitative properties of equations (1.12) and (2.10)

In this section, we give a zero-points analysis as well as the a priori estimatesfor all solutions of equations (1.12) and (2.10).

At the first, analogously with [12, Remark 2, Lemmas 2 and 3] and with thehelp of Proposition 2.2, we have the following zero-points analysis for equation(2.10).

Lemma 3.1. Let f(r) satisfy all assumptions of Theorem 1.2. Let z = z(t) bea nontrivial solution of equation (2.10). Then there exists a sequence tk ∈ IN ofconsecutive zero points of z (increasing if p ≥ N and decreasing if p < N) suchthat |tk − ∂IN | → 0 as k → ∞, where the point ∂IN is defined in (2.2). Moreover,there are k0 ∈ N, ε0 > 0 and positive constants Ci, i = 0, 1, 2, 3, 4 such that for allk0 ∈ N, k > k0 and ε ∈ (0, ε0), we have:

C1 |min{tk, tk+1} − ∂IN |σp ≤ |tk − tk+1| ≤ C2 |max{tk, tk+1} − ∂IN |

σp ,(3.1)

C3

(1

k + k0

) pσ−p

≤ |tk − ∂IN | ≤ C4

(1

k − k0

) pσ−p

.(3.2)

Moreover, there is an index function k = k(ε), k : (0, ε0) → N such that k(ε) > k0,k(ε) → ∞ as ε → 0 and

(3.3) |tk+1 − tk| ≤ε

2for each k > k(ε).

Furthermore, there is a sequence τk ∈ (min{tk, tk+1},max{tk, tk+1}) of consecutivezero points of z′ such that

(3.4) |z(τk)| ≥ C0|τk − ∂IN | σpq ,


where q is the conjugate exponent of p.

From Proposition 2.1, equality (2.7), Lemma 3.1 and ϕN (∂IN ) = b, we easilyderive the zero-points analysis for equation (1.12) as follows.

Lemma 3.2. Let f(r) satisfy all assumptions of Theorem 1.2 and let ϕN =ϕN (t) be a function given by (2.3). Let y = y(r) be a nontrivial solution of equation(1.12). Then there exists an increasing sequence rk ∈ (a, b) of consecutive zeropoints of y such that rk ↗ b. Moreover, there are k0 ∈ N, ε0 > 0 and positiveconstants ci, i = 0, 1, 2, 3, 4 such that for all k0 ∈ N, k > k0 and ε ∈ (0, ε0), wehave:

(3.5)c1∣∣min

{ϕ−1N (rk), ϕ

−1N (rk+1)

}− ϕ−1

N (b)∣∣σp

≤ rk+1 − rk ≤ c2∣∣max

{ϕ−1N (rk), ϕ

−1N (rk+1)

}− ϕ−1

N (b)∣∣σp ,

(3.6) c3

(1

k + k0

) pσ−p

≤∣∣ϕ−1

N (rk) − ϕ−1N (b)

∣∣ ≤ c4

(1

k − k0

) pσ−p

.

Moreover, there is an index function k = k(ε), k : (0, ε0) → N such that k(ε) > k0,k(ε) → ∞ as ε → 0 and

(3.7) |rk+1 − rk| ≤ε

2for each k > k(ε).

Furthermore, there is an increasing sequence σk ∈ (rk, rk+1) of consecutive zeropoints of y′ such that

(3.8) |y(σk)| ≥ c0∣∣ϕ−1

N (σk) − ϕ−1N (b)

∣∣ σpq ,

where q is the conjugate exponent of p.

Analogously with [12, (3),(5), Propositions 1 and 2] and with the help of Propo-sition 2.2, we derive the following asymptotic formula for all solutions of equation(2.10).

Lemma 3.3. Let f(r) satisfy (1.2) and (1.7). Let the interval IN and the point∂IN be defined in (2.1) and (2.2) respectively. Let w = w(ξ), ξ > 0, be the so-calledgeneralized sine function which is a solution of half-linear differential equation,

(3.9)

{(|w′|p−2w′)′ + (p− 1)|w|p−2w = 0, ξ > 0,

w(0) = 0, w′(0) = 1.

Then, for a nontrivial solution z(t) of equation (2.10), there are two functionsV = V (t) and ϕ = ϕ(t) such that

(3.10)

⎧⎨⎩ z(t) = (p− 1)1pq F− 1

pq (t)V1p (t)w(ϕ(t)),

|z′(t)|p−2z′(t) = −(p− 1)−1pq F

1pq (t)V

1q (t)|w′(ϕ(t))|p−2w′(ϕ(t)),

for t ∈ IN near ∂IN , where the functions V and ϕ satisfy:

(3.11)

{0 < limt→∂IN V (t) < ∞,

ϕ(t) < 0 and ϕ′(t) ∼ −F1p (t) as t → ∂IN .

As a consequence of Lemma 3.3, we derive the following a priori estimates forall solutions of equation (2.10) (see also [12, Remark 3]).


Lemma 3.4. Let f(r) satisfy (1.2) and (1.7). Then, for a solution z(t) ofequation (2.10), there are two positive constants c1, c2 such that

(3.12) |z(t)| ≤ c1F− 1

pq (t) and |z′(t)| ≤ c2F1p2 (t) for all t ∈ IN near ∂IN .

Now, from (2.11) and Lemma 3.4 we can easily state a priori estimates for allsolutions of equation (1.12).

Lemma 3.5. Let f(r) satisfy (1.2) and (1.7). Then there are two positiveconstants c1, c2 such that for all solutions y(r) of equation (1.12) we have:

(3.13) |y(r)| ≤ c1f− 1

pq (r) and |y′(r)| ≤ c2f1p2 (r) near r = b.

Moreover, if f(r) satisfies the asymptotic condition (1.8), then there are two positiveconstants c1, c2 such that for all solution y(r) of equation (1.12) we have:

(3.14) |y(r)| ≤ c1(b− r)σpq and |y′(r)| ≤ c2(b− r)

− σp2 near r = b.

Proof. With the help of (2.7), it is clear that

CNϕN−1N (t)

∣∣∣∣dϕN

dt

∣∣∣∣ f(ϕN (t)) = CNrN−1

∣∣∣∣dϕN

dt

∣∣∣∣ f(r).

Next, by (2.6) we obtain two positive constants m1 and M1 such that

m1f(r) ≤ CNrN−1

∣∣∣∣dϕN

dt

∣∣∣∣ f(r) ≤ M1f(r), r ∈ (a, b),

which together with (2.11) gives:

(3.15) m1f(r) ≤ F (ϕ−1N (r)) ≤ M1f(r), r ∈ (a, b).

Also, from (2.6) and (2.7), we get:

(3.16)

∣∣∣∣dydr∣∣∣∣ = 1∣∣∣dϕN

dt

∣∣∣∣∣∣∣dzdt∣∣∣∣ ≤ 1

m

∣∣∣∣dzdt∣∣∣∣ , t ∈ IN .

Putting (3.15) and (3.16) into (3.12), we obtain (3.13). �

4. Proof of Theorem 1.2

Now, we are able to start with the proof of Theorem 1.2. In order to prove theinequality (1.13) for a solution y = y(r) of equation (1.12), it is enough to showthat there exist three positive constants c0, c1, ε0 such that

(4.1) c0ε2−s0 ≤ |Γε(y)| ≤ c1ε

2−s0 for all ε ∈ (0, ε0),

where the dimensional number s0 satisfies (1.14), the Γε(y) denotes the ε-neighbour-hood of graph Γ(y) and |Γε(y)| denotes the 2-Lebesgue measure of Γε(y). Indeed,from (4.1) we immediately have

c0 ≤ Ms0∗ (Γ(y)) ≤ M∗s0(Γ(y)) ≤ c1.

Thus Theorem 1.2 holds.First we consider the case where p < σ < p2.


Proof of (4.1) for the case p < σ < p2. Let y = y(r) be a nontrivial solu-tion of equation (1.12). From Lemma 3.1, it follows that y is oscillatory near r = b.We note that ∫ b

a

[f(r)]1p2 dr < ∞,

by (1.2), (1.8) and p < σ < p2. Theorem 1.1 and Lemma 2.1 in [7] imply thatlength(y) < ∞. By Tricot [14, p. 106], we conclude that

length(y) = limε→+0

|Γε(y)|2ε

.

Hence we havec0ε ≤ |Γε(y)| ≤ c1ε, ε ∈ (0, ε0)

for some c0 > 0, c1 > 0 and ε0 > 0, which means that (4.1) holds for s0 = 1. �

For the case σ > p2, the proofs of the left and right inequalities in (4.1) will bepresented separately in the following two steps.

Proof of the left-hand side inequality in (4.1). For the proof of theleft-hand side inequality in (4.1), the following geometric lemma plays a crucialrole.

Lemma 4.1 (see [3]). Let y = y(r) be a continuous function on [a, b] and letrk ∈ (a, b) be a decreasing sequence of consecutive zeros of y(r) such that rk ↗ b.Let k = k(ε) be an index function k : (0, ε0) → (k0,∞) such that

(4.2) |rk − rk+1| ≤ ε for all k ≥ k(ε) and ε ∈ (0, ε0).

Then

(4.3) |Γε(y)| ≥∑

k≥k(ε)

maxr∈[rk,rk+1]

|y(r)|(rk+1 − rk) for all ε ∈ (0, ε0).

Let now y = y(r) be a solution of equation (1.12). We can suppose, for instance,that ϕN is increasing (p > N and p = N , let see (2.3)). The case when ϕN isdecreasing (p < N) can be considered analogously. From (3.5), (3.6) and (3.8), weobtain:

(4.4)

∑k≥k(ε)

maxr∈[rk,rk+1]

|y(r)|(rk+1 − rk)

≥ c5∑

k≥k(ε)

∣∣ϕ−1N (rk) − ϕ−1

N (b)∣∣ σpq∣∣ϕ−1

N (rk) − ϕ−1N (b)

∣∣σp= c5

∑k≥k(ε)

∣∣ϕ−1N (rk) − ϕ−1

N (b)∣∣σp ( 1

q+1) ≥ c6∑

k≥k(ε)

(1

k + k0

) σσ−p (

1q+1)

≥ c7

(1

k(ε) + k0

) σσ−p (

1q+1)−1

.

Next, let k = k(ε), k : (0, ε0) → (k0,∞) be an index function satisfying:

(4.5) d0ε− σ−p

σ + k0 − 1 < k(ε) < 2d0ε− σ−p

σ − k0,

for some positive d0 and ε0. Then k = k(ε) satisfies the required statement (4.2).Indeed,(

2d0ε− σ−p

σ − k0

)−(d0ε

− σ−pσ + k0 − 1

)= d0ε

− σ−pσ − 2k0 + 1 > 1, ε ∈ (0, ε0),


which means that k(ε) is a well defined natural numbers for any ε ∈ (0, ε0). More-over, with the help (3.5) and (3.6), for k > k(ε) we obtain:

|rk − rk+1| ≤ c2∣∣ϕ−1

N (rk+1) − ϕ−1N (b)

∣∣σp ≤ c2cσp

4

(1

k + 1 − k0

)σp

pσ−p

≤ c8

(1

k(ε) + 1 − k0

) σσ−p

≤ c8d− σ−p

σ0 ε

σ−pσ

σσ−p ≤ ε,

which proves the desired inequality (4.2), where we take d0 such that c8d− σ−p

σ0 < 1.

Therefore, we can use Lemma 4.1. So, from (4.3), (4.4) and (4.5) we conclude:

|Γε(y)| ≥ c

(1

k(ε) + k0

) σσ−p (

1q+1)−1

≥ cε1q+

pσ ≥ cε2−s0 , ε ∈ (0, ε0),

where s0 is defined in (1.14). Thus, the left-hand side inequality in (4.1) is shown.�

Next, it remains to show the right-hand side inequality in (4.1).

Proof of the right-hand side inequality in (4.1). The next lemma willplay a crucial role in the proof of the right-hand side inequality in (4.1).

Lemma 4.2 (see [11, Lemma 3.4]). Let y ∈ C1([a, b)) be bounded on [a, b).Assume that

(4.6) limr→b

∫ r

a

|y′(ξ)|dξ = ∞

and

(4.7) lim supr→b

⎧⎨⎩[(b− r) sup

ξ∈[r,b)

|y(ξ)|] s−1

2−s ∫ r

a

|y′(ξ)|dξ

⎫⎬⎭ < ∞

for some s ∈ (1, 2). Then there exists c2 > 0 such that

|Γε(y)| ≤ c2ε2−s, ε ∈ (0, 1).

Let now y = y(r) be a nontrivial solution of equation (1.12). First of all, with

the help of [7, Theorem 1.1] we know that y(r) satisfies (4.6) provided f1/p2 �∈L1(a, b). On the other hand, from (1.8) we have:

f1/p2

(r) ∼ (b− r)− σ

p2 �∈ L1(a, b) since σ > p2.

Thus, y(r) satisfies the first condition of Lemma 4.2. Moreover, we prove that y(r)also satisfies (4.7) for s = s0, where s0 is defined in (1.14). In fact, from (3.14) wehave: [

(b− r) supξ∈[r,b)

|y(ξ)|] s−1

2−s ∫ r

a

|y′(ξ)|dξ

≤ c2

[(b− r) sup

ξ∈[r,b)

(b− ξ)σpq

] s−12−s ∫ r

a

(b− ξ)− σ

p2 dξ

≤ c3

[(b− r)1+

σpq

] s−12−s

(b− r)1− σ

p2 = c3.


Taking the limit superior on both sides in previous inequality we see that y(r)satisfies the required condition (4.7). Thus, we may apply Lemma 4.2 on y(r) andconclude that:

|Γε(y)| ≤ c3ε2−s0 , ε ∈ (0, 1).

Hence, the right-hand side inequality in (4.1) is shown. �

5. Proof of Proposition 1.1

The proof of Proposition 1.1 in the case of N = 2 is based on the intuitivelyobvious property that the graph of u is locally bi-Lipschitz diffeomorphic withΓ(y) × (0, 1). Hence, using finite stability of the upper box dimension and itsadditivity property with respect to Cartesian product, see Falconer [1], we concludethat dimBΓ(u) = dimBΓ(y) + 1. The aim is to show that in the general case thegraph of u is locally bi-Lipschitz equivalent with Γ(y) × (0, 1)N−1, see Lemma 5.3below.

Let us first consider the case of N = 2. It will be convenient in the rest ofthis section to denote the polar coordinates of a point T (r, ϕ) in the plane R2 by(r, ϕ)p, while the Cartesian coordinates T (x, y) will be denoted by (x, y)c. Hence,in general (r, ϕ)p �= (r, ϕ)c. Let a and b be a two fixed positive real numbers suchthat 0 < a < b, and let Ωp be a set (ring-like sector) defined by

(5.1) Ωp = {(r, ϕ)p ∈ R2 : r ∈ (a, b), ϕ ∈ (α, π − α)}.The indices p and c stands for “polar” and “Cartesian”.

Here we take a fixed α ∈ (0, π/2). Now we define the following rectangle in theplane:

(5.2) Uc = {(r, ϕ)c ∈ R2 : r ∈ (a, b), ϕ ∈ (α, π − α)} = (a, b) × (α, π − α).

The mapping

(5.3) Φ : Ωp → Uc, Φ ((r, ϕ)p) = (r, ϕ)c

is clearly bijective. Furthermore, it maps the circular fibres from Ωp onto verticalfibres of Uc, more precisely, for any fixed ρ ∈ (a, b) we have Φ(Ωp ∩ {r = ρ}) =Uc ∩ {x = ρ}. In other words, this mapping rectifies the sector onto the rectangle.It is intuitively clear that due to a > 0 the mapping Φ is diffeomorphic. Here is amore precise statement.

Lemma 5.1. The mapping Φ : Ωp → Uc is a bi-Lipschitz diffeomorphism.

Proof. Using Cartesian coordinates, we can write (5.3) as follows:

(5.4) Φ ((x, y)c) =(√

x2 + y2, arctany

x

)c,

for (x, y)c ∈ Ωp. The condition α ∈ (0, π) is equivalent to saying that the convexhull of the closure of Ωp, which we denote by Ω′

p = co (Cl Ωp), does not contain theorigin of the plane. Indeed, for any (x, y)c ∈ Ω′

p we obviously have y ≥ a sinα > 0.Extending the definition Φ in (5.3) by the same formula to the set Ω′

p, for any(x, y)c ∈ Ω′

p we have

(5.5) Φ′(x, y) =

⎛⎜⎝x√

x2 + y2y√

x2 + y2

− y

x2 + y2x

x2 + y2

⎞⎟⎠ .


Using the Lagrange mean value theorem, we have that for all (x, y)c ∈ Ω′p,

‖Φ′(x, y)‖∞ = max

{|x| + |y|√x2 + y2

,|x| + |y|x2 + y2

}≤ max

{2b

a sinα,

2b

a2 sin2 α

}=

2b

a2 sin2 αmax{a sinα, 1} =: LΦ,

where we deal with ∞-norm in R2. Hence, the mapping Φ is Lipschitzian with aLipschitz constant equal to LΦ.

Denoting U ′c = Φ(Ω′

p), it is easy to see that the inverse mapping Ψ of Φ : Ω′p →

U ′c is defined by

(5.6) Ψ((x, y)c) = (x cos y, x sin y)c,

for (x, y)c ∈ U ′c. After a short computation we find that it is Lipschitzian with

LΨ = 1 + R. Therefore, Φ is a bi-Lipschitz diffeomorphism. �

Remark 5.1. The values of Lipschitz constants LΦ and LΨ obtained in theproof of Lemma 5.1 are not optimal.

Remark 5.2. Viewing Ωp and Uc as subsets of the complex plane C = R2, themapping Φ : Ωp → Uc in (5.4) can be understood as complex function of a complexvariable defined by Φ(reiϕ) = r+iϕ, i.e., Φ(z) = |z|+i arg z, with arg z ∈ (α, π−α).It is easy to see that the function Φ is not analytic, since it does not satisfy theCauchy-Riemann conditions.

An analogous result as in Lemma 5.1 can be stated in R3. Indeed, let (r, ϕ, θ)be polar (more precisely, spherical) coordinates in R3, so that

(5.7) x = r sin θ cosϕ, y = r sin θ sinϕ, z = r cos θ.

Let us define the following two sets in R3:

Ωp ={(r, ϕ, θ)p ∈ R3 : r ∈ (a, b), ϕ, θ ∈ (α, π − α)

},

Uc = (a, b) × (α, π − α)2,

where α ∈ (0, π/2) is fixed. Then the mapping

(5.8) Φ : Ωp → Uc, Φ ((r, ϕ, θ)p) = (r, ϕ, θ)c,

with the meaning of subscripts anologous to those for N = 2, is a bi-Lipschitzdiffeomorphism. This follows similarly as in the proof of Lemma 5.1, since in theCartesian coordinates we have

(5.9) Φ((x, y, z)c) =

(√x2 + y2 + z2, arctan

y

x, arccos

z√x2 + y2 + z2

)c

for all (x, y, z)c ∈ Ω′p = co (ClΩp), and

(5.10) Φ′(x, y, z) =

⎛⎜⎜⎝xr

yr

zr−y

x2+y2x

x2+y2 0

xz

r2√

x2+y2

yz

r2√

x2+y2−

√x2+y2

r2

⎞⎟⎟⎠ ,

where r =√x2 + y2 + z2. Due to (5.7) we have that for all (x, y, z)c ∈ Ω′

p,

(5.11) y ≥ a sin2 α > 0,


so that ‖Φ′(x, y, z)‖∞ is bounded by an explicit constant LΦ depending only on a,b and α. The inverse mapping Ψ = Φ−1 is

(5.12) Ψ ((x, y, z)c) = (x cos y sin z, x cos y cos z, x cos z)c.

Passing to the general case of RN , N ≥ 3, we introduce polar (i.e. spherical)coordinates

(5.13) (r, θ1, θ2, . . . , θN−1)p,

using the following relations with Cartesian coordinates x = (x1, x2, . . . , xN ):

x1 = r sin θN−1 sin θN−2 . . . sin θ3 sin θ2 cos θ1

x2 = r sin θN−1 sin θN−2 . . . sin θ3 sin θ2 sin θ1

x3 = r sin θN−1 sin θN−2 . . . sin θ3 cos θ2

...(5.14)

xN−2 = r sin θN−1 sin θN−2 cos θN−3

xN−1 = r sin θN−1 cos θN−2

xN = r cos θN−1.

Note that in the case of N = 3 the angle θ1 has the role of ϕ. Fixing α ∈ (0, π/2),we define the following sets in RN :

(5.15)Ωp = {(r, θ1, . . . , θN−1)p : r ∈ (a, b), θj ∈ (α, π − α), j = 1, . . . , N − 1}Uc = (a, b) × (α, π − α)N−1.

Lemma 5.2. The mapping Φ : Ωp → Uc defined by

(5.16) Φ ((r, θ1, . . . , θN−1)p) = (r, θ1, . . . , θN−1)c

is a bi-Lipschitz diffeomorphism.

Sketch of the proof. Let j ∈ {1, . . . , N}, and define the orthogonal pro-jection Pj : RN → Rj by Pj(x) = (x1, . . . , xj). Using (5.14) we see that

r = (x21 + · · · + x2

N )1/2,

θ1 = arctanx2

x1, θj = arccos

xj+1

|Pj+1x|for j = 2, . . . , N − 1.

The intuitive meaning of the angle θj for j = 2, . . . , N − 1 is the angle betweenpositive part of xj+1-axis and the vector Pj+1x. The mapping Φ has the followingform in Cartesian coordinates:

Φ(x) =

(|x|, arctan

x2

x1, arccos

x3

|P3x|, . . . , arccos

xN−1

|PN−1x|, arccos

xN

|x|

).

Here the norm is Euclidean. Therefore, Φ′(x) is given by⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x1

|x|x2

|x|x3

|x| . . . xN−1

|x|xN

|x|−x2

|P2x|2x1

|P2x|2 0 . . . 0 0x1x3

|P3x|2 |P2x|x2x3

|P3x|2 |P2x| − |P2x||P3x|2 . . . 0 0

.... . .

...x1xN−1

|PN−1x|2 |PN−2x|x2xN−1

|PN−1x|2 |PN−2x| . . . − |PN−2x||PN−1x|2 0

x1xN

|x|2 |PN−1x|x2xN

|x|2 |PN−1x| . . . xN−1xN

|x|2 |PN−1x| − |PN−1x||x|2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.


Using (5.14) we conclude that for each x ∈ Ω′p = co (ClΩp),

(5.17) x2 ≥ a sinN−1 α > 0,

so that Ω′p does not contain the origin. Furthermore, for each j = 2, . . . , N and

each x ∈ Ω′p,

(5.18) |Pj(x)| = r sin θN−1 . . . sin θj ≥ a sinN−j α > 0.

This proves that supx∈Ω′p‖Φ′(x)‖∞ < ∞, and hence, Φ is Lipschitzian. The inverse

mapping Ψ = Φ−1 is Ψ(x) = (g1(x), . . . , gN (x)), where

g1(x) = x1 sin xN sin xN−1 . . . sinx3 sin x2 cosx1

g2(x) = x1 sin xN sin xN−1 . . . sinx3 sin x2 sinx1

g3(x) = x1 sin xN sin xN−1 . . . sinx3 cosx2

...

gN−2(x) = x1 sin xN sin xN−1 cosxN−2

gN−1(x) = x1 sin xN cosxN−1

gN (x) = x1 cosxN .

The rest of the proof is similar to that of Lemma 5.1. �Lemma 5.3. Let y : (a, b) → R be a bounded function, and let u : Ωa,b → R

be defined by u(x) = y(|x|). Then Γ(u|Ωp), where Ωp is defined by (5.15), is bi-

Lipschitz equivalent with Γ(y) × (0, 1)N−1, and

(5.19) dimBΓ(u|Ωp) = dimBΓ(y) + N − 1.

Proof. There exists a0 > 0 such that sup{|y(r)| : r ∈ (a, b)} < a0. Let Φ :

Ωp → Uc, see (5.15), be defined by (5.16). Then the mapping Φ : Ωp × (−a0, a0) →Uc × (−a0, a0) defined by Φ(x, u) = (Φ(x), u) is a bi-Lipschitz diffeormophism,

and Φ(Γ(u|Ωp)) is isometrically isomorphic to Γ(y) × (α, π − α)N−1 with respect

to Euclidean metric. Indeed, the function Φ is bi-Lipschitzian due to Lemma 5.2.Since Φ((r, θ1, . . . , θN−1)p, v(r)) = (r, θ1, . . . , θN−1, y(r))c, where r and θj satisfythe following property P :

(5.20) P = {a < r < b, α < θj < π − α, ∀j = 1, . . . , N − 1},we have that

(5.21) Φ(Γ(u)) = {(r, θ1, . . . , θN−1, y(r))c : P}.Performing a permutation of coordinates, this set is clearly isometrically isomorphic(with respect to this permutation) to the set

(5.22) {(r, y(r), θ1, . . . , θN−1)c) : P} = Γ(y) × (α, π − α)N−1.

The claim follows from the transitivity of bi-Lipschitz equivalence and from theproduct formula for the upper box dimension, see Falconer [1]. �

Proof of Proposition 1.1. It is easy to see, due to the relative compactnessof Ωa,b, that for any fixed α ∈ (0, π/2) there exist finitely many sets Ω1, . . . ,Ωk ofthe form Ωp, generated with possibly different coordinate systems with respect tothe common origin, which cover Ωa,b. More precisely, the orthogonal coordinate

systems (x(i)j )Nj=1 defining Ωi may depend on i = 1, . . . , k. The first claim follows


from the finite stability property of the upper box dimension, see Falconer [1], andfrom Lemma 5.3.

If we assume that the graph of y = y(r) in Lemma 1.1 is Minkowski nonde-

generate, then using [17, Theorem 4.1] and the fact that Φ is bi-Lipschitzian, we

obtain that the set Φ(Γ(u)) is also Minkowski nondegenerate. In particular,

dimB Γ(u) = dimB Γ(y) + N − 1.

�

Remark 5.3. Assume that the graph of y in Lemma 1.1 is Minkowski non-degenerate (this condition is satisfied for the chirps considered in Example 1.1,

provided α < β). Using [17, Theorem 4.1] and the fact that Φ is bi-Lipschitzian,

we obtain that the set and Φ(Γ(u)) is also Minkowski nondegenerate.

6. Proof of Lemma 1.1

Let suppose the opposite claim of (i)-Lemma 1.1, that is,

(6.1) lim supr→b

f(r) < ∞.

It together with f(r) > 0 in [a, b) and f ∈ C([a, b)) gives that f(r) is boundedon [a, b]. Consequently, the function F (t) defined in (2.11) is also bounded on IN ,where interval IN is defined by (2.1). That is, we observe that the coefficient F (t)of equation (2.10) satisfies

(6.2) 0 < F (t) ≤ c0 on IN ,

for some constant c0 > 0. Let now consider the next half-linear differential equationwith constant coefficients:

(6.3)

⎧⎨⎩ (|w′0|p−2w′

0)′ + c0|w0|p−2w0 = 0, t > 0,

w0(0) = 0, w′0(0) = ((p− 1)/c0)

1/p.

Then, by the transformation

w(ξ) = w0(t), ξ = (c0/(p− 1))1/p t,

equation (6.3) is transformed into the generalized sine equation (3.9), where w =w(ξ) is generalized sine function. Since w(ξ) has a finite zeros on any boundedinterval (see [12, Section 2]), we observe that

(6.4) w0(t) is nonoscillatory on any bounded interval.

In what follows, we also use the notation ∂IN and ϕN (t) defined respectively in(2.2) and (2.3). Let u(x) = y(|x|) be a radially symmetric solution of (1.1) whichoscillates near x = |b|. Then y(r) oscillates near r = b and satisfies equation (1.12).Consequently, for the solution z(t) = y(ϕN (t)), t ∈ IN , of equation (2.10), by (2.7),we conclude that

(6.5) z(t) oscillates near t = ∂IN .

According to inequality (6.2), we apply the Sturm comparison principle to the half-linear equations (2.10) and (6.3), which shows that between any two consecutivezeros of z(t) there is at least one zero of w0(t). It together by (6.5) gives that w0(t)oscillates on IN . But, it is not possible because of (6.4). Thus, the assumption(6.1) is not possible and hence, the first claim of this lemma is shown.


The second claim of this lemma immediately follows from the fact that f ′(r) > 0on [a, b) and lim supr→b f(r) = ∞ implies limr→b f(r) = ∞.

Acknowledgements

The second and fourth authors would like to thank the University of Messinaand in particular to Dr. David Carfı, for their very successful organization of the“First International Meeting of PISRS, Conference 2011: Analysis, Fractal Geom-etry, Dynamical Systems, and Economics”. Their participation at this conferencewas an impetus to prepare this article.

References

[1] Kenneth Falconer, Fractal geometry: Mathematical foundations and applications, John Wiley& Sons Ltd., Chichester, 1990. MR1102677 (92j:28008)

[2] Philip Hartman, Ordinary differential equations, 2nd ed., Birkhauser Boston, Mass., 1982.MR658490 (83e:34002)

[3] Man Kam Kwong, Mervan Pasic, and James S. W. Wong, Rectifiable oscillations in second-order linear differential equations, J. Differential Equations 245 (2008), no. 8, 2333–2351,DOI 10.1016/j.jde.2008.05.016. MR2446834 (2009k:34078)

[4] M. L. Lapidus, G. Radunovic, D. Zubrinic, Fractal analysis of zeta functions in Euclideanspaces, in preparation.

[5] Michel L. Lapidus and Machiel van Frankenhuijsen, Fractal geometry, complex dimensionsand zeta functions. Geometry and spectra of fractal strings, 2nd ed., Springer Monographsin Mathematics, Springer, New York, 2013. MR2977849

[6] Yuki Naito, Uniqueness of positive solutions of quasilinear differential equations, DifferentialIntegral Equations 8 (1995), no. 7, 1813–1822. MR1347982 (96g:34039)

[7] Yuki Naito, Mervan Pasic, and Hiroyuki Usami, Rectifiable oscillations of radially symmetricsolutions of p-Laplace differential equations, Differ. Equ. Appl. 4 (2012), no. 1, 11–25, DOI10.7153/dea-04-03. MR2952627

[8] Mervan Pasic, Minkowski-Bouligand dimension of solutions of the one-dimensional p-Laplacian, J. Differential Equations 190 (2003), no. 1, 268–305, DOI 10.1016/S0022-0396(02)00149-3. MR1970964 (2004b:34052)

[9] Mervan Pasic, Fractal oscillations for a class of second order linear differential equations ofEuler type, J. Math. Anal. Appl. 341 (2008), no. 1, 211–223, DOI 10.1016/j.jmaa.2007.09.068.MR2394076 (2009d:34084)

[10] Mervan Pasic and Satoshi Tanaka, Fractal oscillations of self-adjoint and damped lineardifferential equations of second-order, Appl. Math. Comput. 218 (2011), no. 5, 2281–2293,DOI 10.1016/j.amc.2011.07.047. MR2831502

[11] Mervan Pasic and Satoshi Tanaka, Fractal oscillations of chirp functions and applications tosecond-order differential equations, Int. J. Differ. Equ. 2013, Article ID 857410, 11 p. (2013).

[12] Mervan Pasic and James S. W. Wong, Rectifiable oscillations in second-order half-linear differential equations, Ann. Mat. Pura Appl. (4) 188 (2009), no. 3, 517–541, DOI10.1007/s10231-008-0087-0. MR2512161 (2010d:34057)

[13] M. Pasic, D. Zubrinic, V. Zupanovic, Fractal properties of solutions of differential equations,in Hagen, W. L. (ed.), Classification and Application of Fractals, Nova Science Publishers,Inc., 2011, pp. 1–62.

[14] Claude Tricot, Curves and fractal dimension, Springer-Verlag, New York, 1995. With a fore-word by Michel Mendes France; Translated from the 1993 French original. MR1302173(95i:28005)

[15] Jingfa Wang, On second order quasilinear oscillations, Funkcial. Ekvac. 41 (1998), no. 1,25–54. MR1627369 (99c:34065)


[16] Darko Zubrinic, Analysis of Minkowski contents of fractal sets and applications, Real Anal.Exchange 31 (2005/06), no. 2, 315–354. MR2265777 (2008b:28015)

[17] Darko Zubrinic and Vesna Zupanovic, Fractal analysis of spiral trajectories of somevector fields in R3, C. R. Math. Acad. Sci. Paris 342 (2006), no. 12, 959–963, DOI10.1016/j.crma.2006.04.021 (English, with English and French summaries). MR2235618(2007g:37016)

Department of Mathematics, Ehime University, Matsuyama 790-8577, Japan


University of Zagreb, Faculty of Eletrical Engeneering and Computing, Depart-

ment of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia

E-mail address: [email protected] address: [email protected]

Okayama University of Science, Okayama 700-0005, Japan


University of Zagreb, Faculty of Eletrical Engeneering and Computing, Depart-

ment of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia



Applications of the Contraction Mapping Principle

John R. Quinn

Abstract. In the setting of complete metric spaces, the contraction mappingprinciple is the crucial tool used to prove the existence of self-similar fractalsets and measures. We discuss this principle and its applications. We willshow that self-similarity is fundamental to much of applied science. We alsoinvestigate the role of the Contraction Mapping Principle in the proofs of manyfundamental results in mathematics.

For one example, we recall that the contractivity of the Picard operator,used to show the existence and uniqueness of the solutions to initial-valueproblems via the Contraction Mapping Principle can be applied to the solutionof certain inverse problems of ordinary differential equations. We review alsothe use of the Contraction Mapping Principle to show that the final coalgebracarried by the set of streams of symbols representing a fractal in a coalgebraicrepresentation theory is a fixed point of a contractive functor and that thusfractality is categorical.

As a novel application, we present a scenario in which position uncertaintyof locations and the geometric contractivity of the causal history of an intervalof spacetime imply that past events have the structure of spacelike fractals.

Contents

1. The Contraction Mapping Principle2. Corollaries, Applications and Implications3. Fractal Method of Solutions to Inverse Problems of ODEs4. Self-Similarity5. A Derivative Corresponding to the Box-Counting Dimension6. Representation Theory of Fractal Sets7. Spacelike Cantor Sets in a Toy Model8. Concluding Remarks and Future DirectionsReferences

1. The Contraction Mapping Principle

Definition 1. For a subset D of a metric space (X, d), a mapping S : D → Dis called a contraction mapping on D if there is a number c with 0 < c < 1, suchthat d(S(x), S(y)) ≤ cd(x, y) for all x, y ∈ D. We call the number c the scalingratio, contraction ratio, or Lipschitz constant of S.

2010 Mathematics Subject Classification. Primary: 28A80, 34A55; Secondary: 28A12,28A15, 34A12, 37C70.

c©2013 American Mathematical Society345

346 JOHN R. QUINN

Banach’s contraction mapping theorem, known variously as Banach’s FixedPoint Theorem, or Banach’s Contraction Mapping Principle, is the most widelyapplied of the class of fixed point theorems [22], which are considered among themost useful in mathematics. These theorems tell us that a function satisfying somegeneral hypotheses has a value in its domain which is fixed under evaluation bythe function. In the case of the Contraction Mapping Principle, the condition onour function is that the function be a contraction mapping; i.e., that the distancebetween points in the image of the function will be less than the distance betweenthe corresponding points in the preimage. We require also that the domain andrange of this contraction be a metric space in which Cauchy sequences converge,that is, we need our space to be complete.

It is this theorem which implies the existence of attractors of iterated functionsystems (IFS), which are families of contraction mappings on complete metric spaces[8], [6], whose attractors form an important class of fractals. Since contractionmappings are automatically continuous, they conserve compactness of the preimage,so that we expect the limiting attractor to preserve compactness. But what isperhaps unexpected is that the limit of the images of the IFS as the number ofiterations tends to infinity will be interesting. After all, the theorem tells us of theexistence of a fixed point, not of a fixed space. Perhaps even stranger, we find thatthe same attractor results when we iterate the IFS starting with any nonemptycompact set in the domain of the IFS. By envisioning these attractors as points ina space of compact sets, endowed with the Hausdorff metric under which this spaceis complete, we will see that the existence of attractors of the IFS is a direct resultof the Contraction Mapping Principle.

We now state and prove the Contraction Mapping Principle and discuss someof its implications in section 3.

Theorem 1 (Banach’s Contraction Mapping Principle). For a complete metricspace (X,d) and a contraction mapping S : X → X, there exists a unique ξ ∈ Xsuch that S(ξ) = ξ and for all x ∈ X the sequence

{xn}∞n=0 := {x, S(x), S(2)(x), ..., S(n)(x), ...}∞n=0,

converges to ξ (where we define S(n)(x) = S(S(n−1)(x)) and S0(x) = x).

This celebrated result of Steven Banach, (possibly going back at least as faras Emile Picard in the case of nonlinear contractions in complete metric spaces)has very many well known applications and implications. A simple and immediatecorollary, proved in the discussion of inverse problems, is the collage theorem whichis often used in fractal image processing and bounds the distance between thepreimage of a contraction mapping and the fixed point of that mapping.

Proof. If x �= y and S(x) = x and S(y) = y, then since c < 1 we have

d(x, y) = d(S(x), S(y)) ≤ cd(x, y) < d(x, y) �= 0.

Thus we reach a contradiction, d(x, y) < d(x, y), so therefore, the fixed pointmust be unique.

For any x, the sequence of iterates under S(x), i.e. the sequence

{x, S(x), S(2)(x), ..., S(n)(x), ...}∞n=0

APPLICATIONS OF THE CONTRACTION MAPPING PRINCIPLE 347

is Cauchy, then by completeness, we will have convergence. Observe that for alln ≥ 1, we have d(S(n)(x), S(n+1)(x)) ≤ cd(S(n−1)(x), S(n)(x)), so that

d(S(n)(x), S(n+1)(x)) ≤ c2d(S(n−2)(x), S(n−1)(x)) ≤ ... ≤ cnd(x, S(x)).

Then, for some m > n, by the triangle inequality we have

d(S(n)(x), S(m)(x)) ≤m−1∑i=n

d(S(i)(x), S(i+1)(x)),

so that

d(S(n)(x), S(m)(x)) ≤ cnd(x, S(x)) + cn+1d(x, S(x)) + ... + cm−1d(x, S(x))

= (m−1∑r=n

cr)d(x, S(x)) ≤∞∑

r=n

crd(x, S(x)) =cn

1 − cd(x, S(x)).

Since c < 1, for any ε > 0 we can find N ≥ 1 such that cN

1−cd(x, S(x)) < ε. Then

if m > n ≥ N we have d(Sn(x), Sm(x)) ≤ cn

1−cd(x, S(x)) ≤ cN

1−cd(x, S(x)) < ε. So

we see that {x, S(x), S(2)(x), ..., S(n)(x), ...}∞n=0 is a Cauchy sequence. Since (X, d)is complete there exists a unique ξ such that xn → ξ as n → ∞. Hence, by thecontinuity of S, ξ is clearly a fixed point of S.

�

2. Corollaries, Applications and Implications

Here we see how this theorem is applied in some classic cases:

Definition 2. We call a finite family S of contraction mappings {Si}Ni=1 (withN ≥ 2), an iterated function system or IFS. An IFS acts on a set A by

S(A) := ∪Ni=1Si(A),

for any subset A of X [8].We call a compact set F invariant under the IFS S = {Si}Ni=1, if F = ∪N

i=1Si(F ).We then refer to F as the attractor of S, and we write F = S(F ), to denote thatF is fixed under S.

Theorem 2. (Existence of attractors of IFS.) For any iterated function systemS on a complete metric space (X, d), there exists a unique invariant set F fixedunder S, and for any nonempty compact subset E ⊂ X, such that Si(E) ⊂ E forall i, the iterates S(n)(E) → F as n → ∞.

Sketch. An IFS defined on a complete metric space (X, d), naturally inducesa contraction mapping in the complete metric space of nonempty compact subsetsof X, equipped with the Hausdorff metric [6]. Thus, by the Contraction MappingPrinciple (Theorem 1), there exists a unique invariant set F . Iteration of S appliedto E ⊂ X as above, results in a decreasing sequence S(n)(E) of non-empty compactsets containing F. Therefore, the intersection ∩∞

n=1S(n)(E) = F . �

Let us recall that a vector ρ = (ρ1, ρ2, ..., ρN ) is called a probability vector

when we have ρi ∈ [0, 1], for all i and∑N

i=1 ρi = 1.

348 JOHN R. QUINN

Definition 3. Let S = {Si}Ni=1 be an IFS, and let ρ = (ρ1, ρ2, ..., ρN ) be aprobability vector with ρi ∈ (0, 1) for all i. We call (S, ρ) the IFS weighted by ρ. Itacts on measures acting on sets by

(S, ρ)μ(E) =

N∑i=1

ρiμ(S−1i (E)).

We call a measure μ such that (S, ρ)μ = μ an invariant measure under (S, ρ).

Theorem 3. (Existence of invariant measures.) For an IFS S = {Si}Ni=1,weighted by a probability vector ρ, there exists μ, a unique Borel regular, unit massmeasure with bounded support, such that μ is fixed under (S, ρ).

Sketch. (S, ρ) is a contraction in the complete metric space of Borel-regularprobability measures, under the L-metric [8]. Thus, existence and uniqueness fol-low, by Theorem 1. �

Analogous theorems for random self-similar fractals [6], and measures [9], mayalso be obtained, but under the weaker conditions of almost sure convergence andequality as distributions respectively.

Theorem 4 (Existence and uniqueness of solutions to a first order initial value

problem of ordinary differential equations). Suppose g(t, x) and ∂g∂x are continuous

functions on some rectangle a < t < b, c < x < d containing the point (t0, x0).Then there is an interval t0 − h < t < t0 + h contained in a < t < b on which thereis a unique solution to the initial value problem x(t) = g(t, x(t)), with x(t0) = x0.

Sketch. For functions g(t, x) with g(t, x) and ∂g∂x continuous on {(t, x) ∈

[a, b] × [c, d]}, the Picard integral operator P (g(t, x(t)) =∫ t

t0g(s, x(s)) ds + x0 is

a contraction mapping on the interval t0 − h < t < t0 + h, and clearly solves theinitial value problem. The result follows from the completeness of R2 and Theorem1. �

This method is referred to as fractal-based in [11] and is suggestive enough thatwe will consider an example later.

Theorem 5. (Newton’s method) For a function f(x) ∈ C2([a, b]), with a sim-ple zero, x ∈ [a, b], there exists a neighborhood Nα(x) ⊂ [a, b], of x, such that for

G(x) = x − f(x)f ′(x) , the sequence {xn}∞n=0 := {x,G(x), G(2)(x), ...}, converges to x.

We call the neighborhood Nα(x), a basin of attraction.

Sketch. For x close enough to x, f(x) = f(x)+f ′(x)(x−x)+O((x−x)2), asx → x. Solving for x gives us a formula for G(x). G′(x) = 0 and G(x) ∈ C2([a, b]),so there is a neighborhood Nα(x) such that G′(x) < 1. Then G(x) is a contractionmapping within Nα(x) ⊂ R. Observe that x is a fixed point of G(x). Therefore,by completeness of R, by Theorem 1, for any x ∈ Nα(x) the sequence {xn}∞n=0 ofiterates of G(x), converges to x. �

Interestingly, for many functions in higher dimensions, we find these basins tohave intricate fractal boundaries [19].

Theorem 6 (Inverse Function Theorem). Let f : Rn → Rn be a continuouslydifferentiable function in an open set containing a, and det(f ′(a)) �= 0. Thenthere is an open set V containing a and an open set W containing f(a) such that


f : V → W has a continuous inverse f−1 : W → V which is differentiable and forall y ∈ W ,

(f−1)′(y) = [f ′(f−1(y))]−1

Sketch. Inverting the Taylor expansion

f(x) = f(a) + f ′(a)(x− a) + o(‖x− a‖)we get that the local inverse f−1 : W → V is differentiable at a with

(f−1)′(f(a)) = [f ′(a)]−1

Then, normalizing a = f(a) = 0 and f ′(0) = In, the identity matrix on Rn, so thatcontinuity of f ′(x) shows that f ′(x) is close to In for x close to 0. Then with thefundamental theorem of calculus this implies that x �→ x−f(x)+y is a contractionmapping on a small ball around the origin for small y. Thus, by the completenessof Rn and by Theorem 1, the inverse exists, and by uniqueness of the attractor, itis given by the above formula. �

Recall also that the proof of the implicit function theorem relies on the inversefunction theorem.

3. Fractal Method of Solutions to Inverse Problems of ODEs

The following result is a well-known and immediate corollary of the Banachfixed point theorem.

Theorem 7 (Collage Theorem [11], [3]). For a complete metric space (X, d)and a contraction mapping S : X → X, with contraction constant c, if ξ is the fixedpoint of S, i.e., if S(ξ) = ξ, then for any x ∈ X,

d(x, ξ) ≤ 1

1 − cd(x, S(x)).

Proof. d(x, ξ) ≤∑∞

i=1 d(S(i−1), S(i)) ≤ d(x, S(x))

∑∞i=1 c

i, by the triangleinequality and contractivity of S. �

This theorem, a simple and well-known consequence of the contraction mappingprinciple (Theorem 1), is a key ingredient in the solutions to many inverse problemsof fractals and contraction mapping techniques. It is sometimes known as the“Collage Theorem” in textbooks on fractals (see e.g. [3] or [11]). Perhaps itsbest known use is to find an IFS that adequately fits a given fractal. While thispossibility guides our treatment of self-similarity throughout this paper, i.e., thatself-similar fractal sets are attractors of IFS or can at least be closely approximatedby such, here we concentrate on the Contraction Mapping Principle’s use in certaininverse problems of ordinary differential equations.

These inverse problems are viewed as a process of approximating a target el-ement in a complete metric space by the fixed point of a contraction mapping,accomplished by minimizing the distance between the target element and its im-age under a suitable contraction mapping, so that the collage theorem then wouldbound the distance between the target element and the fixed point of the contrac-tion mapping [11].

For an initial value problem, x(t) = g(t, x(t)) with x(t0) = x0, we may ap-proximate solutions by use of a contractive Picard integral operator P (g(t, x(t)) =

350 JOHN R. QUINN∫ t

t0g(s, x(s)) ds+x0, for Lipschitz continuous functions f(t, x(t)) (using a more gen-

eral version of Theorem 4). For the first n elements {φi(t, x)}ni=1 of an orthonormalbasis of an appropriate L2 space, we approximate f(t, x(t)) by

∑ni=1 aiφi(t, x), cre-

ating a Picard operator Pa for each a = (a1, a2, ..., an) ∈ Rn. Then we seek tominimize, by classical methods, the L2 distance squared of the difference betweenx(t) and Pa(x): we have

|x− Pax|22 =

∫t∈I

|x(t) −∫ t

0

n∑i=1

aiφi(s, x(s)) ds|2 dt.

Upon the minimization of this quantity, we have solved the following inverseproblem:

Problem 1 (Inverse Problem of ODE’s). Given a target solution curve x(t), fort ∈ I0, where I0 is some interval centered at x0, find a vector field g(t, x) (subject toappropriate conditions), such that the (unique) solution to the IVP x(t) = g(t, x(t))with x(t0) = x0 is as close to x as desired in L2 norm.

Of course given x(t) in closed form, we can often simply differentiate and ma-nipulate it to find g(t, x(t)), yet this technique is applicable to x(t) given a collectionof data points {(ti, xi)}ni=1, interpolated by a smooth curve, or when we wish torestrict g(t, x(t)) to a specific class, such as the polynomials.

4. Self-Similarity

Self-similarity comes in many flavors, including algebraic, analytic, geometricand stochastic. We will concern ourselves chiefly with the geometric and analyticnotions, therefore, we will define a set as being self-similar when it is composed ofscaled isometric copies of itself, we see that this describes the unique attractor ofan IFS e.g.,

F = ∪Ni=1Si(F ) = S1(F ) ∪ S2(F ) ∪ ... ∪ SN (F ).

An example is the unique compact set F which is fixed by the contraction mappingS in the complete metric space of compact sets under the Hausdorff metric. Evenfor self-similar sets that are not formed by iterated function systems, a contractionmapping based algorithm exists to find an IFS fractal arbitrarily close to F [3]. Thusself-similarity and its implications are very closely related to contraction mappingsand the Contraction Mapping Principle.

Definition 4. The box-counting dimension, dB(F ), of a set F ⊂ Rn is defined

as limr→0logN(r)− log r , where N(r) is the number of n-cubes of side length r required to

cover F (for equivalent definitions, see [6]).

Definition 5. We say that the IFS S = {Si}Ni=1 satisfies the open set conditionif there is an open set U such that ∪N

i=1S(U) ⊂ U , and if i �= j then Si(U)∩Sj(U) =∅. For a self-similar set F that is the attractor of an iterated function systemS = {Si}Ni=1, satisfying the open set condition, with N ≥ 2, and with scaling ratios{ri}Ni=1, we define the similarity dimension of F to be the unique real solution dSto the equation

∑Ni=1 r

dSi = 1.

Theorem 8 (Moran’s Theorem [6]). For a set F invariant under an IFS S ={Si}Ni=1 satisfying the open set condition, and with contraction ratios {ri}Ni=1, with


ri ∈ (0, 1) for all i = 1, ..., N , the similarity dimension of F , dS(F ), and the box-counting dimension dB(F ) both equal the Hausdorff dimension dH(F ) (see e.g. [6]for a discussion of Hausdorff dimension).

Example 1. Let S = {Si}Ni=1 be an IFS satisfying the open set condition andwith scaling ratios {ri}Ni=1 such that ri = r for all i = 1, ..., N . Then dS solves

N · rd = 1, so that we obtain dS = − logNlogr .

Example 2. Since classical fractal constructions such as the Cantor set, theKoch curve, and the Menger Sponge, satisfy the open set condition, and are fixedunder contractions with the same ratio in each construction, they have similaritydimension ds given by Theorem 8, and Example 1. Then, by Theorem 8, we knowds(F ) to be the box-counting, and Hausdorff dimensions of each fractal F below, aswell:

Fractal Number Contraction dsSet of Contractions ratioCantor Set 2 1/3 log 2/ log 3Koch Curve 4 1/3 log 4/ log 3Menger Sponge 20 1/3 log 20/ log 3Sierpinski Carpet 8 1/3 log 8/ log 3Sierpinski Triangle 3 1/2 log 3/ log 2

We may informally derive the equations in Definition 5, and Example 1, above,

by reasoning that vol(F ), the ds-dimensional volume of F , is vol(F ) =∑N

i=1 vol(riF ),where each function in the IFS scales F by a factor ri. By the disjointness pro-vided by the open set condition, and by supposing that vol(F ) scales as vol(rF ) =rdSvol(F ); where dS is the similarity dimension. This scaling behavior is at theheart of our notion of self-similarity for functions.

Definition 6. We say that a function f is scale invariant with exponent β ifthere is a number β such that μ(aE) = aβμ(E) or if f(ax) = aβf(x), respectively[23].

We will see that this notion of self-similarity is fundamental to applied science,since scale invariance implies that physical laws are independent of the units usedto measure them.

4.1. Symmetry of Scale and Conservation of Physical Quantities.Fractal research often concerns one particular symmetry in nature: the symme-try of scale. Often we will invoke the beauty of fractal images or the amazingcomplexity of chaos, but we may not mention that this particular symmetry isof fundamental importance in science, especially in the field theories of physics[D-M-S]. Transformations of scale, together with translations, the special confor-mal transformations and the Lorentz transformations, form the group of conformalsymmetries, the global symmetry group of a non-supersymmetric interacting fieldtheory. Loosely speaking then, these symmetries imply that physical laws should bethe same no matter where we find ourselves (translation subgroup), no matter whatspeed we are traveling at (Lorentz subgroup), no matter if we exchange the rolesof the very far and very near (inversions in the special conformal transformations),and no matter the size of the scale of observation (scale symmetry subgroup).

We recall an impressive theorem of Emmy Noether [1].

352 JOHN R. QUINN

Theorem 9 (Noether’s Theorem). If an action admits a one parameter familyof diffeomorphisms, it has a first integral.

This theorem is paraphrased in [23] as saying that “for every continuous sym-metry of the laws of physics, there must exist a conservation law. For every con-servation law, there must exist a continuous symmetry,” referring to the vanishingderivative of the first integral as a conservation law. This theorem is stated in termsof an action e.g. a Lagrangian L = T − V .

Example 3. If the potential V (�r) has the property that for any scalar α,V (α�r) = αkV (�r) for some k, i.e. if V is scale invariant, then under the trans-

formations �r �→ α�r and t �→ βt we have r �→ αβ r and T �→ α2

β2 T , so that β = α1− k2

means that

L(α�r) = T (α�r) − V (α�r) =α2

β2T (�r) − αkV (α�r) = αkL(�r),

so that the Lagrangian is invariant when we assume a symmetry of scale.

This example demonstrates that the action has a scale symmetry, and impliesthe existence of a conserved quantity. The change in scale of the parameters ofthe action corresponds to a change in units of measurement,and the conservationof the relative quantities then results in the validity of the science of dimensionalanalysis. Dimensional analysis has proven useful in the study of difficult nonlinearproblems, once a suitable choice of a “similarity variable” has been made [23].

4.2. Dimensional Analysis. As a consequence of the scale symmetry of theLagrangian of a system, we can justify the use of a tool for forming hypotheses,checking solutions and determining units of relevant quantities, often greatly sim-plifying the analysis of nonlinear problems. Units of measurements are thought ofas measuring what are called the dimensions of a system. In mechanics, these arethe fundamental quantities mass (M), length (L), and time (T ).

The independence of the scale of units demonstrated above implies that mean-ingful physical laws must be homogeneous in terms of physical dimensions, so thatthe same dimensions appear on both sides of an equal sign, and only quantities in thesame dimensions can be added or subtracted. Quantities in differing dimensionsare combined by multiplication, so that monomials MμLλT τ represent elements< μ, λ, τ > in a 3-dimensional vector space over Q with rational powers (MμLλT τ )q

of those monomials corresponding to scalar multiplication of these vectors. In lightof this structure, we can view the choice of fundamental dimensions as a basis of Q3,with the basis {M,L, T} corresponding to {M,L, T}={(1, 0, 0), (0, 1, 0), (0, 0, 1)}but a basis consisting of the dimensions force(F ), length (L), and time (T ) cor-responds to the basis {F,L, T}={(1, 1,−2), (0, 1, 0), (0, 0, 1)} (with respect to thebasis {M,L, T}) since [F ] = [MLT−2].

Example 4. In his 1941 theory of turbulence, A. N. Kolmogorov determinedthat the velocity ul of the flow in an eddy of size l should be a function of the

energy transfer rate ε = d(u2)dt . The relevant quantities have the dimensions [l] = L,

[ul] = LT−1 and [ε] = L2T−3, corresponding to the vectors (0, 1, 0), (0, 1,−1) and(0, 2,−3) respectively. Then we solve (0, 1,−1) = a[b(0, 1, 0)+ c(0, 2,−3)], since wewant to express ul in terms of l and ε. The solution a = 1

3 , b = 1, c = 1 corresponds

to multiplying the two variables and taking the cube root, to obtain ul = c(lε)1/3.


The dimensionless constant c is a result of Buckingham’s Pi theorem, which impliesthat since there is one dimension in our basis unused in our formula, there is onedimensionless constant in the solution [2].

5. A Derivative Corresponding to the Box-Counting Dimension

In effect, the work of scientists who propose fractal geometries to describenatural structures, suggests that self-similarity holds only for a certain ranges ofscales [18], [20]. Like virtually all fractality that is observed in the natural world, itis approximate [26]. This leads us to attempt to develop fractal analysis tools thatcan estimate the fractal dimension of a set at a particular scale. Scale dependenceof structures is called scale covariance, or dependence of a phenomenon on the scaleof observation, which includes self-similarity as a special case.

According to B. B. Mandelbrot, “the familiar box dimension DB simply mea-sures the rate of increase of N(b) with b”, [17], see also [25]. Thus motivated, webegin our study of scale covariance by considering the slope of a log-log plot of thebox-counting function against the scale of measurement [4], as the generalized de-

rivative, d logN(r)d log(r) . Using this derivative, we can find ODE’s to describe fractals as

well as prefractal, almost self-similar structures. The box-counting dimension is thelimit as the scale vanishes of this derivative, essentially a “boundary condition”, forODE’s describing scale covariance. In the remainder of this section, we will employthe convention of using the variable r, to represent the inverse scale.

Definition 7. The box-counting derivative, is the dependence of the logarithmof the box counting function on the logarithm of the inverse scale, a generalizedfunction on the space of inverse scales r ∈ (0,∞),

∂BoxN(r) := rdN(r)dr

N(r),

in the sense of generalized derivatives of distributions.

Remark 1. (1) For N(r) a smooth, non-vanishing, mass distribution,∂BoxN(r) is the corresponding double logarithmic derivative:

d logN(r)

d log r= lim

r0→r

logN(r) − logN(r0)

(log r − log r0)

Also, whenever the limit of this quantity exists as r → ∞, for a given F ,we see that ∂BoxNF (r) approaches the box-counting dimension dB(F ).

(2) For a given F with N(r) a discrete counting function, ∂BoxN(r) is a sumof Dirac measures. This singular measure has support at the singularitiesof N(r), and we recover the “slope of the log-log- plot” by evaluating thissingular measure at its singularities. This quantity estimates the box-counting dimension of F , and may vary over changes in inverse scaler.

Example 5. For N(r) smooth, we can verify that we get the expected power

law solution N(r) = rk, for a fixed constant k, for the equation d logN(r)d log r = logN(r)

log r

by a simple separation of variables and exponentiation. If we suppose that scaleinvariance holds only in a range of scales [a, b], and that F scales as do points

outside of [a, b], then separating and integrating we find that N(r) = eba

k

, that is,N remains constant on [a, b].

354 JOHN R. QUINN

Example 6. Let F be the middle thirds Cantor set. Observing that each incre-ment of (inverse) scale increases the number of “boxes” (line segments in R) neededto cover F by a factor of 2. Then we can compute the counting function of F interms of the inverse scale (or magnification factor) as NF(r) = 2�log3 r�0 , where wedefine �r 0 := max(0, �r ), and �r is the least integer greater than or equal to r.Then

∂BoxNF(r) =r

2�log3 r�02�log3 r�0 log 2

∑∞n=0 δ(r − 3n)

r log 3=

log 2∑∞

n=0 δ(r − 3n)

log 3.

We recover the pointwise “slope of the log-log- plot” by computing

∂BoxNF(r) =

{0 if log3 r /∈ Zlog 2log 3 if log3 r ∈ Z.

We see that ∂BoxNF(r) = ∂BoxNF(3r), thus it has multiplicative period of 3.

In the concluding example above, we observed the log-periodicity of the mea-sure associated to the estimated box-dimension for a given deterministic fractal.This is one example of log-periodic scaling observed in both fractal geometry andin the study of critical systems. Since this periodicity may be different for frac-tals that share the same box-counting dimension, we propose that this techniquecan be useful in determining lacunarity of deterministic fractals and for evaluatinggeometrical models of critical systems.

6. Representation Theory of Fractal Sets

Extending the theory of self-similar fractals to that of self-similar measures, asin [8], has been a natural step toward understanding the algebra of self-similarity,since we may define integral transforms of these measures [28], giving us a type ofrepresentation theory for these objects [15].

Recent work in algebra [7] has developed a representation theory of the streamsof characters comprising the words on the alphabet of indices of the contractionmaps of the iterated function systems that give rise to self-similar fractals. Indeed,the contraction mapping principle is key to establishing a bijection between thefractal set itself and the representation by streams of characters.

Definition 8. An infinite stream of characters is a word a0a1a2... ∈ {0, ..., N−1}ω, the space of infinite words on {0, ..., N − 1}. For each element x in a self-

similar fractal F given by an IFS, S = {Si}N−1i=0 , a stream can be chosen so that for

all i, x ∈ S(ai) ◦ Sai−1 ◦ .... ◦ S(a0)(I), where, without loss of generality, I = [0, 1].

A self-similar fractal F ⊂ I resulting from an IFS (here we work withoutoverlaps, see [7] and [14] for generalizations) can be given a symbolic representationin terms of words σ = a0a1... ∈ Nω, the space of infinite words on the alphabetN := {1, ..., N}, we shall call each such word a stream, after [7]. Given this streamσ we assign a point [[σ]] in an interval I, calling the assignment [[ ]] : Nω → I thedenotation map and see that its restriction to equivalence classes determined by theelements in F which the streams indicate, Nω ∼−→ F is bijective by construction,with inverse F

∼−→ Nω called the representation map.Again following [7], we see that the set of symbolic representatives of F , Nω

carries the final coalgebra ι : Nω ∼−→ N · Nω for the combinatorial specification ofF , the functor N · ( ) : Sets → Sets, reflecting the N-fold recursive construction


of F . We call this final coalgebra the symbolic fractal for F , and note its recursivestructure. It is this view of the alphabet N as an IFS which we identify with analgebra χ : N · F → F . Then the (restricted) denotation map makes the followingdiagram commute:

(1)

N ·Nω N ·[[ ]]χ−−−−−→ N · F∼=G⏐⏐ι

⏐⏐Iχ

Nω [[ ]]χ−−−−→ FIn the absence of any overlap between the images of I under the functions in theIFS, Theorem 1 is used to show uniqueness of the denotation map:

Theorem 10. There exists a unique denotation map [[ ]]χ that makes the dia-gram (1) commute.

The following sketch highlights the use of the Contraction Mapping Principle:

Sketch. The set of morphisms Sets(Nω, I) is a complete metric space underd(f, g) = supσ∈Nω{d(fσ, gσ)}. On Sets(Nω, I) the map Φ : F �→ χ ◦ (N ·F ) ◦ ι is acontraction map. Therefore, by the Contraction Mapping Principle (Theorem 1),it has a unique fixed point.

�

7. Spacelike Cantor Sets in a Toy Model

The hypothesis that the contents of the universe may be arranged in fractalpatterns [18], [20], seems to suggest that space itself may have an underlying self-similarity. The goal of this section is to explore a simplified scenario in whichspacelike fractal sets can occur. The main ingredients are the domain of causalcontact viewed as a contraction mapping on intervals of space as time is reversed,the completeness of Euclidean space, and the almost sure discrepancy betweenmeasurements of positions by differing observers, when the measurement of theposition of the source is taken as a continuous random variable. This classicalscenario blissfully ignores quantum uncertainty, (other) spacetime singularities, andassumes an absolute time and an impartial observer.

Recall that the domain of influence at time t of an initial condition at space-timecoordinates (x, t) = (x0, 0) is the interval [x0−ct, x0+ct] in one dimension of space,as given by solutions to the wave equation uxx − cutt = 0 [27]. We first considerdiscrete increments in time to study the fractality of the origin of a signal in ourscenario. Then, this interval, for a given time of measurement t > 1, with t fixed,in steps of Δt is contracted by a contraction factor 2c(t−Δt)/2ct = (t−Δt)/t < 1.Thus, we observe the contractivity of causally connected regions when looking intothe past.

In our scenario, the emission from a spacelike fractal source, results from thephysical reality that two measurements of our source, with space coordinate in anon-trivial complete metric space, will almost surely yield two different positionsof that source. The idealization of our model suggests that each such front is inde-pendent of which observer will measure it, therefore, compositions of contractionmappings of each front will be taken with respect to each of the measurements ofthe initial position. Let us imagine the signal observed by the two observers at timet0, with t0 fixed, whose positions are located in the space-interval (ξ1 − tc, ξ2 + tc),

356 JOHN R. QUINN

and who detect the source at spacetime coordinates (ξ1, 0) and (ξ2, 0), respectively,with ξ1 = −ξ2 and ξ2 > 0 for definiteness. Then we define contraction mappingson the interval (ξ1 − tc, ξ2 + tc), with time parameterizing the steps of the compo-sition of the resulting IFS. Taking steps in time at the negative integer powers ofthe initial time allows for an infinite number of steps so that the resulting IFS canconverge to a true fractal attractor at the time of emission of the signal. The resultis a spacelike Cantor set in this simplified case of two observations.

Example 7. We embed our images of the interval I = [ξ1−ct0, ξ2+ct0] in I×n

at the times n = 1− log tlog t0

∈ [0,∞] (for t ∈ [0, t0]), in the space of scales of the initial

time t0. We have t = tlog tlog t00 ∈ (0, t0], then t = t1−n

0 so that x = ct �→ x = ct1−n0 .

Iterating the IFS S = {Si}2i=1, in steps of n = 1− logt0 t ∈ N, we obtain a spaceinterval for each contraction mapping at every finite n. We define S1(x) = r1x+ ξ1and S2(x) = r2x + ξ2, for contraction ratios r1 and r2 and for ξ1, ξ2, the initialsource locations as measured by the observers.

Taking the ratio of the lengths of the successive intervals, at times n + 1 and

n, we compute ri :=2ct

1−(n+1)0

2ct1−n0

= 1t0

< 1, thus establishing contractivity (under the

assumption above that t0 > 1, noting that a similar argument works for small t0)and providing r1 = r2 = t−1

0 so that S1(x) = xt−10 +ξ1 and S2(x) = xt−1

0 +ξ2. Thenthe invariant set F = ∪2

i=1Si(F ) will define a fractal attractor on the surface t = 0or n = ∞, by contractivity of S applied to I = [ξ1 − ct0, ξ2 + ct0]. A symmetricalargument supplies a fractal attractor for the future dependency of present events.

We can easily calculate the fractal dimension of the resulting invariant set F .For equicontractive self-similar fractals (eventually) without overlap of the imagesof the contraction mappings (as e.g. for ξ2 = 1

3 and t0 = 3), the box-countingdimension, and Hausdorff fractal dimensions are both equal to the similarity di-mension, by Theorem 8, the exponent dS that solves the Moran equation withri = r for all i. Thus we compute dH(F ) = dB(F ) = dS(F ) = log 2

log t0. We note that

the time of observation is in the denominator of this expression, suggesting thatthe dimension of the fractal source is seen to diminish with the passage of time,and our distance from the source renders it more pointlike in appearance.

8. Concluding Remarks and Future Directions

We see that fractal concepts and methods stemming from the properties ofself-similarity and contractivity [11], [13], are fundamental to fractal analysis. Bygeneralizing our notion of fractal dimension into a notion of scale covariance, wehope to better grasp the transitions of physical models to and from self-similarregimes. By applying the notion of contraction mapping to physical models, wehope to explain the fractal structures now encountered in models of physical sys-tems.

References

[1] V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts inMathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K.Vogtmann and A. Weinstein. MR997295 (90c:58046)

[2] Grigory Isaakovich Barenblatt, Scaling, self-similarity, and intermediate asymptotics, Cam-bridge Texts in Applied Mathematics, vol. 14, Cambridge University Press, Cambridge, 1996.With a foreword by Ya. B. Zeldovich. MR1426127 (98a:00005)


[3] Michael Fielding Barnsley, Superfractals, Cambridge University Press, Cambridge, 2006.MR2254477 (2008c:28006)

[4] C. Brown, L. Liebovitch, Fractal Analysis (Series: Quantitative Applications in the SocialSciences), SAGE publications, Thousand Oaks, 2010.

[5] Philippe Di Francesco, Pierre Mathieu, and David Senechal, Conformal field theory, GraduateTexts in Contemporary Physics, Springer-Verlag, New York, 1997. MR1424041 (97g:81062)

[6] Kenneth Falconer, Fractal geometry: Mathematical foundations and applications, 2nd ed.,

John Wiley & Sons Inc., Hoboken, NJ, 2003. MR2118797 (2006b:28001)[7] Ichiro Hasuo, Bart Jacobs, and Milad Niqui, Coalgebraic representation theory of fractals,

Proceedings of the 26th Conference on the Mathematical Foundations of Programming Se-mantics (MFPS 2010), Electron. Notes Theor. Comput. Sci., vol. 265, Elsevier Sci. B. V.,Amsterdam, 2010, pp. 351–368, DOI 10.1016/j.entcs.2010.08.021. MR2909663

[8] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5,713–747, DOI 10.1512/iumj.1981.30.30055. MR625600 (82h:49026)

[9] John E. Hutchinson and Ludger Ruschendorf, Random fractal measures via the contractionmethod, Indiana Univ. Math. J. 47 (1998), no. 2, 471–487, DOI 10.1512/iumj.1998.47.1461.MR1647916 (99j:60019)

[10] H. E. Kunze and E. R. Vrscay, Solving inverse problems for ordinary differential equationsusing the Picard contraction mapping, Inverse Problems 15 (1999), no. 3, 745–770, DOI10.1088/0266-5611/15/3/308. MR1696910 (2000f:34024a)

[11] H. E. Kunze, D. La Torre, F. Mendivil, E. R. Vrscay, Fractal-Based Methods in Analysis,Springer Science+Business Media LLC, New York, 2012.

[12] Michel L. Lapidus and Machiel van Frankenhuijsen, Fractal geometry, complex dimensionsand zeta functions: Geometry and spectra of fractal strings, Springer Monographs in Math-ematics, Springer, New York, 2006. MR2245559 (2007j:11001)

[13] F. William Lawvere, Metric spaces, generalized logic, and closed categories [Rend. Sem. Mat.Fis. Milano 43 (1973), 135–166 (1974); MR0352214 (50 #4701)], Repr. Theory Appl. Categ.1 (2002), 1–37 (English, with Italian summary). With an author commentary: Enrichedcategories in the logic of geometry and analysis. MR1925933 (2003i:18014)

[14] Tom Leinster, A general theory of self-similarity, Adv. Math. 226 (2011), no. 4, 2935–3017,

DOI 10.1016/j.aim.2010.10.009. MR2764880 (2012f:18006)[15] Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Company,

Inc., Toronto-New York-London, 1953. MR0054173 (14,883c)[16] Benoit B. Mandelbrot, Multifractals and 1/f noise, Selected Works of Benoit B. Mandel-

brot, Springer-Verlag, New York, 1999. Wild self-affinity in physics (1963–1976); With con-tributions by J. M. Berger, J.-P. Kahane and J. Peyriere; Selecta Volume N. MR1713511(2003c:28006)

[17] Benoit B. Mandelbrot, A class of multinomial multifractal measures with negative (latent)values for the “dimension” f(α), Fractals’ physical origin and properties (Erice, 1988), Et-tore Majorana Internat. Sci. Ser. Phys. Sci., vol. 45, Plenum, New York, 1989, pp. 3–29.MR1141390

[18] R. L. Oldershaw, Cosmological self-similarity and the principle of scale covariance, Astro-physics and Space Science, 128 (1986), 449O.

[19] H.-O. Peitgen and P. H. Richter, The beauty of fractals, Springer-Verlag, Berlin, 1986. Imagesof complex dynamical systems. MR852695 (88e:00019)

[20] L. Pietronero,“The fractal structure of the universe: correlations of galaxies and clusters”.Physica A 144 (1987): 257.

[21] J. R. Quinn, “Scale Covariance of Distributions, Sets, and Measures, a Differential Approachto the Counting Function of a Fractal String, with applications to Spacetime Physics”, PhDThesis, UC Riverside, In Press, 2013.

[22] D. R. Smart, Fixed point theorems, Cambridge University Press, London, 1974. CambridgeTracts in Mathematics, No. 66. MR0467717 (57 #7570)

[23] Didier Sornette, Critical phenomena in natural sciences: Chaos, fractals, selforganizationand disorder: concepts and tools, 2nd ed., Springer Series in Synergetics, Springer-Verlag,Berlin, 2004. MR2036307 (2004k:82002)

[24] Michael Spivak, Calculus on manifolds. A modern approach to classical theorems of advancedcalculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965. MR0209411 (35 #309)

358 JOHN R. QUINN

[25] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon Press,Oxford, 1971.

[26] D. Stauffer, H. E. Stanley, From Newton to Mandelbrot: A Primer in Theoretical Physics,Springer-Verlag, Berlin, 1990.

[27] Walter A. Strauss, Partial differential equations, John Wiley & Sons Inc., New York, 1992.An introduction. MR1159712 (92m:35001)

[28] Robert S. Strichartz, Self-similar measures and their Fourier transforms. I, Indiana

Univ. Math. J. 39 (1990), no. 3, 797–817, DOI 10.1512/iumj.1990.39.39038. MR1078738(92k:42015)

[29] Bob Coecke, David Moore, and Alex Wilce (eds.), Current research in operational quantumlogic, Fundamental Theories of Physics, vol. 111, Kluwer Academic Publishers, Dordrecht,2000. Algebras, categories, languages. MR1907153 (2003a:81014)

Department of Mathematics, University of California, Riverside, California 92521-

0135



Economics and Psychology. Perfect Rationalityversus Bounded Rationality

Daniele Schiliro

Abstract. Classical mathematical algorithms often fail to identify in timewhen the international financial crises occur although, as the classical theoryof choice would suggest, the economic agents are rational and the markets areor should be efficient and behave also rationally.

This contribution does not pretend to give a complete answer to thesequestions, but it will highlight some well-known limits of the classical theory ofrational choice and compare this theory of choice with the approach that seeksto combine economics and psychology, focusing on Herbert Simon’s notion ofbounded rationality. The work also makes some references to the literatureof behavioral finance which has given important contributions in explainingthe behavior and the anomalies of financial markets. Finally, following theapproach of Simon, the paper proposes an analytical model to describe thebehaviour of agents which are rationally bounded, risk averse and loss averse,emphasizing the relationship between psychology and economics which helpsto explain the crises in financial markets.

Introduction

The economic and financial crisis has created a climate of great uncertainty.People ask why speculation is constantly present in the markets and why individuals(at least some of them) are incapable of curbing speculative instincts to preservethe common good, the stability of the entire system rather than the (hefty) gains ofa few. Furthermore we wonder why the classical mathematical algorithms often failto identify in time when the international financial crises occur if, as the classicaltheory of choice would suggest, the economic agents are rational and the marketsare efficient and behave also rationally.

This contribution does not pretend to give a complete answer to these ques-tions, but it highlights some well-known limits of the classical theory of rationalchoice and compares this theory of choice with the approach that seeks to combineeconomics and psychology and that has established itself as behavioral economics.In particular, the paper will focus on bounded rationality, that has in Herbert Simonits most influential theorist. The work also makes some references to the literature

2010 Mathematics Subject Classification. 91-2, 91E10.Key words and phrases. Bounded rationality, procedural rationality, rational choice, cognitive

economics.The author wishes to thank David Carfı for his helpful suggestions in devising the analytical

model and Mario Graziano for the discussions and observations. The usual disclaimer here applies.


359

360 DANIELE SCHILIRO

of behavioral finance which has given important contributions in explaining thebehavior and the anomalies of financial markets. Finally, following the approach ofSimon, the paper proposes an analytical model to describe the behaviour of agentswhich are rationally bounded, risk averse and loss averse. These agents are myopicin their behaviors, since they influence with their market sentiment the trend offinancial markets causing losses at global level, while they are trying to protectthemselves locally.

1. Economics and the ‘perfect’ rationality

Economics in its classical conception follows a neo-positivist approach of syste-mic-formal nature. Thus economics takes the form of nomologic-deductive propo-sitions, which are obtained by reasoning, starting from unproven axioms. Withthese axioms we deduce the propositions of the theory, which requires the use oflogic and mathematics. Thus economics presents itself as a rational science in thesense that its propositions are obtained by means of logic, in a way which is similarto rational mechanics. In economics, moreover, rationality is interpreted in termsof consistency not of substance. We have therefore a syntactic and non-semanticnotion of rationality. The agents are rational if they have a coherent criterion ofchoice. The consistency of the choices implies that the agents are represented by asystem of preference. Economics describes the choice as a rational process drivenby a single cognitive process that includes the principles of the ‘theory of rationalchoice’ and it orders the decisions on the basis of their subjective expected utility.

In this view the “homo oeconomicus” appears perfectly rational and has acomplete knowledge, while his economic choices, guided by rationality, are selfcontained in the economic sphere without affecting other aspects of the individual,such as the emotions, or being influenced by the environment.1

1.1. The rational choice theory. Let us start analyzing the rational choicetheory (RCT). The first basic parameter which is taken into consideration by theTRC is the ’preference’. The theory sets several basic axioms on the preference of arational agent. The theory adopts a concept of rationality which can be representedin the following way:

Let X be a set of mutually exclusive alternatives. Economic agents are assumedto have preferences, denoted by ., on this set X:

x . y means “x is at least as good as y”.

The preference relation . is called rational if it satisfies the following twoproperties:

(1) Completeness: For all x, y ∈ X, x . y or y . x.(2) Transitivity: For all x, y, z ∈ X, x . y and y . z implies x . z.

Thus, if an individual’s preferences satisfy appropriate consistency conditions,then it is possible to associate a numerical value to each outcome through an utilityfunction u(·). By means of the utility functions it is possible to decline formallythe principle of maximization. The choice rule implied in the RCT is the following:

1Hogarth and Reder (1986) underline that the paradigm of rational choice provides economicswith a unity that is lacking in psychology.

PERFECT RATIONALITY VERSUS BOUNDED RATIONALITY 361

Let B be a family of non-empty subsets of X (“budget sets”). We call a corre-spondence

C : B → P (X) : B �→ C(B)

a choice rule if, for any member B of the family B, that is B ∈ B, we have

C(B) ⊆ B and C(B) �= 0.

Then, (B, C(·)) is a choice structure.Given ., rational choice theory specifies the choice rule to be

C∗(B,.) = {x ∈ B : x . y for all y ∈ B}.

Thus the preference optimization implies that C ∗ (B,.) picks the best elementsin B; under the assumption: C ∗ (B,.) is non-empty for all B.

Agent behavior is preference-maximizing if (B, C(·)) fulfills the weak axiom ofrevealed preference (warp) (Samuelson, 1938,1948). Let B1,B2 ∈ B and x, y ∈B1,B2. The choice structure (B, C(·)) satisfies the weak axiom of revealed prefer-ence if

x ∈ C(B1) and y ∈ C(B2) ⇒ x ∈ C(B2).

It can be shown that if (B, C(·)) fulfills the weak axiom and B contains all sub-sets of X up to three elements, the choice rule C(·) can be rationalized uniquely bythe preference-maximizing choice rule C∗(B,.). This is achieved through choosingthe preference ordering . such that

x . y if and only if there is a B ∈ B such that x, y ∈ B andx ∈ C(B).

To apply this RCT it is not necessary to make any particular psychologicalassumption, but — as Hogart and Reder (1986) pointed out — the definition ofrationality implied in this theory is broad and lacks specificity.

1.2. The expected utility theory. von Neumann and Morgenstern (1944)proposed an analysis of choice under uncertainty, which depends on strong as-sumptions of a psychological nature. The rationality is now represented by themaximization of the expected utility. The expected utility theory is nothing morethan a criterion that facilitates choice under risk.

According to von Neumann and Morgenstern, individuals generally move inthe reality following predetermined patterns of behavior, at the base of which thereis the assumption that they always prefer to have a greater wealth than less. Thetheory studies the preferences underlying consumer behavior under risk, i.e. whenthe subject is asked to make a decision without knowing with certainty which ex antestate of the world will happen, but he knows the probability distribution, that is, itis known to him a list of possible events, each of which he associates a probabilityof occurrence. This theory assumes that each individual has stable and consistentpreferences, and that he makes decisions based on the principle of maximizationof subjective expected utility. So given a set of options and beliefs expressed inprobabilistic terms, it is assumed that the individual maximizes the expected valueof a utility function u(·). The individual uses probability estimates and utilityvalues as elements of calculation to maximize his expected utility function. Thushe evaluates the relevant probabilities and utilities on the basis of his personalopinion but also using all relevant information available.


von Neumann and Morgenstern have proposed a well-known theorem in whichthey make the construction of an expected utility function possible. Any individualacting to maximize the expectation of a function u(·) will obey to four axioms,which are: completeness, transitivity, continuity, and independence.2 The firsttwo axioms (completeness and transitivity) have been explained in section 1.1.They require respectively that an individual has well defined preferences, whichare therefore complete, and that preference is consistent across any three options,so the consistency requirement reminds us that intransitive preferences lead toirrational behavior. von Neumann–Morgenstern theorem is also based on a thirdaxiom of continuity which states that the preferences of rational agents are orderedand without points of discontinuity. Lastly, the fourth axiom is the independenceaxiom, also called the substitution axiom. This independence axiom says that Iprefer p to p′, I will also prefer the possibility of p to the possibility of p′, given thatthe other possibility in both cases is some p′′. In particular, the axiom says that ifI am comparing αp+(1−α)p′′ to αp′ +(1−α)p′′, I should focus on the distinctionbetween p and p′ and hold the same preference independently of both α and p′′.

The expected utility theory has been generally accepted as a normative modelof rational choice, defining which decisions are rational. If an individual does notmaximize his expected utility he is designed to violate in his choices some preciseaxiomatic principles, which are rationally binding. This theory has also been ap-plied as a descriptive model of economic behavior (Friedman, Savage, 1948; Arrow,1971) so as to constitute an important reference model for economic theory.

2. Psychology into Economics. The cognitive dimension.

Within the scientific community there has been a growing need to consideradequately the complexity of economic phenomena and processes that guide thechoices of the individuals.

During the fifties there have been important explorations along the boundariesbetween economics and psychology. In particular, experimental psychology, con-cerned with the study of actual behavior and aware of the complexity of choices, hadhighlighted the systematic (and unconscious) divergence of human behavior fromthe postulates of economic rationality. Then some economists using experimen-tal results questioned the validity of the classical model of rational choice (Simon,1959). Thus a new line of research, called behavioral economics, started to bedeveloped, trying to relate psychological factors to economic behavior. One impor-tant contribution came from Herbert Simon, who developed the notion of boundedrationality. Bounded rationality depends — according to Simon (1972) — on thelimits of attentive and computational capacity. Thus, he gave start to an approachbased on the heuristics, that are interpreted as a trade-off between the limits ofthe human mind and the computing performance required by complex problems.Simon’s concept of bounded rationality can be interpreted — according to Kahne-man (2003) — as defining a realistic normative standard for an organism with afinite mind. Simon essentially criticized — on the basis of analysis conducted on

2An individual averse, neutral or risk lover has indifference curves convex, linear or concave,according to the form of the expected utility function. But in the von Neumann–Morgensternframework, individual’s attitude towards risk is defined without making any prior assumptionsabout his behavior.


the field — the lack of realism of the neoclassical economic theory based on theassumption of full rationality.

2.1. The Allais’ Paradox. Another major contribution to the critique ofthe rational choice theory came from the pioneering experimental studies of Al-lais, which have given a boost to the cognitive economic approach. Allais’ studiesdemonstrated that preferences of individuals violate expected utility theory, so heproved the systematic discrepancy between the predictions of traditional decisiontheory and actual behavior.

In 1952, Maurice Allais presented in Paris his famous paradox to an audiencecomposed of the best economist of his generation; among others, Kenneth Ar-row, Paul Samuelson, Milton Friedman, Jacob Marschak, Oskar Morgenstern andLeonard Savage.

The paradox consists in presenting a subject in two situations. In the firstsituation (A) the person is proposed to choose between getting for sure $1,000,000(a) and receive a lottery (b) which has 0.1 probability to win $5,000,000, 0.89probability of winning $1,000,000 and 0.01 probability of not winning anything.In the second situation (B) the person is proposed to choose between a lottery(c) which has 0.1 probability to win $5,000,000 and 0.9 probability of not winninganything, and another lottery (d) which has 0.11 chance of winning $1,000,000and 0.89 probability of not winning anything. We would expect that a rationalindividual chooses (a) in the first situation and chooses (c) in the second situation.But this outcome, apparently evident, contradicts the utility theorem. In fact,calculating the utilities for each choice we obtain:

u(a) = u(1M)

u(b) = 0.1u(5M) + 0.89u(1M) + 0.01u(0)

u(c) = 0.1u(5M) + 0.9u(0)

u(d) = 0.11u(1M) + 0.89u(0)

From which:

u(a) − u(b) = 0.11u(1M) − [0.1u(5M) + 0.01u(0)]

u(d) − u(c) = 0.11u(1M) − [0.1u(5M) + 0.01u(0)]

The utility theorem tells us that if the individual prefers (a) with respect to (b):(u(a) / u(b)) in the first situation (A), then the individual must prefer (d) to (c):(u(d) / u(c)) in the second situation (B) and vice versa, hence the paradox.

Therefore the results of laboratory experiments conducted by Allais have shownthat individuals chose inconsistently and that they preferred solutions which didnot maximize the expected utility. In this way Allais have demonstrated thatthe axiomatic definition of rationality did not allow to describe and even predicteconomic decisions.

Later, another paradox has been identified by Ellsberg (1961), who, by meansof experiments, demonstrated another type of inconsistency in preferences, showingthat individuals prefer to bet on a lottery with a chance of obtaining a win alreadyknown than on a lottery with ambiguous results.

This aversion to uncertainty (ambiguity) of the individual is completely ignoredin the expected utility model from a descriptive point of view, while is not consideredacceptable from a normative point of view.


2.2. Bounded Rationality. In economics the concept of bounded rational-ity is associated to Herbert Simon, who proposed the idea of bounded rationalityas an alternative basis for the mathematical modeling of decision making. Simonhas coined the term ‘bounded rationality’ in Models of Man (1957). In his view,rationality of individuals is limited by the information they have, the cognitive lim-itations of their minds, and the finite amount of time they have to make decisions.Bounded rationality expresses the idea of the practical impossibility (not of thelogical impossibility) of exercise of perfect (or ‘global’) rationality (Simon, 1955).“Theories that incorporate constraints on the information-processing capacities ofthe actor may be called theories of bounded rationality” (Simon, 1972, p. 162). Si-mon argues that most people are only partly rational while are emotional/irrationalin the remaining part of their actions. He maintains that, although the classicaltheory with its assumptions of rationality is a powerful and useful tool, it fails toinclude some of the central problems of conflict and dynamics which economics hasbecome more and more concerned with (Simon, 1959, p. 255). Simon identifies avariety of ways to assume limits of rationality such as risk and uncertainty, incom-plete information about alternatives, complexity (1972, pp. 163-164). Furthermore,he asserts that an individual who wants to behave rationally must consider not onlythe objective environment, but also the subjective environment (cognitive limita-tions), thus you need to know something about the perceptual and cognitive processof this rational individual. Simon, therefore, considers the psychological theory veryimportant to enrich the analysis for a description of the process of choice in econom-ics. This is why he adopts the notion of procedural rationality, a concept developedwithin psychology (Simon, 1976), which depends on the process that generated it,so rationality is synonym of reasoning. According to Simon (1976, p. 133), a searchfor procedural rationality is the search for computational efficiency, and a theoryof procedural rationality is a theory of efficient computational procedures to findgood solutions. Procedural rationality is a form of psychological rationality whichconstitutes the basic concept of Simon’s behavioral theory (Novarese, Castellani,Di Giovinazzo, 2009; Barros, 2010, Graziano, Schiliro, 2011; Schiliro, 2011, 2012),in contrast to economic rationality, defined by Simon as ‘substantive rationality’.

Another way to look at bounded rationality is that, because individuals lackthe ability and resources to arrive at the optimal solution, they instead apply theirrationality only after having greatly simplified the choices available. Actually, in-dividuals face uncertainty about the future and costs in acquiring information inthe present. These two factors limit the extent to which agents can make a fullyrational decision. Thus, Simon claims, agents have only bounded rationality andare forced to make decisions not by ‘maximization’, but rather by satisficing, i.e.setting an aspiration level which, if achieved, they will be happy enough with, andif they don’t, try to change either their aspiration level or their decision. Satisficingis the hypothesis that allows to the conception of diverse decision procedures andwhich permits rationality to operate in an open, not predetermined, space (Barros,2010). Real-world decisions are made using fast heuristics, ‘rules of thumb’, thatsatisfice rather than maximize utility over the long run. Thus agents employ theuse of heuristics to make decisions rather than a strict rigid rule of optimization.The agents do this because of the complexity of the situation, and their inabilityto process and compute the expected utility of every alternative action. In fact,there are limits in the attentive, mnemonic and computational capacity binding


the computational load, hence the usefulness of automatic routines. Rationality isbounded by these internal constraints in the uncertain real world. Simon, there-fore, relates the concept of bounded rationality to the complementary constructof procedural rationality, which is based on cognitive processes involving detailedempirical exploration and procedures (“search processes”) that are translated inalgorithms. This is in contrast to the notion of perfect rationality, that is based onsubstantive rationality, which derives choices from deductive reasoning and froma tight system of axioms, an idea of rationality that has grown up strictly withineconomics (Simon, 1976, 1997). For Simon “as economics becomes more and moreinvolved in the study of uncertainty, more and more concerned with the complexactuality of business-decision making, the shift in program will become inevitable.Wider and wider areas of economics will replace the over-simplified assumptions ofthe situationally constrained omniscient decision-maker with a realistic (and psy-chological) characterization of the limits on Man’s rationality, and the consequencesof those limits for his economic behavior” (Simon, 1976, pp. 147–148).

Simon, however, does not reject the neoclassical theory tout court, he describesa number of dimensions along which neoclassical models of perfect rationality canbe made somewhat more realistic, while sticking within the vein of fairly rigorousformalization. These include: limiting what sorts of utility functions there mightbe, recognizing the costs of gathering and processing information, the possibility ofhaving a “multi-valued” utility function.

Simon’s work has been followed in the research on judgment and decision mak-ing, both in economics and psychology. Two major approaches produced importantinsights into perception mechanisms shaping the individual’s internal representa-tion of the problem: the “heuristics and biases” program (Tversky, Kahneman,1974), which has been fundamental to the contemporary development of behav-ioral economics. The other approach, derived from Simon’s work, is the “fast andfrugal heuristics” program (Gigerenzer, Goldstein, 1996; Todd, Gigerenzer, 2003).

Tversky and Kahneman, in particular, offered a theoretical explanation aboutthe observed deviations from perfect rationality. They explored the psychology ofintuitive beliefs and choices and examined their bounded rationality (Kahneman,2002, p. 449). Tversky and Kahneman (1979, 1984, 1986) articulated a direct chal-lenge to the rationality assumption itself, based on experimental demonstrations inwhich preferences were affected predictably by the framing of decision problems,or by the procedure used to elicit preferences.3 One major conclusion of this al-ternative approach is that the susceptibility of people to framing effects violates afundamental assumption of invariance. Kahneman and Tversky (1979, 1984) alsoargued that any individual has a deformation of the probability, which is differentbetween gains and losses and, moreover, the individual has aversion to losses. Aloss, in fact, is more weighted by a psychological point of view than a gain.

Consequently taking into account framing effects and other aspects like lossaversion, money illusion, etc. the model of choice based on perfect rationality withits underlying expected utility theory fails as an adequate descriptive model ofchoice under risk.

3In their ‘Prospect theory’ Tversky and Kahneman have shown experimentally the presenceof inconsistent judgments and choices by an individual facing the same problem presented indifferent frames (‘invariance of failures’). It follows that the frame, or the context of choice,coeteris paribus, helps to determine a different behavior.


The other approach, derived from Simon’s work, is the “fast and frugal heuris-tics” program (Gigenzer, Goldstein, 1996; Todd, Gigerenzer, 2003). These fastheuristics are conscious processes, accessible to introspection in humans. FollowingSimon’s notion of satisficing, Gigenzer and Goldstein have proposed a family of al-gorithms based on a simple psychological mechanism: one-reason decision making.These fast and frugal algorithms violate fundamental tenets of classical rational-ity: they neither look up nor integrate all information (Gigenzer, Goldstein, 1996).The heuristics are determined by a trade-off between the limits of the human mindand the computing performance required by complex problems. The psychologyof choice is to codify these heuristics in humans, to help apply them in situationswhere they work well.

Also Ariel Rubinstein (1998) proposed to model bounded rationality by ex-plicitly specifying decision-making procedures, applying game theory, specificallyrepeated games. Rubinstein has contributed to formalize the theoretical notion ofbounded rationality, and he has put the study of decision procedures on the researchagenda.

3. Behavioral Finance

Theory of expected utility is also applied to financial investment decisions, thusthe individual is following a preference-maximizing choice. Financial decisions forthe rational optimizing economic theory are based on the hypothesis that peoplecalculate their rational advantage and then act consistently with that.

Yet, research in psychology have supported the view that emotional reactionsto situations involving uncertainty or futurity often differ sharply from cognitiveassessments of those situations, and that when such differences occur, it is oftenthe emotional reactions that determine behavior. From the seventies onwards therehas been an increasing interest towards psychological and sociological aspects inthe analysis of financial behavior. Then there has been the development of a newbranch of finance: the behavioral finance, which in itself combines aspects of cog-nitive psychology and financial theories in the strict sense. In practice this newapproach seeks to explain the so-called financial market “anomalies” by analyzingthe behavior of economic agents. However, the adoption of heuristics by individualsis necessary to solve the problems of everyday life, but in the financial sector it canlead to biases which have proved very expensive (Tversky, Kahneman, 1974).

3.1. Behavioral finance: anomalies and biases. In the reality of financialmarkets the fact that the price of a stock should coincide with its fundamental valueseems to be more the exception than the rule. The “anomalies” in the behavior ofprices and yields, in contrast to the hypothesis of efficient markets, are numerousand show that the securities are by no means in line with their fundamentals.

So there have been models which departured from economic rationality andform the idea of efficient markets. Usually these models do not abandon completelythe rationality model as the basic framework, but they focus on some particulardeviation that explains a family of anomalies.

In particular, the models of behavioral finance, used in the valuation of assets,usually criticize the efficient market theory based on the idea of “informationalefficiency of markets”, underpinned by Fama (1970), that a market is efficient inthe sense of information if at all times the stock prices fully and correctly reflect allthe available information. The theory of market efficiency has been challenged, for


instance, by the discovery of some anomalies that would produce excess returns.De Bondt and Thaler (1985) have shown that bonds, characterized by particularlyhigh yields (so-called winners), record in the aftermath the worst yield and viceversa. This depends on investors’ overreaction to an event. Over the time theinvestors realize the error and correct their assessments causing a reversal of returns.Odean (1998), instead, have designed a stock market in which all traders believethey are above average. Bernatzi, Thaler (1995) represents in their model a stockmarket in which traders are myopic and loss-averse. Furthermore, Thaler andShefrin (1981, 1988), who gave major contributions to behavioral finance, presentedtheir behavioral life-cycle theory arguing that economists who wish to analyze theconsumption-saving decision must address the bounded rationality and impatienceof consumers. The behavioral-life cicle theory models consumers as responding topsychological limitations by adopting rules-of-thumb, such as mental accounts, thatare used to constrain the decision making of the myopic agent.

Kahneman, Knetsch and Thaler (1991) analyzed the topic of loss aversion.4

They carried out a significant experiment based on the “endowment effect” wherethese authors demonstrated that the individuals feel a great sorrow when they loosethe objects they possess, more than the pleasure would cause them to acquire thosesame objects, if they do not already possess them. So the “endowment effect” isan anomaly that causes a statu quo bias (a preference for the current state thatbiases the individual against both buying and selling his object). The “endowmenteffect” is connected to the particularly pervasive phenomenon of loss aversion, forwhich the disutility of a loss is greater than the utility of a win of the same size.

However, there is another approach, sympathetic to behavioral economics,which is neuroeconomics, a discipline at the turn of neuroscience and economics.This relatively new approach aims at studying the processes underlying the decision-making choices and that reveals what instincts are activated when you have to dowith the risk, the gains and losses. Neuroeconomics tries to offer a solution throughan additional set of data obtained via a series of measurements of brain activity atthe time of decisions. Neuroeconomic theory proposes to build brain-based modelscapable of predicting observed behavior (Brocas, Carrillo, 2010). The underlyingidea of neuroeconomics is that the brain is a multi-system entity with restrictedinformation and conflicting objectives characterized by bounds of rationality, sothe decision-maker must be modeled as an organization. So the financial modelsmust take into account the neuro-cognitive constraints, i.e. the mechanisms put inplace by the brain in response to certain environmental stimuli, and the influenceof emotions on the choices of investment. Thus, neuroeconomics can be consideredanother development of Simon’s intuitions, that tries also to explain the collapse offinancial markets and of confidence, as an effect unconscious decisions.

4. Bounded rationality and risk aversion: a model of behavioral finance

In this section I outline an analytical model which follows the approach ofSimon regarding agents’ bounded rationality and also takes into account behavioralconcepts such as loss aversion (Bernatzi, Thaler, 1995; Kahneman, Knetsch andThaler, 1991) and a strong aversion to risk. An excessive perception of, and aversion

4The literature of behavioral finance includes the lack of symmetry between decisions toacquire and maintain resources and the strong aversion to the loss of some (emotionally) valuableresources that could be completely lost.


to, risk on the part of investor is, in fact, the major source of current global economicproblems. This aversion has resulted in an excessive desire for liquidity and relativesafety. This behavior, which is partly rational, has brought to a situation in whichthe fear of risky investment has exceeded. This has led, in turn, a greater propensityto hold liquidity by the investors as they tried to protect themselves, but thisbehavior, pushed by psychological motivations, has caused losses at a global level.

4.1. Modeling bounded rationality. We start from Simon’s idea that anagent has constraints in his information-processing capacities. Since individuals areonly partly rational, Simon (1972) assumes limits of rationality such as risk anduncertainty, incomplete information about alternatives, complexity.

A Decision-maker which should take some decisions at a certain time 0 is mod-eled by a pre-ordered space (X,≤):

- X is the set of all possible choices;- ≤ is a binary relation everywhere defined on X which is (by assumption)

reflexive, total and transitive.

We could generalize and direct this absolute-rational model of a decision makerin the following way:

(1) the assumption that ≤ is everywhere defined is unrealistic: a decisionmaker is very often not able to define a preference on the totality of hisstrategy set X;

(2) the assumption of totality (according to which every strategy pair is com-parable) is unrealistic even if the relation is not everywhere defined;

(3) the decision-maker very often should decide not just at a time 0, butduring an entire time interval [0;T ], in which the conditions of the marketare changing in time.

In the first case, we remember that any strategy not comparable with any otherstrategy is a Pareto maximum, so that any such x is a possible choice of the decisionmaker.

In the second case, since a decision maker is erroneously supposing that hisown preference is total, it is likely for him to obtain a violation of transitivity.

A way to overcome the third problem is to consider a family of preferences(≤t)t∈[0;T ], any preference ≤t holds in charge at time t. We, at this point, could

have different problems:

- the agent could erroneously think that any member ≤t of the family ispreserving the preferences ≤t′ , with t′ < t, and this is another form ofbounded rationality;

- the agent could think he has one unique preference for the entire decisionprocess;

- the agent could think that the family of preferences is continuous in sometopological sense, so excluding dangerous “choice fractures” in the decisionproblems.

We now consider a financial market modeled in a state preference context.In particular, we consider, for simplicity, a market with a unique financial asset.Moreover, firstly, we consider the classic case of financial model with m-possiblestates of the world, the same states, for any future time t > 0; for this reason we


could identify the space S of any possible future state of the world with the set ofthe first m integers, which we shall denote by m.

Assume the asset has an initial unit price p0 (at time 0);

then, assume that at any future time t, the price p(t) of the asset is a vector inRm:

the i-th component p(t)i of the vector p(t) is the unit price of the asset, if thestate of the world i has occurred.

So that the vector p(t) is a random scalar (more precisely a random price).

Indeed, we could interpret our vector p(t) as a function

p(t) : m → R

associating to any possible state of the world i in m the possible price p(t)(i).

In this model, when we define a probability measure μ(t), at any time t, on thespace S of all states of the world m, we can evaluate the probability that a certainpossible unit price could occur, at a certain time t.

Observe that a probability measure μ in this simple model can be assimilatedto a positive unit m-vector with respect to the 1-norm (that is a positive m-vectorμ such that

∑μ = 1).

Note that an amplitude ψ(t) can be associated with any probability measure:the amplitude such that μ = ψ2, where ψ2 = ψψ, is the component-wise productof ψ times itself.

If we consider discrete time, we could assume that the discrete dynamical prob-ability amplitude

ψ : N → Rm

follows an evolution law of the type:

ψ(t + 1) = U(t; t + 1)ψ(t);

for every t in N where

U(t; t + 1): Rm → Rm

is a unitary operator, for every time t.

A rationally bounded decision-maker could erroneously think that

U(t; t + 1) = U(1);

for every time t.

4.2. Risk aversion and bounded rationality in financial choices. An-other form of bounded rationality is related to the situation when every agent in afinancial market is risk averse.

Instead of adopting a utility function that represents risk aversion, as, for ex-ample, the commonly used hyperbolic absolute risk-aversion function (HARA), wecould represent risk aversion, of a certain agent, as a reaction function r : E → Fsending any price p of a certain security into a decision r(p) in R indicating howmuch to buy (in algebraic sense) of that security, for example r could be definedby the function “integer” int as it follows:

r(p) = int(pp0);


for every p belonging to a certain bounded neighborhood U(p0) of p0 (and we couldbe interested only on this range U(p0)).

But we have at least two problems:

- presumably, “usual” agents could have very similar reaction functions;especially agents of the same type;

- also banks have such reaction functions, the risk aversion of the banksbrings to a situation of credit crunch.

In this context, agents’ behavior is likely dominated by psychological sentimentof fear. This sentiment influences the financial investment choices through thestrong risk aversion of the investors.

Therefore, the price p of a certain security is a reaction function to the actionsof the agents on the markets, assume, for simplicity that p is a reaction function tothe aggregate quantity of bought security and indicate by r the aggregate reactionof the agents, we have a kind of reaction chain (since we have a decision-form game(r; p)): an initial price p0 determines r(p0) which determines p(r(p0)) and then

r(p(r(p0)))

and so on . . . ; but also p is an order preserving function, so that both the values ofprice and bought security tend to the minimum possible level, leading to a crisis.

In other terms, we have a dynamical system tending to a state which determinesthe worst possible gain (or loss).

Conclusions

Financial crises have raised many questions and created new problems for eco-nomic theory. It is not all certain that the mathematical algorithms devised by theclassical theory can predict in time when the international financial crises occur,but, as this paper tried to argue, we can enrich our knowledge of the complex real-ity of financial markets through the fertile contribution of Simon and of behavioralfinance.

Firstly, the present contribution has discussed the notion of perfect rationalitywhich has been confronted with the concept of bounded rationality as formulatedby Herbert Simon. It has been underlined the relation between bounded rational-ity and procedural rationality which is the form of psychological rationality thatconstitutes the basic concept of Simon’s behavioral theory. Moreover, the work hasexamined the criticism to the classical theory of rational choice and to expectedutility coming from the approaches derived from Simon’s bounded rationality. Inparticular behavioral finance has highlighted anomalies and biases in the behav-ior of the economic agents in financial markets, although the critical part of thebehavioral theory seems more convincing than the positive and proactive part ofthe same theory, leaving a significant degree of indeterminacy in defining solutions.In the last section, this work has suggested an analytical model to describe thebounded rationality of the agents following Simon’ approach and that also takesinto account loss aversion and a strong aversion to risk to demonstrate that the be-havior of investors, influenced by psychological elements, leads to crises and lossesat the global level.


References

[1] Allais M., Le comportement de l’homme rationnel devant le risque: critique des postulatset axiomes de l’ecole americaine, Econometrica 21 (1953), 503–546 (French). MR0058952(15,455c)

[2] Arrow Kenneth J., Essays in the theory of risk-bearing, North-Holland Publishing Co., Am-sterdam, 1970. MR0363427 (50 #15865)

[3] Aumann Robert J., Rationality and bounded rationality, Games Econom. Behav. 21 (1997),no. 1-2, 2–14, DOI 10.1006/game.1997.0585. MR1491259

[4] Barberis N., Thaler R. H. 2002. A survey of behavioral finance, NBER working paper 9222.[5] Barros G., 2010. Herbert A. Simon and the concept of rationality: boundaries and procedures,

Brazilian Journal of Political Economy, 30 (3), pp. 445–472, July-September.[6] Benartzi S., Thaler R., 1995. Myopic loss aversion and the equity premium puzzle, Quarterly

Journal of Economics, 110, pp. 73–92.[7] Brocas I., Carrillo J. D., 2010. Neuroeconomic theory, using neuroscience to understand

bounds of rationality, VoxEU.org, March. http://www.voxeu.org

[8] De Bondt W. F., Thaler R. 1985. Does the stock market overreact?, The Journal of Finance,40(3), pp. 793–805.

[9] Ellsberg D., 1961. Risk, ambiguity, and the Savage axioms, Quarterly Journal of Economics,75(4), pp. 643–669.

[10] Fama Eugene F., Efficient capital markets: a review of theory and empirical work, Frontiers ofquantitative economics (Invited Papers, Econometric Soc. Winter Meetings, New York, 1969),North-Holland, Amsterdam, 1971, pp. 309–361. Contributions to Economic Analysis, Vol. 71.With comments by William F. Sharpe and Robert A. Schwartz. MR0436921 (55 #9855)

[11] Friedman M., Savage L., 1948. Utility analysis of choices involving risk, Journal of PoliticalEconomy, 56 (4), pp. 279–304.

[12] Gigerenzer, G., Goldstein D. G., 1996. Reasoning the fast and frugal way: models of boundedrationality, Psychological Review, 103 (4), pp. 650–669.

[13] Graziano M., Schiliro D., 2011. Rationality and choices in economics: behavioral and evolu-tionary approaches, Theoretical and Practical Research in Economic Fields, II (2), pp. 183–196.

[14] Hogarth R. M., Reder M. W., 1986. Editor’s comment: perspectives from economics andpsychology, Journal of Business, 59(4) pp. S185–S207, October.

[15] Kahneman D., 2002. Maps of bounded rationality: a perspective on intuitive judgement andchoice. Prize Lecture, December 8, 2002, Princeton University, pp. 449–489.

[16] Kahneman D., 2003. A psychological perspective on economics, AEA Papers and Proceedings,vol. 93 (2), pp. 162–168, May.

[17] Kahneman D., Knetsch J. L., Thaler R. H., 1991. The endowment effect, loss aversion, andstatus quo bias, Journal of Economic Perspectives, 5(1), pp. 193–206, Winter.

[18] Kahneman D., Thaler R. H., 2006. Anomalies: utility maximization and experienced utility,Journal of Economic Perspectives, 20 (1), pp. 221–234.

[19] Kahneman D., Tversky A., 1979. Prospect theory. An analysis of decision under risk, Econo-metrica, 47 (2), pp. 263–292.

[20] Kahneman D., Tversky A., 1984. Choices, values and frames, American Psychologist, 39,pp. 341–350.

[21] Kindleberger C., 1978. Manias, panics, and crashes: a history of financial crisis, New York,Basic Books.

[22] Novarese M., Castellani M., Di Giovinazzo V., 2009. Procedural rationality and happiness,MPRA Paper 18290, University Library of Munich, Germany.

[23] Odean T., 1998. Are investors reluctant to realize their losses?, Journal of Finance, 53 (5),pp. 1775–1798.

[24] Rabin M., 1998. Psychology and economics, Journal of Economic Literature, 36(1), pp. 11–46.[25] Rubinstein A., 1998. Modeling bounded rationality, Cambridge, MIT Press.[26] Samuelson P. , 1938, A note on the pure theory of consumers’ behaviour. Economica, 5,

pp. 61–71.[27] Samuelson P. , 1948, Consumption theory in terms of revealed preference. Economica, 15(60),

pp. 243–253.


[28] Schiliro D., 2011, Decisions and rationality in economics, MPRA Paper 29477, UniversityLibrary of Munich, Germany.

[29] Schiliro D., 2012, Bounded rationality and perfect rationality: psychology into economics,Theoretical and Practical Research in Economic Fields, vol. III, issue 2(6) Winter 2012.

[30] Shefrin H., Thaler R., 1988. The behavioral life-cycle hypothesis, Economic Inquiry, 26(4),pp. 609–643.

[31] Simon H. A., 1955. A behavioral model of rational choice, Quarterly Journal of Economics,

69 (1), February, pp. 99–118.[32] Simon H. A., 1956. Rational choice and the structure of the environment, Psychological

Review, 63 (2), March, pp. 129–138.[33] Simon H. A., Models of man, social and rational. Mathematical essays on rational human

behavior in a social setting, John Wiley & Sons Inc., New York, 1957. MR0113732 (22 #4565)[34] Simon H. A., 1959. Theories of decision making in economics and behavioral science, Amer-

ican Economic Review, 49 (3), June, pp. 253–283.[35] Simon H. A., Theories of bounded rationality, Decision and organization (a volume in honor

of Jacob Marschak), North-Holland, Amsterdam, 1972, pp. 161–176. Studies in Mathematicaland Managerial Economics, Vol. 12. MR0418864 (54 #6899)

[36] Simon H. A., 1976. From substantive to procedural rationality, in Latsis S.J.(ed.), Methodand appraisal in economics, Cambridge: Cambridge University Press, pp. 129–148.

[37] Simon H. A., 1979. Rational decision making in business organizations [Nobel MemorialLecture], American Economic Review, 69 (4), September, pp. 493–513.

[38] Simon H. A., 1991. Bounded rationality and organizational learning, Organization Science,2 (1), pp. 125–134.

[39] Thaler R. H., Shefrin H., 1981. An economic theory of self-control. Journal of Political Econ-omy, 89 (2), pp. 392–406.

[40] Todd P. M., Gigerenzer G., 2003. Bounding rationality to the world, Journal of EconomicPsychology, 24, pp. 143–165.

[41] Tversky A., Kahneman D., 1974. Judgement under uncertainty: heuristics and biases, Science,185, pp. 1124–31, September.

[42] Tversky A., Kahneman D., 1986. Rational choice and the framing of decisions, Journal of

Business, 59(4) pp. S251–S278, October.[43] Von Neumann J. and Morgenstern O., Theory of Games and Economic Behavior, Princeton

University Press, Princeton, New Jersey, 1944. MR0011937 (6,235k)

DESMaS “V. Pareto”, University of Messina; CRANEC, Catholic University of

Milan


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601 David Carfı, Michel L. Lapidus, Erin P. J. Pearse, and Machiel vanFrankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure andApplied Mathematics II, 2013

600 David Carfı, Michel L. Lapidus, Erin P. J. Pearse, and Machiel vanFrankenhuijsen, Editors, Fractal Geometry and Dynamical Systems in Pure andApplied Mathematics I, 2013

599 Mohammad Ghomi, Junfang Li, John McCuan, Vladimir Oliker, FernandoSchwartz, and Gilbert Weinstein, Editors, Geometric Analysis, MathematicalRelativity, and Nonlinear Partial Differential Equations, 2013

598 Eric Todd Quinto, Fulton Gonzalez, and Jens Gerlach Christensen, Editors,Geometric Analysis and Integral Geometry, 2013

597 Craig D. Hodgson, William H. Jaco, Martin G. Scharlemann, and StephanTillmann, Editors, Geometry and Topology Down Under, 2013

596 Khodr Shamseddine, Editor, Advances in Ultrametric Analysis, 2013

595 James B. Serrin, Enzo L. Mitidieri, and Vicentiu D. Radulescu, Editors, RecentTrends in Nonlinear Partial Differential Equations II, 2013

594 James B. Serrin, Enzo L. Mitidieri, and Vicentiu D. Radulescu, Editors, RecentTrends in Nonlinear Partial Differential Equations I, 2013

593 Anton Dzhamay, Kenichi Maruno, and Virgil U. Pierce, Editors, Algebraic andGeometric Aspects of Integrable Systems and Random Matrices, 2013

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591 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, VladimirMaz’ya, Simeon Reich, David Shoikhet, Gilbert Weinstein, and LawrenceZalcman, Editors, Complex Analysis and Dynamical Systems V, 2013

590 Ursula Hamenstadt, Alan W. Reid, Rubı Rodrıguez, Steffen Rohde,and Michael Wolf, Editors, In the Tradition of Ahlfors-Bers, VI, 2013

589 Erwan Brugalle, Maria Angelica Cueto, Alicia Dickenstein, Eva-MariaFeichtner, and Ilia Itenberg, Editors, Algebraic and Combinatorial Aspects ofTropical Geometry, 2013

588 David A. Bader, Henning Meyerhenke, Peter Sanders, and Dorothea Wagner,Editors, Graph Partitioning and Graph Clustering, 2013

587 Wai Kiu Chan, Lenny Fukshansky, Rainer Schulze-Pillot, and Jeffrey D.Vaaler, Editors, Diophantine Methods, Lattices, and Arithmetic Theory of QuadraticForms, 2013

586 Jichun Li, Hongtao Yang, and Eric Machorro, Editors, Recent Advances inScientific Computing and Applications, 2013

585 Nicolas Andruskiewitsch, Juan Cuadra, and Blas Torrecillas, Editors, HopfAlgebras and Tensor Categories, 2013

584 Clara L. Aldana, Maxim Braverman, Bruno Iochum, and Carolina NeiraJimenez, Editors, Analysis, Geometry and Quantum Field Theory, 2012

583 Sam Evens, Michael Gekhtman, Brian C. Hall, Xiaobo Liu, and Claudia Polini,Editors, Mathematical Aspects of Quantization, 2012

582 Benjamin Fine, Delaram Kahrobaei, and Gerhard Rosenberger, Editors,

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581 Andrea R. Nahmod, Christopher D. Sogge, Xiaoyi Zhang, and Shijun Zheng,Editors, Recent Advances in Harmonic Analysis and Partial Differential Equations, 2012

580 Chris Athorne, Diane Maclagan, and Ian Strachan, Editors, Tropical Geometryand Integrable Systems, 2012

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/conmseries/.

This volume contains the proceedings from three conferences: the PISRS 2011 Interna-tional Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics,held November 8–12, 2011 in Messina, Italy; the AMS Special Session on Fractal Ge-ometry in Pure and Applied Mathematics, in memory of Benoıt Mandelbrot, held January4–7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis onFractal Spaces, held March 3–4, 2012, in Honolulu, HI.

Articles in this volume cover fractal geometry and various aspects of dynamical systemsin applied mathematics and the applications to other sciences. Also included are articlesdiscussing a variety of connections between these subjects and various areas of physics, en-gineering, computer science, technology, economics and finance, as well as of mathematics(including probability theory in relation with statistical physics and heat kernel estimates,geometric measure theory, partial differential equations in relation with condensed matterphysics, global analysis on non-smooth spaces, the theory of billiards, harmonic analysisand spectral geometry).

The companion volume (Contemporary Mathematics, Volume 600) focuses on the moremathematical aspects of fractal geometry and dynamical systems.

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