111

<,

<,

<,

(2.21

(2.1)

!!

,

3jq]

o "

'_"'~~i.>4...A...

x"

o

J) ODi)J

l~ "1"1\x*, ... 1

F,

>('1 IFl

_1

l~ 11Xb tl_1I .

XI_

I_

,

0./ .......... 1 E"2

x·

Xl~"'ll I.l't l. /"

Fo

FlO. I. An example of I binary tree and tbe labelins used inthis paper. A trlnJition from leneration 10 geqeration i' associ­Ited witb sbining Ihe t, 's u sbo....n. Eatb point of the sct Uobtained rrom tbe seed x· U I rauh or a compolition of tbepresentation rUneliOIU F" in the order .ho'WII, i.e.,XI.I(t••... ,tll-F'lo ... of,. (x oJ. NOle, however, tint the.'.....luQ obt.ained in Ihi, "'Ir need not be ordered &Jenl: tbe x t,:..lS.

e points in the sel, with Jc integer, Jc ~ 2. Then everypoint in tbe nIh seneration un be given an address(t~'€._I'···'(ll, where f, is an indel. that takes on J:values 10, I, . , , ,Jc - I). (See Fig. I for a binary tree n_ample.) This address can be also converted into anum.ber 1,11.1}

t = i fJ*.~-!.J- ,

Assume also that tbere are Jc given functions F" tbat canbe used to find the poinl$ of the (n + nth generation fromthe points of the nth generation

Finally, assume that there exists a unique "seed" pointx 0 of the teroth generalion, from which all points of tbenih generation can be constructed. TbiJ cboice of gen·eration tbrough the functions F" c.alJ«I. e1sewhere21 "prc-

'\

(I.SI

F'EIGEl"BAUM, PROCACCIA, AND TEL

U. EIGL"VALUI EQUATIONS

e--GI,!J= ~ If·,-,

wrile,

scaling faclors. To avoid confusion with ql1") we: shalladopt a different notation for this rate of ,rowth. a nOla·lion that follollo's standard thennodynamics. We shall

:;360

We consider hcre point-sets that can be organiled onregular trees, such Ihat at the /lth level there are enctly

I"f

G(tJ) mi~ht depend on the pan ilion. For poiDt sets or­ganized on regular trees, we shall use the coveragedefined in Sec. II. (We stress that for problems for whicbp/=const there is a correspondence 1".... -(3, G«(3) il--q1-8) InaJ Otherwi~, we are dealing \\;Ih dilJ'erent :functions, cr. Sec. III D. In facl, for the gentrating par­tition of hyperbolic s)'stems, -G«(3) is ll. quantity calledpressure in the mathematical literature of thennodrnam­ic formalism and has ulensivel)' been studied.I~.1 Thesum (1.5) has alrClldy ~n in\'estigBted for nonhyperbolicsyslems, too. and it has been found that GIIJ) can havenonanal\'ticilies lphase: transitions) as .....ell.70

The theet} developed belo..... is aimed at calculating the~function G (.8). It will be seen that one can write eigen­value equations using an operator whose largest eigenval­ue is e -COIIJl. The eigenfunctions are: interesting, andtheir analysis \\;11 shed light on the free energ)' GWI "ndon other eigen\'alucs in this formalism. The theorydeveloped below is valid for scts that can bt organiled onregular trees-binary, ternary, or higher~rder trees.There is no requirement that the trees be complete; thenumber of legs can gro..... at a rate slo.....er thin 2", 3", etc.m such cases we shall ne«l. the rules of pruning of tbetrees in order to .....rite a c1~-form theory. ~

.: 1 In 5«. II we derive the eigenvalue equations for com·; '1' plete trees. Section III discusses applications to dynami·:;::; cal systems which are: (Smale) complete maps of the inter-

'. g.t: val, In Sec. IV a detailed discussion of the singularities of• -:~. the eigenfunctions is perfonned. It yields an understand.

~.;: ing of the space of functions in which the eigenvalue~~~ equations operate, and the nature of the discre:te and con-

.: ::. tinuous components of the spectra. Section V is devoted'e., ...... to tbe study of intermittent maps and the interesting'1 :;,; phase transition in their thermodynamic functions, It is,t1j shown that this phase transition is identical to the one;:.~. fou!1d in the context of mode lockings of quasiperiodic~;~. systems, and that it bas an equivalent number theoretic. ~. representation. The eigenvalue equations allow us a solu-- ~;. tion of the phase transition, which turns out to be of;. 'i:. infinite order. Section VI treats incomplete maps of the

.J interval where the pruning of the trees is vital. It is euyto accomplish a calculation of the thennodynamic func­tions when the itinerary of the critical point is periodic orprcpcriodic. However, when the itinerary is aperiodicone has to resort 10 indirett methods. The method sug·gested in Sec. VI 'relies on calculations at parametervalues for \\'hich the: itineraries are periodic, but wilh in·creasing length of the period. It is shown tbat such cal·culations converge. Section Vll is a summaf')' and discus­sion.

~A \

Pi ~

1 -~·il~e

/,,/

• 11.4)

..; (1.3)

--....:

MAY 15,1989

,-I"Cft\.CI.<6

(~I~J ~ ~ FI~)~\ ;

@)1989The American Pb)"ieal Societ)'

No"f

td _ L't'·,-,

~ -~GUll_ I ~ -~E, ,,where GI(3) is the free energy Idensity) muhiplied by Iheinverse temperature (3. Indeed, this resemblance gave riseto the develo~ment of the thermod~ll.mic formalism ofmultifractals.•,1) In particular 0,1' c:an be calculated asthe largest eigenvalue of a transrer matrix of an appropri­ate spin model whose thermOdynamiCS is equivalent tothai of the given fractal measure, Nonanalyticilies inql-r) could be interpreted LS phase.lransitioos..·- tt

If the panitions arc not ~uimeasure partitions, 911')cannot ~ calculated in this way. Still, the rate of growthof the sum I.,l,-' is an important piece of information ontbe multifract:l1 sct, shedding light on il$ geometric re-

S)S9

tion of the saling properties offnctal measures.-Simple and eleganl theories '0 calculate 1"(9) [or, in

fact, its inverse function ql1"ll, have been developed whenthe partition is an equimeasure panition, i.e., wherep,-const = lIN. II-I) Such I situation OCCUI1, for exam­ple, at the accumulation point of period doubling, whereN=r at the nth generation of refinement of the parti­tion. Since in practice one calcuJates q{T) by requiringthat rlq,1"I-l, one can rewrite Eq. 11.11 in such a~ as

N ,;rft,l_ 1: 1,-' , 11.2),-,

Inspecting Eq, (1.2) or (1.3) one nOlices the resem·blances to the statistical mechanical relalion

In general, the number of balls in a partition is not in·crensing with the generations necc:ssarily like 2\ but rath­er like o~, where 0 can be any real number. Then Eq.11.2) turns to

..4-

12

n.1l

VQLVME 39, NUMBER 10

""(I-) F,'i.

\.

../*~--

Itamar Procaccia and Tamas Tel·Dtporrmtnl ojChrmkol Physics. 17Ir Wti~m(llltl /rutifliit ofScit/lrt, RtllolJOl 76100. flrod

lRecc.h'cd 17 November 198B)

The calculation of the saling propt'nia of mullifract&.l KU is presented a5 AD dgcnva.lut prob­lem. The dgCll\'Jluc equation unities the \raiment ohcu that C&n be OteJ.Diud OD regular Iret$_be the)' complete or incomplete (pruned) trees. In panicu!.a.r. this approach unilieslhc multifractaluW~ of seu at tbe borderline of ct,aO' 'loith tbose: of chaotic sets. Phase transitions in tbc lhct·modrnaI:lic formalism of multifracuh ITC identified as I cfouing of the luSest diacrclt eigenvalue\\;Ib the continuous pan of the spcclrum of the relevant operator. An lUIllysis of the eigenfunctionsis presented and loevCral examples arc solved in dellil. Of plrti~ular interest is the analysis of i~ter­

mittent ma~, ...·hich shows the e.tisten~ of an ;nfinite-order phue U'llUitiora, IDd of (Sll1.llel in·comp1ct: m3;K of the ;ntervll with finite Ind infinite rules of pruninl of the ml.lllifraeL&!lrccs,

it can be shown that in the appropriate""-. ce limit thispaJ1ition function is either zero or infinity depending onwhether 1">nq) or 7<-:-(ql. This defines rlql, II basbeen shown further that the funclion -:1q) furnishes im­ponant infonnation about the scalin!; propeJ1ies of frac·tal mea.~urC$. In paJ1icular a Legendre transform of rlq~

yields the Ita) funclion, a very con\'enient rcprescnta-

Scaling properties of multifrncta1s as an eigenvalue problem

Ever since it h:J.S been recogniz.odl-~ that fractal ob­jects appearing in complex and nonlinear systems arc not.... ell characterized by a single scaling upoeeet, bUI ralh­el b)' a spectrum of scaling elponents, Ihere has been anel.plosive interest in such objects, which were termedmultifraculs.) MuJlifractals pia)' a dominant role asurznge aUracto~ of chaotic dynamical sySlems,S dissipa·lion fields'of lurbulent Bows,' in growth patterns,'" non·linur resislor networks.' etc. The aim of this paper is towork out in detail a powerful technical tool for the study(If multifractals, a tool that allows calclll3tions of rde\'antscaling propenies from solutions of appropriate eigenval­ue equations. For dynamical systems, this tool unifies thetreatment of sets at the borderline of chaos with that ofsystems in their chaotic regime.

1E.e,objects under study are usually fractals that sup·.~rt some measure. Thus for example in chaotic dynami.cal sysums one has B limit set-a slrange attractor­which is Bfractal sct. The natural invariant measure pro­vides the probability tb.e.t a typical orbit visits various reogiuns of the attractor_ IO Having such "fractal measures"one can consider a coverage of the set by a panition into/I' balls of radii II/Ir"l' each of which having a measurep,. If one introduces a plr.ttition function:'·

Nnll.l",~)- k p!II:,,-,

Mitchell J. Fc:igc:nb:lumRockt/t/itr UnilW/'fir)'. /1J!J )'o,k At~llut. Ntw rork, Ntw York 1001/

L I!'TRODUCTION

~H tSrCAl. REVmW A

'-

~

!I"'"i,_

J

I

I'

\........

,

•

,,

42

-L'r/l.If>~

-20 .1,4 -1.0 .0.6

o-2

-4

2~~

0

§'" -,

-"

,

o§

" -2

C. Fair 1nrII.l..IIa about mtullJ'Cl

The natural in\'ariant measures of maps of the family().1) all have singularities It x =0, J such that for Z -f or

·2 0' 2 ..

/ PFIG, 2. GliB Yl (J for the quadntic, tolOplete map. u ob­

wned from '-1.(,81 of Eq, 13.7). In the huel the vicinity of (hephuc transitiOIl is/enlarged I.lld the dashed line is tbe CUCI

'Olu(ion for -7-6110&1-

/ 12P2P ln2 P< -)Gi)= (P-lllnl .8>-1 . 13.S)

and tlie numerics agree very well. Convergence 15 fasl ex­cept!near the pb.ase transition point .8. """ -1 where tbediiference between tbe numerics and O,S) is still seen (andis of the order of 4%) even after 16 iterations.

Eu..mple Hi). Here we take z =4. There is no anal)1.icsolution for tbe whole .8 axis, and tbe numerical solutionis displayed in Fig. 3. Notice that the left branch appc.anto be straight as in Fig, 2, whereas the right brancb iscurved. The tbeory of Sec. rv will \'erify these numericalfindings.

FlG. 3. G({JI \IS P ror the quartic complete map with z -".obtained from '-ll({Jl. The cl&shod line, ....hich is limply{/J-Illog2, is put 10 IDC$$ (he ract that the right bBDCb iscurvin&-

13.51

13,7)

FEIGENBAUM, PROCACClA, Al't'D TELS362

Onee v.rilten. this equation is immediately recogniz.edas a lleneraliZ3tion of the Frobenius-Perron equation!1which is ohtained in the limit fJ=l. }.II)=I. Differenlextensions of the Frobenius_Perron equation hive beenused in the context of transient chaotic behavior.1J Fornonhyperbolic maps Eq. (3.4) has been introduced in Ref.24 wbere it was shown that for smooth initial functionsI!to there exists one and onl)' one coefficient }.(8) so thatthe ~ries of 0~61 con\'erges fexponentiaU)' fast) toward Ifinite ¢.I61(z). i.e., both AlfJl and ~161 are unique. andfunhermore, that for hy~rbolic maps }.IB) )ields thespectrum of generalized entropies~ for both permanentand transient chaos. [We call a map b)'perbolic if1< 1/,1;()1 < 00 holds.]

The operator L 161 of Eq. (2. J2) wbicb is now a specialcase of Ruelle's Frobenius-PetTon operator. I. takes onthe fonn

L '6IQ()'I= }: ~.~erl", V'(xtl

where

Equation 0.4) can be solved numerically in a straighl.forward way. Since the limit is unique. we can start withan initial function '#0= I. At all levels '#~ is nonnaliudsuch that ~~IX)=1 for some suitable X. We approach~(6lfx) through a sequence of functions ,#,,(;(l whichtransfonn according to .

B. J",'umerical solutiollJ

The largest eigenvalue of it ;.1.8), is connected via Eq,(2.13) with Gl.8) Oflhe length scale panition sum 11.5) onthe co\'erage defined by <2,41. It is 10 be DOted Ibat (2.4)provides a panial co\'erage of the attractor which, there­fore, differs slightly from the co\'erage of Ihe generatingpanition used for one.hump maps, I •. \} The thermo-­dynamic~ for these panitions are, however, expected tobe tbe same.

Below we shall find it ad\'antalleous 10 consider thewhole Spectrum of L IBI and eigenfunctions belonging toeigenvalues Other than the largest. Before we do that,however. we examine some numerical solutions of Eq.(3A).

The ~riC5 of }.~ tends simultaneously to the largest eillen·value ).(.8). This transformation is numeric.ally accom­plished through a reeursiye function call. In the next sec­lion we discuss the eigenfunctions ¢IB1lz ) tbemselves.Here we displa)' the results for ;.IB) of tbe procedure de·scribed above.

Example m. As a first enmple we talce z = 2 in Eq.13.1). This parabolic case. yields wcll-known thenno-­dynamics and we present il here: as a checlc.

Figure 2 displays GI.8) versus (3. The analytic solutiolJin this~ reads",1l·1.

)

•

•

,.

[

,

U.71

(3.4)

(3.11

13-)

S361

x'=I-lI-2J:I' •

where '1= I, ... k -I and S""''1-1.This calculation of a ball diameter relies on the fact

thai F. is contractive. Thus distances between nearestpaints arc naturally obtained by Eq, 12.4) as a COO$C'

quence of the ~ponentiaJ growth of stabilit)· of a compo­silion of F.'s. For n sufficiently large Eq. 12.4) behavesas)'mplolicalJy according to the chain rule as

11•

'{'1,f._I"" ,f l )

-IF;lIF'I.F.J

••• IF~,IF.)oF••.. '1"'1. (2.51./

We arc no...... in position to derive the eigenvalue equa-tion. Consider the sum

S(n.tn= .L )..- ..'8)(/I·,I'1,f._I.···.fll}1I· (2.6).,. ....·.-1

m. APPUCAnONS TO DYNAMICAL SYSTEMS:COMPLETE MAPS OF TI{E Th."TERVAL

where f takes on the t......o values 0 and I.To write the eigenvalue equation in this ca.se. We have

to find F;fxl. For the problem at hand these derivativescan be written u

peel fBllx) 10 occupy, and whit kind of spectrum ).1.8)belongs to it. To tbis aim we tum nnw to SOme concrelec.u.mples.

on tbe interval [0.1] for z::!: t, A set thac conforms .....iththe assumptions of Sec. II is the set of aU the preimagesof tbe critical point x.=t, x·=xr The two functionsFo and F I which transfer x, into XlI and x ll + 1 are thetwo inverses of the function [(x). For Ihe family 13.\)tbe functions F,(x) are

F,(x)=++lf"-+Ml-x)"'. 13.2)

A. PrdimIJW')' detallJ

Consider tbe family of one·hump maps [(x) whichmap the interval onto itself (Smale complete maps) andgenerate chaotic dynamics. A particular branch of thisfamily is given by the maps

(2.9)

GA) Using Eq.12,5) we can estimlte Sln.8) as

(2.10)

12.12)

a.B)--

SCALING PROPERTIES OF MULTlFRACTAU AS A."l , ,

S(r.,PI_ ... LI.-II.8>1F~,{F".1···)11I L }.-I(tnIF;._IIF" ..F',.l "'116 ...

'. "-1

(2.31

!>Cntation functions," hss dynamical foundations whichare expounded in Rd. 21. :n~ connecti~ns ,ensure thatthc roachine!')' developed Ul thIS paper IS directly per­tioent to a IUle varielY of significanl dynamical prob­lems. Giyen the fUDC.lions F" we can wrile

Where tbe symbol 0 denoles a composmoD, FoFlx)=FIF(.t}}. We sh.Jll assume that a panition at the nthgeneration is obtained from balls of diamekr , ...11'1.f .. _l' _ .. ,f l ),

To proceed. adopt the following notation:

f{J.61IF.,oF',.lo'" )

= L )..-lltnIF;._,IF"oF., .. lo ... )16 (2.S)

"-I

With this notation Ihe terms shown on the risht-handside of f2.7) )ield precisely ~~B~ Ilx). Also, written explic­itly, these tenDS read

Finally, assuming convergence in the limit r _ 00. We can"'rite tbe functional eigenvalue equation

)..(Pl~(61(xl=:I !F;(x)j,ll,t61IF.{x» . (2.1I)

The number )..{p) is an eigenvalue. Defining the linearoperator L 1,Il1 via

1·..ll'1,f._ I •·•· ,Ell

= iF" of,-::0 ••• of•• _I(F,Ix·))

-F'loF.,o "'oF'._IIF,lx·l)l.

we realize that ).(.8) is the wgest eigenvalue of L lBI. Ontbe other band. the statement that fnr Iur r.\~, + 1-!/I.is equivalent to requiring that I,}.-"Ilf· ll -l'jr [cr. Eq.(U)]

Al,8)=e -GtBl •

"".

lI.'herc G (.8) is the thermodynamic function associated",itb the COYenge (2.41. We shlU be interested both inIhe dgenvalucs ).(,8) and in the eigenfunctions ,,(6)(xl.

e have to scud)' no",' Ihe space of functions thlt we ex·§--

e---------.J

and the auociated eigenvaJues are invariant to conjuga­tion. It is of particular imponance that one can choose afunction h such that tbe conjugation leads from nonby­perbolic to a hyperbolic map as pointed out in Ref. 28.HavinJl; a nonhyperbolic map f(x) whose invariant densi­ty is 1/f11(X), one defines

h(X)= J~tP"l(y)dy (4.18),and then conj"ugating as in Eq. (4.161 leads to a hyperbolicmap g(x)=f(x). The discrete spectrum of hyperbolicmaps is always associated with smooth (Le., nonsingularldgenfunctions t/7ifl. The conjugation maps the spectrumof the hyperbolic map onto the spectrum of the nonhy­perbolic map. while via Eq. t4.11) the eigenfunctions willgain the singularities typical of class A. Since tP1l1lx} bassingularities of strength n- I /%), the eigenfunctions willhave singularities of strength (1-1 /z){J as calculated inSec. II A.

The considerations abo~e make it possible to establisba relation between AI(8) and the generaliu:d entropiesKfI·" The latters can be e.xpressed in terms of the conju­gate map ].n,n In the specia.l case of hyperbolic maps,tbe eigenvalue of (3.4) obtained hy staning with anysmootb initiaJ function bas been shown to be directly re­lated to K 6.H,n,Xl More generally, for any ehaotic mapone ohtains

B. DUcret.e~ &rid Il:f e1aenf'wletfou

The: discrete spectrum belongs to eigenfunctions ofclass A. Initial conditions ¢'o(xl, which are of this clusand are non.negative everywhere, remain in tbe class Un­der iterations.

The analysis of tbe discrete spectrum and the eigen.functions is facilitated hy studying conjugations todifferent maps and in panicular to hyperbolic maps17(i.e., maps which must haVe a maximum of order: = ll.The details of the conjugation and its effects on the eigen_functions are presented ill Appendix A. Here tbe mainresulu are summa.rized.

The complete maps f(xl and g(x) of the interval arccalled conjugate if tbere: exists a function h (x) such thar

f(x)=h -Iogoh (x). (4.161

If I#J'II(X) and ,If'lx) are eigenfunctions belonging to 13.41with f and g, respectively, then

¢'1'(x)=¢~61oh(x)lh'lx1l6, (4.17)

wise. ir lurns automaticall)' to a function of c1a.u A.Then, alw, sianing "ith a function of class A the itera_tion s~)'s Wilhin Ihe: class. We turn now 10 an anal}'sis ofthe discrete Spectrum, stressing that as a function of Bone can observe transitions belwee:n solutions belonging10 the twO classes. These transitions are strongly relatedto phase transitions in the thennod)'namic formalism.

lnAI(81={I-{J)K/l, (4.19)

where AI{P) tS the eigenvaJue of the operalorH={",UI)-8Lc61(tPfll )6 reached from a constant InJtlalfunctlon,17 t.e., lD the formulation of the present paper,

(4.10)/'

(4.12)

{4.14)

n = 2: Upon tbe second iteration the singularity oftP~lIl(y) for y_l is DOt cbanging; at the other comer wefind /

!/J~61(Yl - (yl-a, a=m

73.Xfa ,01. (4.1]),-,

~est: types of singulan

7'ties are nOt ailered upon funher

Iteratlons.

We. can thus conc~ude tbat there are two types of initialfunctions \l'o{x), which we designate as follows.

Oass A: al...~; this class of functions iterates to func­tions having the: same singularities at both ends.

Oass B: a l > a; this class of functions iterates to func­tions having two different tyPes of singularities, a¢o attwo ends.

Notice that since a is P dependent these classes arcalso 8 dependent. In Sec. IV B we focus on class A. Herewe sbow that class B belongs to the continuous spectrumofthe operator 0.5).

. To see this we consider #61(y) obtained afte:r manyiterations In _ 00) in tbe limit y-O,

A-I J-I",181(y)-_6 t/7161(y/cl=-=--et/7ltJI{I-y/cll.

< <,

Since A-I can be chosen arbitrarily, this eigenvaJue, if itexists, belongs to tbe conrinuollS Spectrum (The latter is anew type of continuum different from what is called thecontinuous spectrum of tbe Frohenius-PeJTon operator inthe mat~~atical literature. I.). On the other hand, ifa =0, A cannot be expressed 10 terms of c. These func­tions, which belong to class A, belong to tbe discrerespectrum.

The last issue of analysis is tbe question wben can onereach functions of class B upon iterations starting with!Po(x) belonging to the class. Denoting the eigenvalues inthe discrete Spectrum by A,I{JI, with J'I({J) being tbe larg­est, we note that only if A of Eq. (4.14) is larger, Le.,

ca-/l>J·I({J), (4.15)

then the function I/Jo(xl remains within the class. Other-

0<

FEIGE"'BAUM. PROCACCIA. AND TEL

where a =( J- J/:IP is wbat we call the "naturaJ" singu­larity,

1/J\~I(yl _ }.-II A; (y/d-C1g+ B;IY!cll-ooj 14.91,_0 C C I

In the last upres.s.ion one of the singularities dominatesas y -0, and we write

From Eq. (4.5) we know tbat tbe right-hand side hassingularity of strength o. The equation can be balancedwith a left singularity of strength a only if

), -,1- cB- a =0 (4.13)

•

(4.8)

14.6)

14.5)

(4.4)

(4.3)

(4.2)

5363

o

x =.l, "

f'(xll=c ,

!'(x1l= -CI

Hal

and walch what happens upon iterations.It =1:

",\tJl(y) _ (I-yl-",-,

With these preliminaries we can ask what is the fate of aninitiaJ function V'n(x), under L l81 with a given A. In par­ticular, we want to know how I/Jlfl(y) behaves for y-O,l.

Using Eq. 14.1) and (4.2). we can conclude that ifV'lf'lx) is smooth around Xc then

81 2A-I1fr~IXc)ll_ )-U-11J161/r~ .. t(Y) ..:1 (%0111)6 y

On the other hand, in the limity-O, we find

[

!JI.61(y/c) ¢161(1_y/C) I\y'61 () A-I _._"--+ " ,•• 'Y ..:o cfl cf

Allow now "'oIx) to bave singularities of strength aoand 00 at the two ends,

~'o(x) - Ao(xl-"o, "'o(xl - Bo(l-xl-"o (4.7)~_o a_I

xJ=I-L.<,

The derivatives at these points are

The deri\'atives f'lx) at these poinu are

1f'(x I,J)Iz:o llJz!l-yll-lI, .

xAG.'. A skelch of a typical one-hump map of the interval

10,Il. The theory depends crucially nn the order of the mu­imum: and on the slopes c and Cl'

(ii) The inve~ for y_O: For y_O the two preimages

."

(4.1)

SCALING PROPERTIES OF MULTIFRACJ"ALS AS AN ..

rv. SINGULARITY A.~ALYSlS

OFTH[ [IGE.:....t1.f!"cnONS

x -1-c, ¢'lltxl_f- Il - 11:1(d. Sec. IV). Thw. an)' ~arti­

tioD C"'M twa boxes of size J at tbe extremes of the Inter­ysl ,,'bose measure PIll scales as PIll-I" llIilh a=l/:.All other boJ:C5 have Donsingular measures "ilb a= I. Inthe noution of Ref. 4, 11ql of Eq. lUI reads ((or: > I)

. :;r lq/z. q>:/l:-II,-:-(q)=qo-/(a)= 7 1, q<:/I:_II. 13.9)

As nOled before for cases where the panilion is anequimea.sure partition, G (,8)= -q (-,8)1D 2. This isindeed the case for ,i-ample W, but is nOI so for exampleHi). The feason (qf the different thermodynamics is thatthe partition \'ia;.ibe preimages of the critical point con·tains boxes with singularly nonCOnSlanl natural measuresat both ends) We reiterate this fact 10 stress thedifference in general between G(fj) and q(7'I,

,,12

The understanding of the tbermodynamics and the d­gen\'alue equations calli for some careful examination oftbe nature of tbe eigenfunctions, the spaces of functionsfrom which the)' are drawn, and the spectrum of dgen­vaJues which characterizes the operator U.5l. The in­sight provided by anaJyses of the Frobenius·PetTon equa­tion llhe (J-I limit) cannot be taken over in cases whereusc is being made of the fact the lfrllltxl is a normalizeddensity. Rather, we shall see that in geoeral the singular­ities of ¢l61(;:r) shed imponaot light on the nature ofclassification needed for these eigenfunctions. In addi­tion, we shall see that eigen\'aJues different from the larg­est one own eigenfunctions whose nature reveals aspectsof the thermodynamics in a vel)' useful way.

A. Space of Irg,al e[itnfuncdons

In this subsection we show that functions that have

•fferent singularities at x =0, 1 belong to a special class,hich gives rise to a continuous component of the spec­

trum. Functions with identical singularities belong to tbediscrete spectrum. The distinction will be shown to benaturaJ since initial functions wbich are smooth are or­thogonal to the functions of the special class.

To see this, consider tbe general famil)' of one-humpmaps 00 tbe intervaL We sball assume that the max­imum of tbe map at x =xc is of the order l, i.e., near themaximum flx)= 1-0 Ix -xci', a >0. In addition, theslopes of the function at x =0, I will be denoted by c,c I,respectivdy, see Fig. 4. It will be seen tbat the qualitativeanal}'sis depends soldy on the order of the maximum andthe vaJues of these slopes. Our analysis is an extension ofwhat was applied 10 the Frobenius-Perron equation.2t

• Inparticular, to analyz.e ¢181{y} via Eq. n.4} we shall needthe inverse functions at y-O, I.

(i) The inverse for y-I: For ),-1 the tWO preimages.,.,

•

*ii.,iiiiiII...'I__ii

? r'

15.5}

15.6)

IS.8I

(5.9)

(5.101

(5.1 1)1/),,;;:z,

...... .....---.- ..

~11m =0-).=2, \bIOI = I). For all Olher f3 (¢-ml2 """IIthe ei~enfunclio"nof IS.)) possesses a branch cut alon~"thenegauve real ax's l/J= +1 is of this son, and, so far as weknow, no analytic SOlution has been obtainedJ.

Lc~ .us proc~ 10 delermine tbe nature of Ihe phase:tranSItIon. NotIce by substituting I/x for x in 15 3) .). > 0, that . ,Since

!f1161(l/xJ=x 28!fJ61l x ) .

Setting x = 1 in (5_3) we~ that

).~I'Il=2-28""\of,l81,~).

Should ~i6l(11 diverge. then so too must ¢!1I1(+l. Seninx = -l-, so 100 must either (or both) of ~16J( I ) J611:) In"d:'I,_d6l. T'T-­

UC!l\e y, or I;J- to be finlle on the rationals in 10 II itfollows that ~161( I} is finite. Taking no",' the li~il'asx -0-, we have

setting x =x +n in (5.10) and multiplying by z~+l, wehave

Z·",llll(x +n)-z-"'I",I~J(x +/1 + I)

-~"'''I~)Ix +n)28 x +n

= =_+1 161 [ x+. ]Ix+n+ljl8tJr ~ , (5.12)

where we again used (5.5) in tbe last transformation. Ifwe now sum (5.12) Over n =0, ... ,00, by (5.8) we hivelafler setting n -/I - I I

¢11I1(X)= i --'_'_",16111 __'_). 15.1311 Ix +n)28 x +11

Since lfi6l is diJferentiable al x = I, we can re",'rile 15.13)

'"

/3

15.2)

FEIGENBAUM, PROCACCIA. AND TtL

o

GI/3)

"..

SIO;C

~IG. 6. A 1I.:eteh of the c:iEerJuncrJol1$ pc:ru.illinS to di5ercQtrepolU of 8. G IPl. for I us.c: ..ilh: > I.

coupl~ nonlinear oscillators, conforms euctly ",ith thatof the Intermittent map.

In principle, an)' map of tbe interval Ihat starts asx +x-.+'" can be used to obser'Ye interminency. The~aIYSIS of the number-theoretic Farcy model in Appen­diX B suggCSts the usc of Ibe map

O.-lh>16110"')=¢!6'(II, (5.7)

so tbal for P< I (and hence ;. > I), ~O+) is also finile sotbat b)' (5.5) ,

15-'''''10"'")--:;;a

and 1/.1161 .......,..-<_ ._z_ X <f 161" ,.~---anCflenslOnto{l,oo). Forf3=I-tl2,I-x' ¢ dep~s from .1 Ix at large x by an abrupt change of

x'= .!=.!., 15.1l asymptotic behaVIOr, and at x =0 by becoming finile ofx x >t· order I/(A-I). In panicuJar, it remains smooth al

. x = I, and we adopt the normalization

SLnC~ g(x).=x/ll-xl is known to be the fUed point of II) ¢16IOl=1the lD~ennlttency renormalization group,l1 we can USC it I . - .!o den.ve stalements that have unh"ersal applicability 10 (Indeed, by differenliating (5.5), we lea.m that r/J'(IIJ( I)lnterml~tent maps.. As far as intennittency is concerned = -,8].the.chOlcc.ofthe ngbt-hand branch is irrcle\'anl as long E I "as It pro~ldes reinjection to x -0. The choicc in Eq. mp Oy the symmetry (5.5) to rewnte 15.3) as(5.Il, .WhlCh . is motivated by the number theoretic 1111 I I Ia.nalys~, prOVides, ~owever, imponant symmetry proper- ),,(f31y. (Xl=¢llll( 1+ x J+ """"'i8¢11l1 1+1. .ties which are used In the SOlution below D . Z x

D, 0'· 10' I b ° . enotmg. n Ing '1 )' and It,ll by 1, we can wrile thelDVerscs of Ihe functions in (5.1) as

F,(x)=_X_I+x'

FI{xl=_I_=I-F {xlI+x o·

All we nc:cd do now is write down tbe funclional eigen­value equalion for (5.2) which reads

A(8)l/Jtlll(xl

~()+X)-"'I~(" [,~X I+~(" ['~Z ]I 1'.3)

N?t~ce firsl thai for 8= I, !/JIll simply has a pole at Ibeongln,

P=I-lfIlI~xl=IIx_A.=1 (SA)

Nut, for IJ= -m 12, m =0,1, ...• 15.3) possessessolutions in the space of polynomials of d~gree m

..H6S

2,/3

o

-- .:

/., // I

.,/. JI

II

II

-2

-2

2

o,f---­"'-

AG. S. The observed thennodyn-.mics for the quartic map.The continuOla brancbes are dctermined by lhe lea.clinl eigen­values '-0 for 1he left bra.nch and h, for tbe riChl branch. The hIbrancb is continued in a duhed-dolled line. The whole 1.MJIbranch corresponds 10 the &cnert.liud enlrOpiC5 K I' d. telLThe dashed line is -,1-111102, which is the thermod)'Tl&Inicsof the invariaol measure, and is a.ddc:d 10 iUwlrale the ....minEofS=. Inc- The oonanalyticily appean al a differeDt value ofPin 1he two C&.W:L

this sectioo we c1isc\W tbe eigenvalue equation andthe thermodynamics for comple:te maps of the inler'Yalwhich arc strongly nonhypcrbolic, in the seosc. lhal aCued point is marginally stable: and the orbits are inler­mittent.]l The sct obtained displays a phase: Iransilionwhose nature has been hitherto c:1usive. This phase tran­sition is of more general interest because an idenlical kindof phase transition appcan in other, seemingly unrdated,problems. In Appendix B it is shown Ihal the thermo­dynamics of the sct provided by the rational numbers inthe Farey modd, which is also the thermodynamics oftbe sct which appears in the Irelltment of mode-locking of

(4_21)

(4.20)

SCALING PROPERTIES OF MULl1FR.ACTAl..5 AS AN."

t-GI61=},IPI= mu.(~Pl,AI{81).

j.,fBJ is the largen eigenvalue or the discrete spcetrumi~'1Cjated ",itb funclions of claM AI.

Thus. an essentially different behavior can be observed ontwo sides of 8 critical u:mperature 8. for which

;'118.I=c -~. . (4.22)

!bercrore, tbe phenomenon can be interpreted as a (first­orded phase: transilion and the region PIP. > I can becalled the condeMed phase.

.Ji.fact, tbis is • special ~ of (4.14), Since ~({J) isWluc at all 8. where it tusts, we can say that the con·tlDuum spectrum restricted 10 the subclus appean as an"declive" discrtlt;bra.ncb '-o({3I. This mUlU that, for(T <0, }'O(P) is tor added to tbe discrete spectrum whenslaTting with smocllh initial functions. In tbe range""hen ~(Bl <"'A\IPl, tbe lalter dominates. In Eq. (4.6)}.=A1fPI>c-". tberefore, .pCfI(O) llI;U tcnd to uro andthe limiting function ~"ll:c) ll.ill belong 10 clus A. If.however. ~(fJl>A1(PI. Ao dominates, and ¢161 will belong10 the subclass introduced above. The o~rved tb~rmo­

dynamics is connected v.itb the actuaUy largest eigenval­ue, i.e.,

As we saw. initial functions of class A do not lea\'e Ihisclass. In other words. the lar"est eigenvalue influencingtheir evolution is always i. l lf3). This provides a usefulmethod for calculating ).,IP), i.e., the Spectrum of gen­eniliz.ed entropies. By starting ",ith, e.g., Jolxl

We are now in position 10 uaminc the thermodynam. =x-"'(I -X)-III, the analoguc of D.6) and 13.7) defines aics which is obtained upon ilerating tbe funetional equa- series ~.(x) tending 10 a tfinitd function of cla.u A (sec

tion with a smooth initial function. Of importance is lbe also Ref. 27). Alt~·can Ihen be oblained as the limit ofcompetition belween the discrete and continuous pan of the A. as given 9Y (3.7). It is wortb emphasizing that intbe specltum. lhis ?rocedurCltlo critical s1o\\oing down appeus at 8. in

By starting ",ilb a smooth function "4:rJ.x J, itS fint contrasl. torhal one sees ",itb a smooth initial function.iterale ",ill possess a singularil)' of strength 0 at x =1 as The critijiJ .value can then be delermined ",ith a higb ac-expl&ir:ted in Sec. IV A, bUI will be COIlSta.nt at the left c~)' the intersection of 1""0 branches (see (4.21l).corner. The singularity It x = I will DOl be removed by Th ranches arc shown on Fig. 5 for lbc eumple liil offunher iteratioIlS. Lc.l us follow ""hat blppca to ¢~(Ol. . m. We conclude by sbo",ing I qualitative sketch in-By a~l~ing Eq. 14.6J for II = I we see lbat the behuior dicaling lbe change of behavior of lhe eigenfunctionsof ..~~I at x __ 0 depends 011 the actual sign of D. For :aJODg differenl branches on Fig. 6. We note lhat one cao-0> 0, the second term dominates on the right.hnd sid not exclude lbe situation wbere ~(IJ) is always less Iban

•this leads to a singularity of strength 0, also:ft 1~).~'L~P5)~·:1:0~':U~'~h~=;:;:~~'h~'~';'~'-.'0~U~I;d~b<;;n;O:::P-.b;"';:;::':....:-:~;~.;~O:O~.__~__

0, ""hich stays there forever. Thus, smooth iD.ltiaJ \functions are mapped after 190'0 steps on class .4,/and, MODYNA.\-{ICS OF L'oTERMIlTDiT MAPStherefore, tbeir e~olution is governed by the discretespe:::::tcum. In panicular, the asymptotic behavior ischaracterizc:d by A1(8) in the case where 0 > 0 lB> 0 forz > I).

The situation is quile ditrerent for u <0 since 1/1\6)(x)then docs not diverge for x_I, but ratber vanishes.Therefore, 1fJlJ v.ill stay constant at x =0. Its furtherfate depends on the actual value of8,

Smooth initial functions in tbe range (f <0 are, thus,iterated nol in class A but rather in a subclass of class B,namely in the class of functions which arc constanl atx =0 and have singularities of strength (f at x = I. It fol·lows then from 14.6) that the eigenvalue inside this sub­class is

,-

~ eN\ .'\:;\\ ..;> -\- r'~ I T

""

,\

1\

,\.

....LA . ~\.{

~\c') "~ r::

'-'

)"

1

5.0,--,---,---,---.,

2.5

S0.0

"N ·25

·5

-7.5

·5 0 5Il

--- 0 -- _

other line in Fig. 9 is obtained from the initial function(6.6).. The. phase transitiOIl is obvious. Other Misi.u~eWJcz pomts can be treated under similar footing andwill not be detailed here.

,;,.,

FlG.~. The~odynamiC$ at the first Misiure-.iez point. TheI~OOI.h ~~ avoub Ibe phase: tnnsitiollS by taking a lingular ini.Iial COoWUOD. The un.ighl line Kgment belonp to Ibe COD­

de:1lSC:d pba.sc.

o

B. SII.pent::ahlt orbits

When the itinerary of the critical poi~t is periodic Wehave a superstable orbit. The set of interest is then not Inanractor but a repeller.],-n.2J An example is the mapI_~l at a] = 1. 754 88 ... , whie:h supports a superslable3:orblt. We denote the region (0-01,0) b~' 0 and the rt­glon (O~I) by I: Observation shows (see Fig. 101 that 0has prelmages In I only, whereas) has preimagcs both inoand 1. We can thus write immedialely

}.(61\lP:ff(x)"""IFi(Fo(x))IB¢\BI(Folx)). (6.71

}.(Dh/AIl1l x )= IFolF I (x »IB¢lfl(FI(x))

+IF;(FI (x))iB,,\8ICFI(x)), (6.81

FlG. 10. The liluation at a ptnme:lC-f value: .'hue period 3 is'upustable.

FEIGENBA.UM. PROCACClA, AND TEL

--0

S36S

FlG. S. The situation lithe: fint MiJ,iure_ie:l poiDt.

I. Accordingl~.. Eq. (6.31 translates immediatc:1y to theset of coupled equations

i.IDh.I.:f'(xl= IF;(FolxJ)jBt,I;~6'(Fo(x)1 •

}.(.8l¢\BI(xl=IF;(FIIXIl181,!1~BIIFllxll. 16.41

t.(DlI!i~BI(X 1= ]FoIF~(x1JlBt.';:fIIF2{x) I

+IF;IF~lxI)IBI!I\BIIFl(X)) .

NOlicing that F I {x)=F2{x)=-Fo{x), we can, denotingFI(x)=Flx), rewrite Eqs. 16.4) as

t.1(fJl¢16J{xl

= IF'(x IIB[ IF'IF(x I)IB¢IBJIF(F(x) I)

+]F'(-F(xl)!B.pIBI(F(-F(x)))]. (6,S)

w~ere ~(~I(x)=\bYJI(f(x)). Equation (6.S1 can be: solvedUSUlg Similar techniques (like: recursive functions ca.lls) asthose used preYiously.

To speed up the convergen~one can start with an ini­tial function that is singular at x = I-Ii and x = I, e.g.,

;!(xl=lIx-ll-al)ll-xWBn. 16.6)

~uch singulari~ies should huild up anyway by the itera­tion, and slamng with them in ~xl speeds up the con­vergen~.

From the .~int of view of scaling behavior, we expect aphase trBOSmon at fJ negative, due 10 the dilation ofJength.s at the critical point. Any point being 6 close tox ~ I comes f~o~ I distanc.: SII'2 near the critical point.This slron.g dilatlon results In making the neighborhoodof Xo atypical, and for fJ sufficiently negative this atypicalbc:havior dominates.. .~e can re~ove the phase transition by suning with anlnlUal fune:tlon that is sinfular also at xo• i.e.,!lx -{I-a))O-xlix -xolI- 1'2• ....ith such a functionand 13 negative, We remo\'e all the influence: of the dilationat Xo' There is no path for 6<0 that lead$ back to thefued point. Indc:ed, having such an initial fune:tion. wecompute the smooth function depicted in Fig. 9. The

•

-~------~-==;;:::=======~\I

16.2)

5367

Denoting now

¢181IFf(x))=Ib~Blf.X) ,

Eq. 16.11 takes on the fonn

t.(Pl¢~~I(X)=L IF;(Ftllx))18¢~BI(Ffll.X)). 16.3),

A. MWun1Fiez points

The simplest situation of this t)'pe is when Ihe: criticalpoint is mapped (in 1.....0 steps) onto the fixc:d point of theiteration. For maps of the type x'= I-ax] this occurs ata = 1.S43 688 ... The situation is depicted in Fig. 8. Forconvenience, we denole s~'mbolicall~'1heinlerval (l-a,O)b~' £=0. and the inlel",ls 10,xo) and (xo.l) b)' £= I and£=2. respectively. De:noting the inverses of the maps. re­Slricted 10 these inlervals b)' Fo, F l , and F 2• respecti\'e1y,it is easy to check that points in 0 and I havc inversesonly in 2, whereas, points in 2 hlye in\'erses onl)' in 0 and

16.1)

In tbis chapter ....·e examine tbe fonnalism developedabove in the conlext of sets thai cannot be organized oncomplete trees. For concreteness .....e draw our examplesfrom tbe dynamics of maps of the interval tbat aize notSma.ie-complete. For one-hump maps, ODe knows tbatby ordering the itineraries (symbolic dynamicsl of all thepoints in a proper way, the itinerary of the critical pointis the largest allowed one. Thus there are fe ....·er than 2­allowed itineraries of It symbols and the binary tree onwhich we can organiu the preimages of the critical pointis severe!)' pruned.Jl

We shall examine a number of pruned uees. The firstcase ....ill ueat parameter \'alues for which the criticalpoint falls on an unstable periodic orbit IMisiurewiczpoints).}4 The second case will deal with situations inwhich the itinerary of the crilieal point is periodic (su­pe~table orbit) and the last case is the rather general sil­uation where the itinel'3l)' of the critical point is aperiod·ic.

To deal effectiYely wilh all these ca.sc:s, we have to re·tum 10 the: derivation of the eigenyalue equation, andwrite a slightly more general equBlion that would alloweffective usc: of the: rules for pruning.

Conside:r again Eq. (2.11). Choosing x =F'I(r) ~e: get

}.(fJh!"lBI(F" U))= L IF~(F" U 11IIBI1,~(BI(F,IF'1U))) .,

\'1. ~CO~tPLETE TREES

The rules for pruning will be given as allo .....ed sequencesHI in this equation. It will be see-n that all the examplestreated here can be brought to such a form, allo .....ing E totake on a sufficient number of values O. I •... , k - I.

Finally, it is worth noting that a sped&! family of inter­minent maps, those characteriz.ed by a cusp [recursion13.11 ....ith : =! is an uample), might possess nonnaliz­abk invariant densities.~1 This family exhibits a first or­der phase tranSilion l• at fJ"'" I.

(S.17)

IS.16)

(S.ISI

~I

SCALING PROPERTIES OF MULTIFRACTALS AS AN.

GIIlI

o.,

Gln(-Gl-kll-fJ) .

:=t-" ,

It is worth noting that IS.l6) appears numerically as afirst-order transition since

z- . Olll+'"(x +nl16.. l

(5.14)

FlG.7. A ,ketch of the thennod)T1amics of the intermitteDtmap (S.II. "'t\be half intelel'l (iDdieatcd poinu.I ....e pouc:s$ an­al)"tic K11utiDI1$.

it is straightforward to yerif)' that the singularity in Tj inthe first tenn is TjInTj. so that with Tj= -G, and for someconsUnt k.

where the ellipsis represents less singular terms, llI"itbOt 1J linear in f=2( I-pI.

The: imponant obscrnlion about 15.131 is that with1.l.{ 1- 1/1J: +n I)-cO) for large" the radius of con ...cr­genet of the series is z = I. Thus, (or all 8> I:=I_i.=t_G({JI=O, 50 Ihat after reaching J.=I atfJ= I, G remains pinned at Gil )=0, and G'IPl=O for8> I (sec: Fig. 11. It follows that there is a phase transi·tion (b~eakdoQ,"Tl in analyticit)' of G in fJl al fJ= 1. Mostimponantly, the transition occurs althe radius ofconnr­gc:ne.c \;nually guuuleeing an infinite: order transition.Tb~ phenomenon can be tracked back exactly 10 the ·'in·termineDc)'" of Fo at x =()(FoIOl=t). Since z =c GtlJl ,ezldfJ )ie1ch G'ID). If we: attempt 10 differentiate {S.14}~ B at 8=1, tbe fLnt lenn In.za-Id:ldfJ) becomes 10ga-

mica1ly diyergml. while the second (the remainder) isfinite (differentiable). Thus, the singular part of tS.14) atthe transilioll is tbe explicit first lenn. Setting x = I in(5.141 and substituting

G"PI-IIDI,k_PJI

so that smooth extrapolation from fJ < 1-( yields a finiteG'( 1-) from the slowness of ,'ariation of the logarithmand its nonanalyticity. With the presence of logarithms,

e1!ess the asymptotic fonn is known, all extrapolation isnsky and misleading.

, .,12 SCAuNG PROPERTIES OF MULTlFRACfAl..S AS AN ... 5369

-1.s37Q FEIGENBAUM, PROCACClA, A."ID TEL

(AS)

IA4)

IA7)

IA61

(All)

(A131

lAl2I

Vill(h (x'nlh'(x')I"

If'lx'H'

with 1; being a monotonous and differentiable function,with h (O}=O, h (1)= I. The invariant densities of f andg are related hy

VJTlx}=,,~~llohlxllh'lxll . (A21

There exiSIS a special conjugation lilx) between fix 1and ](x) sucb that Ihe invariant density of ]Ix) is theconstant function. This conjugating function Ii(x) is ob­tained as

Glx}= f~ "'/'Ixld.x (A31•It has been shown that ](x) is everywhere expanding (hy.perbolic) 9I;th a muimum at which tbe derivative of ](x)is discontinuous (tent-maplike maximum).»

Consider 0091' the eigenvalue equation for two conju.gale functions! and g.

"_ ~,!\x))./~,!'YI- :}; 1/'1 )1' '

~Ef 11,.1 x

161 _ t!I-!'(x)A,!?, (y)- J; Ig'(x)I'"

~e, lerl

We shall show that the following statement is true:

Idh(xl I'¢11(xl=¢!flohlxl~

Af="A, .

"A,l/;!fl(yl= 1:.!~'Ie,-llrl

!,(x)=(h-l)'ogoh(x)](g'oh(x)h'lxl) , (ASI

Ih-l)'ogoh(x) I {A9}h'oh logoh(xl

Perform now the change of variables x """h (x') in Eq.(AS),

To s.ec this use the identities

l/!!fl(h (x'))

18'lh (x'IlI'

y =goh{x')=hof(x'). (A 10)

Multiplying now the oumerator and denominator hyI(h -l)'ogoh (x')h'(x')IP and using (A9) we find

A, ",~'I(y)Jh'o !(x')jP

This equation is then satisfied hy t#J1' where

andA,=Af ·

In panicular, using tbe special conjugation 1i Eq, {A3}, wefind(A))

e.l$

•2o-2

§..,N

Two functions on [0,11, f and g are conjugate if

f(xl=h -logohlxl,

The introduction of an eigen\'alue equation for the cal­c.ulation of tbe thermodynamic functions of multifractalsets has led 10 se\'era] results.

Ii) An efficient melhod of computation via re(:ursivefunction calls is available.

liil The nature of first-order phase transitions in thethermodynamic formalism has been clarified as a switchbetween Ibe discrete and continuous pans of Ihe spectraof the linear operalor L.

(iii) The phase transition of Ihe intermittent map,which is in the same class as Ihe Farey model, the sub­critical mode locking of coupled nonlinear oscillators,and a certain one-dimensional Fermi gas with logarith­mic interactions, could be fully understood.. The transi­tion was shown to be of infinite order

(iv) The thermodynamics of multifractal sets which lieon incomplete trees can be efficiently computed once thepruning rules are given, The usefulness of this approachfor other problems in nonlinear physics where muilifrac­tals appear will be tested in Ihe future. Preliminary usesin the conlnt of fractal aggreg.to in two dimensions in­dicate that the approach is efficient and worthwhile.60

VIL SUMMARY

FlG. 12. TbermodYTl.1.mics at lupcrsuble orbits or lengtb 3,.s, 8, alld 13. The t,.·o Il5t ones arc indislinguisbable on Ibegrapb. The parameters used in x'-I-cu:: are o""I.7S4U,1.62HI, 1.711 08, and 1.70770, respectiVely.

ACKNQWLEDGMEJI,,-n;

One of us (I.P.I acknowledges the partial support of theU.S.-Israel Binational Science Foundation, the IsraelAcademy of Sciences-Commission for Basic Research,and the Minerva Foundation, Munich, West Germany.T_ T. is grateful 10 A. Csordas and P_ Supfalusy for in­forming him about results of their paper prior 10 publica.lion. Valuable discussions 9I;th A. Csord&5, P. Sup­falusy. D. Auerbach, Z. Kovacs and R. Zeital.: are ac­knowledged.

APPE!"r>'DlX A.: CONJUGATION

2

-,j'-·.,1

:'l'j-'1',~,;

;.,,~''!i:··.D·~~

"."i,

(6,15)

~.----,---,--,-

FlG. II. Same uRI. 10, for supcnuble period.s.

wilh FI=F]=F,=F, Fo=-F.A similar analvsis can be done for superstahle orbits

8.U, _.. .F", al~ays yielding F" -I coupled equations.The point is that GID) appears: to converge nicely to alimit, s.ec Fig. 12. We should slress tbat in the limit weobtain an infinity of coupled equations; it is our hope thaisuch an infinite set can be reprc:sc:nted by a single, non­linear e:igen\'alue equ:uion. Work to find such nonlinearequations is in progress, and wiD be reponed elsewhere.

C. Apc.riocLie ItiDtrariti

When the itinerary of the critical point is aperiodic, ~t

is not easy to write down a closed form theory. ThIsproblem can be circumvented by caleulating the t~erm(}­

dynamic functions at a series of parameters for whIch theitinerary is periodic, a series that converges to the puam­eter at which the itinerary is aperiodic. As an exampleconsider the parameter value for which the itinerary ofthe critical point is the same &5 the itinerary of a gOlde~;

mean rotation, properly reordered for ooe-hump maps.As is well kno9>'D, the golden mean is the limit of ratios ofFibonacci numbers F~ IF~ -+ I_ Thus a strategy suggestsit..self: we can look at the thermodynamic functions at pa­rameter values corresponding to superstable orbits oflength J,5,8,13, .. " and these should converge to thethermodynamic functions of the map with the aboveaperiodic itinerary. The period 3 case was discussed inSec. VI B, The period 5 problem is sketched in Fig. II.It is easy to verify that the following equations are ob­tained:

A({Jl¢lfl(X)= IF;(Fo(x))I"¢~J(Fo(xli ,

A(/3)¢\"I(x)= IFi:(FI(X)II"l,!J~"I{FI(XJ) ,

A(/3)~(X)= IFa(F21x)>1"VJlfl(F2(x I)

+ IF;(F:lx))I'~"I(F]lx)) ,

"A(/3lr~l(x) = IFa{F,Ix lIj"VJIf'(F,Ix))

+ IF~(Fl(X))I'yrtl"I(Fl(xll ,

/6.91

/6.14)

(6.10)

16.l2}

GlIl=a,

dG{P} I -AdD I-I '

GWol=O _

!,!JIBI(y) _ (I_y)-I,m"-,, _I I_y)-om",-,

where again

I j,n

I-xFl{x)= -Q- ""-Folx).

Thes.: equations can be solved using similar Icchniquc:s to&11 prc\;ous equations. ..

It is interesting to Dotice that the numenca1 solutionfor i .• of ().71 converges in an oscillatory fashion, \l>itb aperiod 3. The reason is obvious. and stems from the factthat arbitrary initial conditions arc~ "imbalanced" withrespect to the invariant measure. A simply convergentseries can be obtaiDed by considering the II.ven.ges

These quantities converge well and approacb }.t81.The convergence can be. sped up again by star1ing with

a singular function. 11 is easy to sec from Eq. (3.4) Ihatthe required singularity is now DOl tbe "natural" one

_=PI2.sinccror Y-J ","chive

~!'II~n¢~~I()'1 = I 1M,_1 .!...=.l.

Q

Ifwe start with ~J:l_J:-g, we get upon iteration

~BJ(y) _ (I_yl- .. n:-... (6.11J,-,However I after three funher steps this singularity istransferred to x =0, and, therefore,

The ditrerent behavior for x --0 from above and frombelow stems from the fact that for y <0 t.bere is only oneprei.:nage, whereas for y > 0 there are two. This corre­sponds to an initial condition r/{xl=t<F;(xl), i = 1,2 inEqs. (6.7) and (6.8).

Upon iterations, starting with tbis initial function, fastcon...ergence is observed. We find quantitative agre(:ment",--jth previous calculations" of the C$C8pe nile from therepeller a, the Lyapunov exponent restricted to the re­peller A and its fractal dimension Do. These quantitiesare calculated froml~

In the limit the singularity becomes 20 =p. Thus thesingular funetion that is ehosen to speed up convergence

• ,g,

jx- B forx>O?t!x)=[[x-{)-al)(I-xl!-6X 1 forx<O. (6.13)

;;\

(BI~

mI.)

Noti~ that Fo has derivative + I at its fued point x ==0,so that IF~(xll < I is marginally violated {Le., a multipij.Cative scaling converging to I at an end pointl. Indeed,(B I S) is tbe find point of the "intermittency reoorm.a.1i.z.a­tion group". so that up to our proviso on measures. weare, in this model, also working out the tbermod)"tIamicaof intermittency (tbe precise form of the "reinjection"given by F I I is inessential).

Each branch is a hyperbole with

of smallest denominator (and hence largest mode-1ockin&intc.rvalll)'ing between t.....o gi\'en rationals numben. I.Ddhas bttn discuued e1sewhere!I.IS The connection to W.terminency. and the solution of the model rcst upon tiltFs of (B3). which we no..... work out.

Denote [c I' ... ,Cl] by x in (B3). It follows from CD2)thai

xFolx)= I+x •

FI(x)=_I_=l-Fo(xl.

,+X

1'1'. Gl"IUbergcr. R. Ba.dii, and A. Politi. J. SLat. Pb)'$. 51. I))119gll.

20M. J. Feigenbaum lunpublWlccl); T. Bohr and M. H. JelaeD,Pbys. Rev. A 36. 4904 U91n.

21M. J. Feigenbaum Complo. Objt'Cu all Rtgulor TrTtS,Proceedings of tbe 19&7 NATO Summer Scbool (Pleoum,New York. 19871.

Up. Collet and J. P. Eck.Il:u.n.n.ltU'tJttd Mops 011 the lll/trwsl (lJ

DyIlQmleol S}'J/t/flJ CBirkbauscr. Ba.sc:1. 19801•IIp. Supfalusy and' T. Tel. Pbys. Rev. A 34. 2520 (986); T. Tel,

Pbys.. 411. A 119. 65 (1986); T. Tel. Pbys. Rev. A 36, 15020987l.

2"T. Tel, Pb)'S. Rev. A 36, 250711987).:.sp. Gl"IUbcrger and 1. Pro.:accia, Phys. Rev. A 2:8, 2591119831.2'0. Gyore; and P. Szepfalusy. J. Stat. Pbys. 34, 451119S4).17.... Csorw and P. Szipfalusy, Pbys. Rev. A 38, 2582 \191&1.21Q. Gyorgyi and P. Szipralusy. z. Pbys. B 55, 179119S4).ltf. Szipfalusy and T. Til, Phys.. Rev. A 35, 1,77 09871.»r. Bohr &.lid T. Ttl, in Dim:ti(l1lS III Ch=, edited by U. Bai·

CLin Hao (World Scienti6c:, Singapore, 1988), Vol. II.

lip. Manneville aDd Y. Pomeau, Pbysica D 1. 2!91198O).nJ. E. Hinch. M. NallCtlbers;, and D. J. Sc:alapino, Pb)". Len­

S1A, 391 tl982).3lp. Cvit.lnovie. G. H: GUUl'Ilnc:, and 1. Proc.accia, Pb)'$. Rev.

A 38, 1503119881.:WM. Misiurewic~ Pub\. Matb. DiES 53. 17(1981).nD. Ruelle, ErSod. Th. Dynam. Syst.l, 99 (1982).36M. Widom. D. BerWmon, L P. Kadanoll'. and S. J. Sbenker,J.

SLIt. Pbys. 32. 443 09131.31L P. Kadanof[" and C. TlnS, Proc. Nltl Acad. Sci. U.s.A-II.

1276(19&41.31H. Kanu and P. Grusberser, Physics D 17, 75 (19851."1. Procaccia. S. Thomae. and C. Tresser, Pbys. Rev. A 35,

1880411987).0):. Proc.accia and R. Zeiu1. Pb)'$. Rev. Lett. 60, 2511 (19881.~IM. H. JelUCn. P. Bak. aDd T. Bobr. Pb)'$. Rev. Lett. SO, 16Ji

119831: Pbys.. Rev. A 30. 1960 119841.

(BI3)

FEIGENBAUM. PROCACCIA. AND TEL

Since by (88) the c;'s are the distances bet .....un adjacentpanicles. we see that Ibe Farcy model at level II is alai­ti~ gas of varying particle number (second quantized) ona lattice of length n with long range logarithmic interac­tion SIllUrating at the nearest particle. NOli~ that thesmallest value of H ~ is obtained with C I =n -H" _ 2Inn.where. for all Ci= I, H. -211 Inp-I [.....here p-I=IVS + I )12] possesses the largest value of H•. Thus, ifthe model possesses a phase transition (with II-cc>, ofcourse), its nature is that there is 0 density below a criti­cal temperature, and finite densit)· above it, with each siteoccupied (densit)· Il as T _ -0 18- - ac:). Thus, themodel is reminiscent of a second-quantized Fermi gas(repulsive, with occupation number per site no more thanone). Indud. it is shown in Sec. V that there is a phasetransition (ar fJ= Il of infinite order in Ihis model.

The connection of this model to mode locking is aconsequence of Farcy addition fe.g .• tet=( I +2)/13+S)=t, look at Fig. 131 which determines that rational

5372

By {BI01, .....e now have a5ymplotieally that

H~t[c,•...• cl. 2)1-2 i Inc,+O 1-1-1i-I ciCi

'Permanent addrcu: Institute of Theoretical Physics, WIVQS

University. H·IM8 Budapesl VlII. Hun£U}'.1M.. J. Feigenbaum, Commun. Muh. Pbys..1I9801.1M. G. E. Henuebel aDd I. Procaccia. Pbysica D 8, 435 119831.'u. FriKh and G. Parisi, io Turbulellu olld Pndiclobl/iry ill

Gt:OphysiCQl Fluid DYllomics olld C/imo/t DYllomics, cclited hyM. Ghil, R. Benzi. and G. Puisi (Nonh·Holland, Amsterdam19851.

.,.. C. Halsey. M. H. JalKn. L P. Kld&nnlf. I. Procaccia, aDdB. t. Scbn.iman. Pb)'s.. Rev. A 33, 1141 (1986).

'G. H. GUlW"Itne and I. Procaccia. Phys. Rev. Lett. 511, 1377(987).

6R. Bcnzi. G. Paladin, G. Parisi, and A. Vulpiani, J. Phys. A 17,3S2119&41.

IT. C. Halsc:y. P. Meakin. and I. Procaecia, Pbys.. Re\·. Lett.. 56.8S411986l.

Ie, Amitrano, A. Coniglio. and F. diLibcno, Pbys.. Rev. Leu.57. 1016(1986).

'A. Coniglio. Pbysica A 104, 51 11986l.10J. P. Eckmann and D. Rudie. Rev. Mod. Phys. 67. 617 098S}.11M. J. Feigenbaum. J. Stat. Ph)'$. 46, 919 (1987l; 46, 925 (1987).1lM.. J. Feigenbaum, M. H. Jensen. and I. Procacci&, Phys. Rev.

Lett. 56, 1503119861:13M. H. Jensen. L P. Kadanof[", and I. Procaccia, Pbys.. Rev. A

/ 36. 1409(1987).V(~D' Ruelle. T'humodynomic Formalism (Addison·Wesley.

Reading. 19781; R. Bowen, Equilibrium Stolts ond the Er.godic Thror>, of AIlOJOO Dijfromorphisnu, Vol. 470 of lAcrunNOlts ill MOlhemolics ISpringer. Nt"" York. 1975l. p. I; Y..Sinai. Ruu. Matb. SUO'. 166,21119721.

V ~. Bohr and D. Rand. Ph)'sica 0 2!:, 387 (19871."P. Cvit.lnovie, in P1'ot:ndiFigs of rhe XV }lIltTntJliOIlOI (:QUo­

t{Ilim 011 Group Thr-owicol Mt/hods ill Physics., cdiled by R.Gilmore (World Scientifie, Singapore. 19871.

"D. Katun and I. Procaecia. Pb)~ Rev. Leu. S1.11907}.ltp. Supfalusy. T. Tel. A. Csordu. and Z. KO'"lcs. Pbys. Rev.

A 36. 3525 119871.

•

:.

(BI2)

1891

(Bill

(B1)

186}

{BS}

'B3)

5311

I

so thai

The thermodynamic sum (canonical partition sum) of the

model lit level n is

(C,. __ . ,cl.2) and [c l •· ... ct+2l,and that the interval between them is oflcngth

1J1~1({CI"" .cl,2]1l

{2'Cl.···. C ll2

I ]2· .. {C C ]1{c]2.t {2.Cl •... ,cd Ct.··· .c I 2' \ 1

IBI01

il{cl ••.• ,cd)

~o ...0IQ......Jl10 ...01 ... }Q......JlIQ......Jl ......-;=r- <1_'-' ;;:-;=1 <:-1 ',-'

(B8)

It is natural to regard the binary string of. (B8) .as aconfiguration of k -1 gas panicles on a one-dimensl?nallattice of length n. It is of variable n~mber. s~ncek=I•...• n+t. (for k=n+l there IS a UOlqueconfiguration of all I's corresponding to (I, I•.•.• 1,2]with n I·s). It can be shown that a pair of rationalsdefining an interval. as depicted on Fig. 13, are of the

ro~

it follows that

xl.IIl"••... l"\)=F.\DF.:D "'DF•• lx"l.

where x"={2]. Equations (54) and (BS) now determine

the binary index o([c\, ... ,cd as

and ordered by the rule

F<(;x:·II=x~~!1 ,

Denoting a rational at level n by its binary index written.,

The tree of Fig. 13 is thus a complete enumeration of the

rationals.Let us construct the (n + Ilth set from the nth by the

follo9ting pair of rules for Fo and F I' respectivd)':

(c i •...• Cl)- [c l + I,Cl' ... ,Cl] •

[c I.···'Cl]-[I,CI'···.Cl}·

Since the sum of the c;'s is incremented .bY 1 by each F,by (BI) e3ch image is in the next level; S\flce CI ~ I. eachrational of levd II prodUces twO distinct nttion.als al.level11 + I. Thus, starting 9t;th [2] (level 01. any ratlonalls o~

tained as

< -, rl-I F'l-I-IF F<,-lI21(c

l.... • cd=Fo

1 F,Fo Fl'" 0 I 0 •

(B4)

(B2)

(Bil

with

<.

<,

,.,

SCAUNG PROPERTIES OF MULTIFRACTALS AS AN.

"

[CI.··· .cd +[ 1 .C, Cl'''' .Cl

_/l~i ~

1\ /'\!\l\ 1\ 1\:'k-~ ;-l ;-; f-5 n"~

FIG. 13. The Flrey model. The numben on the fiSure arethe J: vllues obtained vii Eqs. (B7) Ind (BI4). Ind the COrTe·spondin& Idd~ (f"(._h··, ,(I)' Note. tbl! In J:-<lrdtredrei)fUetll.ltion of tbe tree is shov.'ll bere Ind tbis is ~h~ tbe K'

quctr...c of the f/S is re ned, d. Fil. I. The arro-:, mdlClte thebalb UICd in the CO\'ef1l e (see (BIOllnd \2.411 It dlf["trenl stages

of Ihe construction.

•2:CI=O +2,,

and each pair encountered having. ?ne dementCt =2, the other with Cl > 2. By definltlon,

I

.-. ."

APPENDIX B: nu: FAREY MODELA...."I) RELATIO!"i TO I!'IoI'ER.o\.iITT'ESC'f

The Farey mood (also a subcritical treatment ofmode-locking of coupled nonlinear osci1lalo.rs;~1.~6 alsointermittency) is I binary tree thermodynamIcs "':Ib tbeIItb sct of intervals the distances between tbe neIJ~~rs

• icted in Fig. 13. Thus, al level II = 2 thuc are 2 In­

Wvals of respective lengths t-t and f-j· Readingfrom left 10 right at level n, tbe rationals enoountered arein increasing order the 2- continued fractions [c I' ... ,cllof variable length k with Cl ~ 2 (for uniqueness of repre­

senlAtion) satisfying

I d~'X} I'ilrIX)=¢Y1.fIX) l-a;-=Vr1lo,lj(';{111Jl'jll(';(lj6. (AI4)

For b}'pcrbolic systems ¢11 is smooth, and. tber~rore.

\'I!'(xl hu Ungularilies which arc indu~d b)' the smgu·larilies of tb~l alone (see also Ref. 27). Since l!J}1 has thesame singularities on both ends of strength I-I (1., weconclude: thai tbe eigenfunctions belonging 10 the discretespectrum belong to class A.

•

..

j