dynamics of one-dimensional maps

Dynamies of One-Dimensional Maps

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL

Centrefor Mathematics antI Computer Science, Amsterdam, The NetherlantIs

Volume 407

Dynamics of One-Dimensional Maps

by

A.N. Sharkovsky S.F. Kolyada A.G. Sivak and v. v. Fedorenko Institute o[ Mathematics. Ukrainian Academy o[ Sciences. Kiev. Ukraine

Springer-Science+Business Media, B.Y.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-4846-2 ISBN 978-94-015-8897-3 (eBook) DOI 10.1007/978-94-015-8897-3

This is a completely revised and updated translation of the original Russian work of the same title, published by Naukova Dumka, Kiev, 1989. Translated by A.G. Sivak, P. Malyshev and D. Malyshev

Printed on acid-free paper

All Rights Reserved @1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997. Sot1:cover reprint of the hardcover 1 st edition 1997

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

conTEnTS

Introduction vii

1. Fundamental Concepts of the Theory of Dynamical Systems. Typical Examples and Some Results 1

1.1. Trajectories of One-Dimensional Dynamical Systems 1 1.2. ffi-Limit and Statistically Limit Sets. Attractors and Quasiattractors 18 1.3. Return of Points and Sets 25

2. Elements of Symbolic Dynamics

2.1. Concepts of Symbolic Dynamics 2.2. Dynamical Coordinates and the Kneading Invariant

2.3. Periodic Points, 1;-Function, and Topological Entropy 2.4. Kneading Invariant and Dynamics of Maps

3. Coexistence of Periodic Trajectories

3.1. Coexistence of Periods of Periodic Trajectories 3.2. Types of Periodic Trajectories

4. Simple Dynamical Systems

4.1. Maps without Periodic Points 4.2. Simple Invariant Sets 4.3. Separation of All Maps into Simple and Complicated 4.4. Return for Simple Maps 4.5. Classification of Simple Maps According to the Types of Return 4.6. Properties of Individual Classes

v

35

35 40

44 49

55

55 64

69

69 74 78 86

100 107

vi Contents

5. Topological Dynamics of Unimodal Maps 117

5.1. Phase Diagrams of Unimodal Maps 5.2. Limit Behavior of Trajectories 5.3. Maps with Negative Schwarzian 5.4. Maps with Nondegenerate Critical Point

117 124 137 150

6. Metrie Aspeets of Dynamics 161

6.1. Measure ofthe Set ofLyapunov Stable Trajectories 161 6.2. Conditions far the Existence of Absolutely Continuous Invariant

Measures 165 6.3. Measure of Repellers and Attractors 170

7. Loeal Stability of Invariant Sets. Struetural Stability of Unimodal Maps 183

7.1. Stability of Simple Invariant Sets 183

7.1.1. Stability of Periodic Trajectories 183 7.1.2. Stability of Cycles of Intervals 187

7.2. Stability ofthe Phase Diagram 190

7.2.1. Classification of Cycles of Intervals and Their Coexistence 190 7.2.2. Conditions far the Preservation of Central Vertices 194

7.3. Structural Stability and Q-stability of Maps 196

8. One-Parameter Families ofUnimodal Maps 201

8.1. Bifurcations of Simple Invariant Sets 201 8.2. Properties of the Set of Bifurcation Values. Monotonicity Theorems 205 8.3. Sequence of Period Doubling Bifurcations 207 8.4. Rate of Period Doubling Bifurcations 216 8.5. Universal Properties of One-Parameter Families 223

Referenees 239

Subject Index 259

Notation 261

In TRODUCTIon

Last decades are marked by the appearance of a permanently increasing number of scientific and engineering problems connected with the investigation of nonlinear processes and phenomena. It is now dear that nonlinear processes are not exceptional; on the contrary, they can be regarded as a typical mode of existence of matter. At the same time, independently of their nature, these processes are often characterized by similar intrinsic mechanisms and admit universal approaches to their description.

As a result, we observe fundamental changes in the methods and tools used for mathematical simulation. Today, parallel with well-known methods studied in textbooks and special monographs for many years, mathematical simulation often employs the results of nonlinear dynarnics-a new rapidly developing field of natural sciences whose mathematical apparatus is based on the theory of dynamical systems.

The extensive development of nonlinear dynamics observed nowadays is explained not only by increasing practical needs but also by new possibilities in the analysis of a great variety of nonlinear models discovered for last 20 years. In this connection, a decisive role was played by simple nonlinear systems, discovered by physicists and mathematicians, which, on the one hand, are characterized by quite complicated dynamics but, on the other hand, admit fairly complete qualitative analysis. The analysis of these systems (both qualitative and numerical) revealed many common regularities and essential features of nonlinearity that should be kept in mind both in constructing new nonlinear mathematical models and in analyzing these models. Among these features, one should, first of all, mention stochastization and the emergence of structures (the relevant branches of science are called the theory of strange attractors and synergetics, respectively).

The theory of one-dimensional dynamical systems is one of the most efficient tools of nonlinear dynamics because, on the one hand, one-dimensional systems can be described fairly completely and, on the other hand, they exhibit all basic complicated nonlinear effects. The investigations in the theory of one-dimensional dynamical systems gave absolutely new results in the theory of difference equations, difference-differential equations, and some dasses of differential equations. Thus, significant successes were attained in constructing new types of solutions, which can be efficiently used in simulating the processes of emergence of ordered coherent structures, the phenomenon of intermittence, and self-stochastic modes. Significant achievements in this field led to the appearance of a new direction in the mathematical theory of turbulence based on the use of

vii

viii lntroduction

nonlinear difference equations and other equations (c1ose to nonlinear difference equations) as mathematical tools.

It is c1ear that iterations of continuous maps of an interval into itself are very simple dynamical systems. It may seem that the use of one-dimensional dynamical systems substantially restricts our possibilities and the natural ordering of points in the real line may result in the absence of some types of dynamical behavior in one-dimensional systems. However, it is weIl known that even quadratic maps from the family x ~ x 2 + A. may have infinitely many periodic points for some values of the parameter A.. Furthermore, for A. = - 2, the map possesses an invariant measure absolutely continuous with respect to the Lebesgue measure, i.e., for this map, "stochastic" behavior is a typical behavior of bounded sequences of iterations. ActuaIly, the trajectories of one-dimensional maps exhibit an extremely rich picture of dynamical behavior characterized, on the one hand, by stable fixed points and periodic orbits and, on the other hand, by modes which are practically indistinguishable from random processes being, at the same time, absolutely deterministic.

This book has two principal goals: First, we try to make the reader acquainted with the fundamentals of the theory of one-dimensional dynamical systems. We study, as a rule, very simple nonlinear maps with a single point of extremum. Maps of this sort are usually called unimodal. It turns out that unimodality imposes practically no restrictions on the dynamical behavior.

The second goal is to equip the reader with a more or less comprehensive outlook on the problems appearing in the theory of dynamical systems and describe the methods used for their solution in the case of one-dimensional maps.

To understand distinctive features of topological dynamics on an interval on a more profound level, the reader must not only study the formulations of the results but also carefully analyze their proofs. Unfortunately, the size of the book is limited and, therefore, some theorems are presented without proofs.

This book does not contain special historical notes; only basic facts given in the form of theorems contain references to their authors. Almost all results are achievements of the last 20-30 years. The interest to the qualitative investigation of iterations of continuous and discontinuous functions of a real variable was growing since 1930 s when applied problems requiring the study of such iterations appeared. However, these investigations were not carried out systematically till 1970 s. The results of many authors worked at that time are now weIl known. We would like to mention here less known works of Barna [1], Leonov [1-3], and Pulkin [1, 2], which also contain many important results.

In Chapter 1, following Sharkovsky, Maistrenko, and Romanenko [2], we give an elementary introduction to the theory of one-dimensional maps. This chapter contains an exposition of basic concepts of the theory of dynamical systems and numerous examples illustrating various situations encountered in the investigation of one-dimensional maps.

Chapter 2 deals with the methods of symbolic dynamics. In particular, it contains a presentation of the basic concepts and results of the theory of kneading invariants for unimodal maps.

In Chapters 3 and 4, we prove theorems on coexistence of periodic trajectories. The

Introduction ix

maps whose topological entropy is equal to zero (i.e., maps that have only cyeles of periods 1,2,22, ... ) are studied in detail and elassified.

Various topological aspects of the dynamics of unimodal maps are studied in Chapter 5. We analyze the distinctive features of the limiting behavior of trajectories of smooth maps. In particular, for some elasses of smooth maps, we establish theorems on the number of sinks and study the problem of existence of wandering intervals.

In Chapter 6, for a broad elass of maps, we prove that almost all points (with respect to the Lebesgue measure) are attracted by the same sink. Our attention is mainly focused on the problem of existence of an invariant measure absolutely continuous with respect to the Lebesgue measure. We also study the problem of Lyapunov stability of dynamical systems and determine the measures of repelling and attracting invariant sets.

The problem of stability of separate trajectories under perturbations of maps and the problem of structural stability of dynamical systems as a whole are discussed in Chapter 7.

In Chapter 8, we study one-parameter families of maps. We analyze bifurcations of periodic trajectories and properties of the set of bifurcation values of the parameter, ineluding universal properties such as Feigenbaum universality.

Unfortunately, in the present book, we do not consider the maps of a cirele onto itself and the maps of the complex plane onto itself. Some results established for maps of an interval onto itself are related to the dynamics of rational endomorphisms of the Riemann sphere: The beauty of the dynamics of the considered maps ofthe real line onto itself

from the family x ~ x 2 + A, A E lR, becomes visible (in the direct meaning of this

word) if we pass to the farnily z ~ Z2 + A, where z is a complex variable and A is a complex parameter (see Peitgen and Richter [1]).

We hope that our book will be useful for everybody who is interested in nonlinear dynarnics.

1. FunDamEnTaL conCEPTS OF THE THEORY OF DynamICRL SYSTEms.

TYPICaL ExamPLES ROD somE RESUL TS

Dynarnical systems are usually understood as one-parameter groups (or semigroups) P of maps of aspace X into itself (this space is either topological or metric). If t belongs

to 1R or 1R +, then a dynarnical system is sometimes called a flow and if t belongs to

~ or ~+, then this dynamical system is called a cascade. These names are connected with the fact that, under the action of f, the points of X "begin to move" (x H f~x)),

and the space "splits" into the trajectories of this motion. A pair (X, f), where f is a mapping of the space X into itself, defines a dynamical

system with discrete time, i.e., a sernigroup of maps {f', n E ~+}, where f' = f 0 f'1-!, n = 1, 2, ... , and jO is the identity map. If the space X is the realline 1R or an interval I C 1R, then this dynamical system with one-dimensional phase space and discrete time is, in a certain sense, the simplest one; nevertheless, in many cases, it is characterized by very complicated dynamics. In some aspects, e.g., from the viewpoint of the descriptive theory of sets, one-dimensional dynarnical systems can be as complicated as dynamical system on arbitrary compact sets.

1. Trajectories of One-Dimensional Dynamical Systems

The main object of the theory of dynamical systems is a trajectory or an orbit (in what follows, we use both these terms). The set

~

orb(x) = {x, fex), f2(x), ... } = U fnex) n=O

is called the trajectory of a dynarnical system (X,f) passing through a point x E X (it is sometimes convenient to regard a trajectory as a sequence of points x, f(x), j2(x), ...

but not as a set because this point of view is closer to the concept of motion along the

trajectory governed by the map n H fn(x)). The trajectory passing through a point x

2 Fundamental Coneepts of the Theory of Dynamieal Systems Chapter 1

is denoted either by the symbol orb (x) or by orb/(x). In most cases, it is necessary to c1arify the behavior of a trajectory (or a family of trajectories) on a bounded or unbounded time interval. In what terms and in what form one can answer this or similar questions?

In the theory of dynamical systems, the asymptotic behavior of trajectories is usually characterized by OJ-lirnit sets. A point x' E X is called an OJ-limit point of a trajectory

{x,J(x), ... , r(x), ... } if, for any n' > 0 and any neighborhood U of x', there exists

n" > n' such that frl'(x) E U (i.e., there exists a sequence ni < n2 < ... ~ 00 such that

f'l (x) ~ x'). The set of all OJ-lirnit point of the trajectory passing through the point x is denoted by OJ/(x) or simply by OJ(x). This set is c1osed. Moreover, if X is compact, it

is invariant and nonempty (if X is not a compact set, then it is possible that OJ (x) = 0, i.e., the trajectory eventually leaves X). Thus, if X is a compact set, then OJ (x) is the smallest c10sed set such that any its neighborhood contains all points of the trajectory

{rex)} beginning with some n (depending on the choice of a neighborhood). The most simple behavior is exhibited by periodic trajectories or cyc1es. A point

Xo E X is called a periodie point with period m if fm(xo) = Xo and fn(xo)::j:. Xo for

0< n < m. Each point xn = fn(xo), n = 1,2, .,. , m - 1, is also a periodic point with pe

riod m, and the points xo, xl' ... , xm-I form aperiodie trajeetory or a eycle with peri

od m. Periodic trajectories play an important role in the theory of dynamical systems. For one-dimensional dynarnical systems, they are of particular importance.

The OJ-lirnit sets of periodic trajectories coincide with these trajectories. Generally speaking, if the OJ-lirnit set of a trajectory is a cyc1e, then this trajectory is either periodic or asymptotieally periodie, i.e., it is attraeted by a periodie trajeetory.

There exists a simple graphic procedure for constructing trajectories of dynamical systems defined on an interval. This procedure can be employed, e.g., in studying the behavior of trajectories in the vicinity of a fixed point or a cyc1e.

Consider a mapping x H fex) defined on an interval land a point Xo E I. The procedure of graphie representation of the trajectory of the point Xo is called the Kö

nigs-Lamerey diagram and can be described as follows: In the plane (x, y), we draw the graphs ofthe functions y = fex) and y =x. The trajectory of the point Xo is represented by a broken line MINIM2N2M3N3 ... whose chains are parallel to the coordinate axes (see Fig. 1). The abscissae ofthe points MI' NI and M 2, N 2 and M 3, etc., are

the successive iterations of the point Xo equal to xo, Xl = f(xo), x2 = f(xI)' ... respectively. Theordinatesofthepoints MI and NI' M 2 and N 2, M 3 and N 3, etc., are equal to Xl = f(xo), x2 = f(xI)' x3 = f(X2), ... , respectively. Thus, to construct the broken line MINIM2N2M3N3 ... , one must start from the point Xo and successively move along its trajectory.

The fixed points of the map f are associated with the points of intersection of the

graphs of the functions y = f (x) and y = x. In Fig. 1, these are the points ßo and ßü· Moreover, the point ßo is repelling and the point ßü is attracting, since the trajectories of the points c10se to ßo recede from ßo, and the trajectories of points c10se to ßü approach this point.

Section 1 Trajectories o! One-Dimensional Dynamical Systems 3

!I

!J

x x

Fig.l Fig.2

The closed broken line M INIM2N2 ... , where Mn+1 = MI' corresponds to a cyc1e

of period n. In Fig. 2, we present an example of a c10sed broken line with n = 2. It cor

responds to a cycle ofperiod 2 that consists ofthe points ßI and ß2 such that !(ßI) = ß2 and !(ß2) = ßI' This cyc1e is attracting because broken lines close to the closed broken line corresponding to this cyc1e approach this line.

For the maps whose graphs are displayed in Fig. 1 and Fig. 2, the ro-limit set of every trajectory can be defined quite simply: Any trajectory is attracted either by a fixed point or by a cycle with period 2. If a map possesses a cyc1e with period greater than 2, then the behavior of trajectories near this cyc1e can be studied by using a computer. However, in many cases, both the Königs-Lamerey method and numerical simulation fail to detect any regularities in the behavior of the trajectories: Thus, one observes no convergence to fixed points or cycles; furthermore, the behavior of trajectories is completely different even if these trajectories correspond to initial points lying at very short distances from each other, etc. The reader can readily check this fact by analyzing (e.g., with a calculator) the trajectories of the maps

x H ft..(x) = Ax(l-x) (1)

for different values of the parameter A > O. It seems useful to choose Xo E (0, 1) and

successively consider the values A E {1.5; 2.9; 3.4; 3.57; 3.83; 4}.

The maps in family (1) are defined for x E IR. Moreover, !t...(0) = ft..(1) = 0 and

4 Fundamental Concepts of the Theory of Dynamical Systems Chapter 1

for A > O. Therefore, for A E (0, 4], the interval [0, 1] maps into itself. By using the Königs-Lamerey method, one ean easily show that, in this ease, the trajeetories of the points that do not belong to [0, 1] approach infinity. Consider trajeetories of the points from [0, 1]. We are now mainly interested in periodie points and the eycles formed by them.

A eycle B = {ß" ... , ßm} of a mapping f: I ~ I is ealled attracting if there exists

a neighborhood U of this eycle such that f (U) C U and

n r(U) = B. n;:;:O

In this ease, we have ü) (xo) = B far every point Xo E U and the trajeetory orb (xo) splits into m sequences convergent to the points ß,,· .. , ßm' respectively.

A cycle B is called repelling if there exists its neighborhood U such that any point

of the set U\B leaves U after a finite period of time, i.e., for any XE U\B, there

exists n = n(x) such that rex) ~ U. These definitions can also be used in the case of an arbitrary topological space. If f is differentiable, then one can use the following simple sufficient conditions that

enable one to distinguish between attracting and repelling eycles: It is necessary to eompute the quantity

which is called the multiplier of a eycle B. If I f..L(B ) I < 1, then B is an attraeting ey

cle and if I f..L(B ) I > 1, then B is a repelling eycle. For I f..L(B ) I = 1, the cycle B is called nonhyperbolic. In this case, it may be either attracting or repelling. One can also observe a more complicated behavior of trajectories in its neighborhood.

The examples presented below illustrate the changes in the behavior of trajectories of a map ft. .. from family (1) for various values of the parameter A. In these examples, we

write f instead of ft... wherever this does not lead to any ambiguities.

1. 0< A ~ 1. In this ease, the interval I = [0, 1] contains a single fixed point x = 0

and this point is attracting. Sinee fex) < x for XE 1\ {O}, we ean write

00 n rU\{O}) = {O}, n=O

i.e., for any point XE l\ {O}, we have fn ~ 0 as n ~ 00. Hence, every trajectory

arb (xo) is attracted by the fixed point x = 0 (Fig.3).

Seetion 1 Trajectories of One-Dimensional Dynamical Systems 5

.!J

Fig.3 Fig.4

2. 1 < A ::;; 3. For A> 1, the fixed point x = 0 becomes repelling (J/(O) > 1) and a

new fixed point ßI = l-lIA appears in the interval I (Fig.4). Since j'(x) = A(I-2x), the multiplier Jl(ßI) is equal to 2 - A and, therefore, the fixed point x = ßI is at

tracting for 1 < A < 3. For any point Xo E (0, 1), we have r(xo) ~ ßI as n ~ 00.

Note that Jl(ßI) > 0 for 1 < A < 2 and the trajectory orb (xo) monotonically ap

proaches ßI. For 2 < A < 3, we have Jl(ßI) < 0 and the trajectory orb (xo) approaches ß I oscillating about this point and taking, in turn, values greater and lower

than ßI. For A = 3, the fixed point x = ßl is still attracting although, in this case, I Jl(ßI) I = 1.

3. 3 < A ::;; 1 + {6. As the parameter A becomes greater than AI = 3, we observe the appearance of a new bifurcation, namely, the fixed point x = ßI becomes repelling

(I Jl(ßI) I > 1 for A > 3) and generates a new attracting cycIe with period 2. The changes in the behavior of the map f in the vicinity of the point x = ß I are displayed in

Fig. 5, where we present the graphs of the function y = f(J(x)) for the parameter A crossing the value AI = 3.

A cycle of period 2 (Fig. 6) is formed by the points

The values ßil) and ßi2) are defined as the roots of the equation j2(x) = x that differ

from the roots of the equation f(x) = x that defines the fixed points of f Thus, for ß~l)

and ß~2), we arrive at the equation A2 xl - A(A + l)x + (A + 1) = o. Since

we have

for 3 < 'A < 1 + {6 '" 3.449 .... For these 'A, the eyde { ß~l), ß~2)} is attraeting. In

deed, forany point XOE l\({0,1} U {f-n(ß1)};=o)' thetrajeetory orb(xo) isattrae

ted by the eyde { ß~I), ß~2)} so that the subsequenee {f2n(xo) 1; = 0 eonverges to one

point of this eyde and the subsequenee {f2n+1 (xo)}; = 0 eonverges to another point of

this eyde.

Fig.S Fig.6

We ean speeify the eharaeter of eonvergenee of a trajeetory to the eyde by using the

multiplier f.l( { ß~I), ß~2)}). As the parameter 'A inereases from 3 to 1 + {6, the mul

tiplier inereases from -1 to 1. Henee, for 3 < 'A < 1 + ß and f.l> 0, the subsequenees

{f 2n (xo)} and {f 2n+ I (xo)} are monotone beginning with eertain n. Furthermore, one

of them is inereasing, while the other one is deereasing (sinee r(x) < 0 for x = ß~I) and x = ß~2)). For 1 + ß < 'A < 1 + {6, we have f.l < 0 and the subsequenees

{f2n (xo)} and {f2n+1 (xo)} approach ß~I) and ß~2) oseillating about ß~I) and ß~2),

respeetively, so that the subsequenees {J4n (xo) }, { f4n+2 (xo)} , {f4n+ 1 (xo) }, and

{f4n+3 (xo)} are monotone.

4. 1 + {6 < A, < 3.569 .... As the parameter A, crosses the value A,2 < 1 + {6 '" 3.449 ... , we observe the appearanee of the next bifurcation: The eyde {ß~l), ß~2)}

beeomes repelling (for A, > 1 + {6, we have 1 Jl( {ß~l), ß~2)}) 1 > 1) and generates a

new attraeting eyde of period 4. This new eyde attraets all points of I exeept a eountable set of points

If the parameter A, inereases further, then, at A,3 '" 3.54, the eycle of period 4 also becomes repelling and generates an attraeting eyde ofperiod 8 (which attracts all points of the interval exeept eountably many points). The proeess of eonsecutive doubling of the

periods of attraeting eydes oeeurs as the parameter A, inereases to A, = A, * '" 3.569 ....

Fig.7

5. There exists a eonvenient graphie representation of the qualitative reeonstruetions of cydes oeeurring as the parameter A, inereases. It is ealled the bifureation diagram

(Fig. 7). The bifureation eurves of this diagram eorresponding to ß~l) and ß~2) diverge

as the branehes of a parabola aeeording to the forrnula for ß~l) and ß~2), namely,

At the same time, the fixed point ß 1 drifts slower: 1 ßl (A,) - ß 1 (A,l) 1 = 0(1 A, - A,l 1).

A similar picture is also observed in the neighborhood of the subsequent bifurcation va

lues A2' A3' .... As noted by Feigenbaum [3}, if we compute the values An with sufficiently high ac

curacy and construct the ratios

n = 1,2, ... ,

then On ~ 0 = 4.66920 ... as n ~ 00, i.e., the rate of appearance of cycles with doubled

periods (as n increases) is characterized by the constant O. There exists another con

stant a "" 2.502 ... that characterizes the sizes of emerging cycles. Let ß;n be the first

point to the right of x = 1/2 belonging to a cycle with period 2n (this point appears for

A> AJ and let ß~n = f2 n-

1 ( ß;n)' Then

2.502 ... as n ~ 00.

6. For any A< A, *, the dynarnical system given by the map x ~ Ax(1-X) has a relatively simple structure on I = [0, 1]. Each trajectory is asymptotically periodic. For any A, there exists a unique attracting cycle of period 2m (rn depends on A), which attracts all points of I except countably many points "pasted" to repelling cycles with periods 2i , i = 0, 1, ... , m - 1).

What happens for A ~ A *? In this case, dynamical systems have more complicated structure. In particular, for any A ~ A *, there are trajectories that are not attracted to any cycle and, therefore, the ())-lirnit sets of these trajectories are infinite. Here, we do not analyze all possible situations (map (1) is investigatcd in more details in what follows and, in particular, in Chapter 5). Let us now consider the dynamical system for the

following values of the parameter: A = A * "" 3.57 ... , 3.83, 4 and >4.

7. For A = A*, map (1) already possesses cycles with periods 2 i , i = 0,1,2, ... (all these cycles are repelling), but have no cycles with other periods. The set K =

(Per (f))' of lirniting points for the set of periodic points Per (f) is a nonempty nowhere dense perfect set, i.e., it is homeomorphic to the Cantor set. This set K does not contain

periodic points, i.e., K n Per (f) = 0. The dynamical system is minimal on K. Indeed, for any point x E K, the trajectory

orb (x) is dense in K, i.e., ())(x) = K. The set K contains the point x = 1/2 (and,

hence, K = ()) (1/2)). All points ofthe interval I, except the countable set

~

p = U ri(Per(f)), i~O

Section 1 Trajectories of One-Dimensional Dynamical Systems 9

are attracted by the set K. Indeed, if XE I\P, then w (x) = K. We discuss the proofs of these statements in Chapter 5.

Fig.8

8. A, = 3.83. As the parameter A, increases further, we observe the appearance of new cyc1es and, in particular, cyc1es whose periods are not equal to 2i , i = 0, 1, 2, ....

For A, = 3.83, the map already has cydes of all periods mE N. The cyde B3 of peri

od 3 formed by the points ß~I), ß~2), and ß~3) (Fig. 8) is attracting. In addition to the

attraeting eyde, there is a repelling eyde of period 3: {ß~I), ß~2), ß~3)} (points of these

eydes ean be eomputed as the roots of the following sixth-degree polynomial: (P(x)

x)/(f(x) -x». What points are attracted by the attracting eyde B3? Let 10 denote an open interval

whose ends are the preimages of the point ß~3), i.e., the points ß~2) and 1 - ß~2), 10 =

(1 - ß~2), ß~2)). By using a computer, one ean check that

(a) P(lo) C 10 (it suffiees to show that p(l 12) E 10);

(b) the interval 10 contains a single fixed point ß~2) of the map P, and this point is

attraeting; the map PlI has no eyc1es of period 2. o

Therefore, for any Xo E 10, we have f3n(xo) ~ ß~2) as n ~ 00, i.e., the point Xo is

attracted by the eyde B3 and the interval 10 belongs to the basin of attraetion of this eyde. It is dear that any trajeetory attracted by the eyde B3 also passes through the interval 10 . Henee, the set


consists of the points of I attracted by the cyc1e B3. The set P is open and dense in I

and mes P = mes I = 1 (see Theorem 6.3). Hence, B3 attracts almost all points of I.

!I .l=4

Fig.9

The set I\P consists of the points that are not attracted by the cyc1e B3. This is a perfect nowhere dense set, i.e., it is homeomorphic to the Cantor set. The fact that the

set I\P is perfect follows from the fact that any distinct (maximal) open intervals which

form f-i (10) have common ends neither for different nor for equal i;?: 0 (the same is true for the ends ofthe interval I (i.e., for the points 0 and 1». We also note that the points x such that

ffi(X)=P (I\P) n [f2 (1 /2)'/(1/2)]

and

are everywhere dense in the set I \P. This dynarnical system is studied in more details in Chapter 5. The problem of the appearance of sets homeomorphic to the Cantor set is

discussed below (see case 10 with A > 4).

9. A = 4. In this case, maxf(x) = f(1/2) = 1 and, therefore, f(I) = I (Fig.9). In XE]

order to understand the properties of the dynarnic system defined by the mapping

x ~ fex) = 4x(l-x), (2)

Section 1 Trajectories of One-Dimensional Dynamical Systems

we use the fact that this mapping is topologically equivalent to the linear mapping

x ~ g(x) {2X' 0 ~ x ~ 1/2,

2(1- x), 1/2 < x ~ 1.

11

(3)

Two maps gl: XI ~ Xl and g2: X2 ~ X2 are called topologically conjugate or equivalent if there exists a homeomorphism h: X I ~ X2 such that the diagram

g,

XI ~ XI hJ- J-h

g2 X2 ~ X2

is commutative, i.e., ho gl = g2 0 h.

For maps (2) and (3), we have XI = X2 = land the conjugating homeomorphism

h: I ~ I is given by the function hex) = ~ arcsin .[X. If two maps are conjugate, then the dynamical systems generated by these maps are

also conjugate (or equivalent) (if hof = g 0 h, then hD r = gn 0 h for any n > 0). Every trajectory of a dynamical system is associated with a trajectory of another dynamical system (this correspondence is established by the function h; the trajectory of the map f passing through the point Xo is associated with the trajectory of g that passes through the point h(xo)). The corresponding trajectories have the same asymptotic

properties (the co-limit sets of the trajectories {f n(xo)} and {g n(h(xo))} are homeomorphic; if one of these trajectories is attracted by a cyde, then the other is also attracted by a cyde, and so on).

Therefore, we can study the dynamics of map (3) instead of map (2) because this is much simpler.

Map (3) is expanding, i.e., it increases the distance between dose points because the modulus of its derivative is everywhere greater than 1. This means that, for any open (in

/) interval J C I, there exists a number m> 0 such that gm (J) = I.

The proof of this fact is almost obvious: If 1 /2 ~ J, then 1 (g (J)) = 21 (J), where

1 ( .) is the length of the interval ; if 1 /2 E J, then there exists c > 0 such that g (J) ~

[0, c] and gm ([0, c]) = [0, c· 2m] for c· 2m < 1 and gm([o, c]) = I, otherwise. A similar assertion can be established for any other map topologically equivalent to

(3). In particular, it holds for map (2).

Lemma 1.1. For any open (in l) interval JC I, there exists a number m such

that r (J) = I.

This lemma does not seem to be obvious because map (2) strongly contracts intervals

in the vicinity of x = 1/2 (f'(l/2) = 0). Nevertheless, in view of the fact that ho fn

= gn 0 h for any n ~ 0, where h( x) = "* arcsin .[X, we condude that, under the ac-

tion of the map f, the interval J will also cover the interval I after about

m= log 1/1 (h(J»

log 2

steps (because h (J) is an interval). This lemma enables us to establish many important properties of the dynamical sys

tem generated by map (2).

Proposition 1.1. Periodic points are dense in I. Moreover, any open interval contains periodic points with arbitrariZy Zarge periods.

Proposition 1.2. There exists a trajectory everywhere dense in I. Moreover, aZmost alt trajectories are everywhere dense in I (these trajectories form a set of the second Baire category in /).

We prove Proposition 1.1. Let J be an arbitrary open interval and let m be such

that fm(J) = I. Then there are points x', x" E J such that f m (x') = 0 and fm (x") = 1.

Due to the continuity of f (and, consequently, of fm), one can find a point Xo lying be

tween x' and x" such that fm (xo) = xo. The point Xo is periodic and its period is a

divisor of m. In order to prove that the interval J contains periodic points whose peri

ods are greater than mo, it suffices to consider the map fm on J with m = mo! There

is an open interval J' C J such that fm (x) * x for any XE J'. Therefore, J' does not contain periodic points with periods 1,2,3, ... , mo. At the same time, according to

what has been proved above, J' contains periodic points and, hence, their periods are greater than mo.

To prove Proposition 1.2, we take an arbitrary countable base on I, e.g., the base

formed on I by open intervals J l , J2, ... , Js ' .... The fact that the family of Js forms

a base means that, for any point x E I, one can indicate a sequence of intervals Js,:J

JS2 :J ... such that

00 n Js; = {x}. i=1

Thus, one can choose a basis in the form of the family of intervals whose ends are binary

rational points on I. It is dear that a trajectory that visits all intervals !!S, s = 1, 2, ... ,

is dense in I. Let us show that one can find a point Xo E J I such that {i (xo) };:,o n Js * 0 for any s = 1, 2, .... By virtue of the lemma, there are positive numbers ml,

m2, ... such that fms(Js) = I for s = 1,2, .... Since ]'(Jl) = I:J J 2, one can find

an open (in I) interval J(I) C !ft such that ]' (J(I}) = J2. In view of the fact that

]2(J2 ) = I:J J3 and j' (J(I}) = J2 , one can find an open interval J<2} C J<l l such

that ],+m2(J(2) = J3. Since f m3 (J3) = I :J J 4, there exists an open interval

J(3) c J(2} such that fml+m2+m3 (J(3») = J4' and so on. We arrive at a sequence of

enclosed open intervals J I ~ ;<1) ~ f2) ~ f3) ~ ... ~ ;<s) ~ ... such that

It is clear that, for each point of the set n:=1 J(S) , one can indicate a trajectory which

passes through this point and is dense in J. The second part of Proposition 1.2 holds for dynamical systems in a general (Baire )

space X:

If X contains a dense trajectory, then the points of the trajectories dense in X form a set ofthe second Baire category in X.

This is a consequence of the fact that the set of these points is a Go-set, i.e., it can be

represented as an interseetion of countably many open sets (Birkhoff [1]). This Go-set is

dense in X (because it contains a trajectory dense in X). Therefore, it is a set of the second class. Thus, almost all points of the space X (and, in particular, I) generate tra

jectories dense in X. Here, the notion "almost all" is understood in the topological sense. For map (2), the Lebesgue measure of these points is equal to mes I = 1, but one can

find Cl-mappings of I onto itself which have trajectories dense in I and are such that

the Lebesgue measure of all trajectories dense in I is less than 1.

For map (2), the Lebesgue measure of the set of points generating dense trajectories in I is equal to mes I. Nevertheless, for general continuous maps on I that have trajectories everywhere dense in I, this condition may be not satisfied (generally speaking, it is often quite difficult to verify this fact (see, e.g., Lyubich and Milnor [1], Keller and Nowicki [1])).

All stated above for map (2) is true for the equivalent map (3). Consider the following important property of maps (2) and (3):

A measure Jl defined in the space X is called invariant under a map f: X ~ X if,

for any Jl-measurable set A C X, we have Jl U-I 0.)) = Jl (A). The Lebesgue measure is invariant under map (3). Map (2) possesses an invariant

measure which is absolutely continuous with respect to the Lebesgue measure, namely,

1 dx Jl (dx) = dh(x) = - --.j

1t x(l- x)

The existence of a finite invariant measure whose support has a positive Lebesgue measure means that, in order to characterize the properties of dynamical systems after a long period of time, one should use the language of probability theory.

In particular, for maps (2) and (3), even the statement of the problem concerning the construction of a trajectory that passes through a point Xo E I must be made more preeise. Thus, it is possible to determine 5 or 10 points of the trajectory of Xo by using a

computer: xl = f(xo), ... , xn = r(xo). At the same time, the exact computation of a

sufficiently large segment of the trajectory, e.g., up to n = 100, is impossible for standard precision of computers used for this purpose and, hence, the problem of constructing large segments of trajectories is incorrect. To explain this idea, we note that,

for map (3), any interval of length e covers [0, 1] after m'" (log (l/e)) I log 2 steps.

If our computer is capable of disceming e = 10-20, then it makes no sense to ask at which point of [0, 1] the trajectory under investigation is located for m > 20log2 10 ('" 70). Maps (2) and (3) "forget" initial conditions (xo) very quickly and, for large m,

one should ask: With what probability can the trajectory be found in a set I' CI? For example, if I' = (a, ß), then this prob ability is equal to ß - a for map (3) and, for map (2), we can write

ß 1 f dx ~ ~ x(l- x)

3:. (arcsin{ß - arcsinja) 7t

h(ß) - h(a).

IX

Sometimes, it is used to say (see Blokh [2] and Guckenheimer [2]) that maps (2) and (3) are characterized by highly sensitive dependence on the initial conditions (on 1). For

such maps, every trajectory is unstable in Lyapunov's sense, for any x E land e > 0,

there exist x' such that 1 x - x' 1 < e and n > 0 for which

p(x, x') = max Ifi (x) - fi (x') 1 > 1/2 0<;; i <;; n

(this is a consequence of the lemma on expansion). Any two trajectories with distinct but dose initial points diverge, and the rate of divergence is characterized by the Lyapu

nov exponent equal to

I d Illn lim In - fn(x) I _

n~~ dx X-Xo

at the point Xo (if this limit exists). Hence, the Lyapunov exponent is the parameter that enables one to estimate the maximum length of a segment of the trajectory the consideration of which makes sense.

The divergence of dose trajectories in the bounded interval I leads to the situation where the number of trajectories with different asymptotic behavior becomes too large.

As a quantitative measure of the variety in the behavior of trajectories, we can take topological entropy defined as folIows:

Let X be a compact topological space and let f: X ~X. If .91. is a family of subsets of X, then

{n-l n-l} .9I.n =.9I.'} = nf-i (A i )IA i E.9I.fori=0, ... ,n-1 and nf-i(Ai );t:0 .

i=O i=O

Section 1 Trajectories of One-Dimensional Dynamical Systems 15

If JI. is an open covering of X, we denote by 9{(JI.) the minimal possible cardinality

of a subcovering extracted from JL Then

h (f, JI.) = lim.!. log 9{( Jl.j) n~oo n

is the topological entropy of f on the covering JI.. The topological entropy of f is then defined by (Adler, Konheim, and McAndrew [1])

h(f) = sup {h(f, JI.) I JI. is an open covering of X}.

Let us also present the Bowen's definition of topological entropy (see Bowen [2] and Dinaburg [1]), which is equivalent to that given above. Let (X, p) be a compact metric space. A subset E of X is called (n, E)-separated if, for every two different points x,

YEE, thereexists O~j<n with p(fJ(x),jJ(y) >E. Aset Fex (n,E)-spansan

other set sex provided that, for each x ES, one can indicate y E F such that

p (ß (x),ji (y) ~ E for all 0 ~j < n.

For a compact set sex, let rn(E, S) be the minimal possible cardinality of a set F

which (n, E)-spans S and let Sn(E, S) be the maximal possible cardinality of an (n, E)

separated set E contained in S (we write r n(E, S,f) and Sn(E, S,f) to stress that the

relevant quantities depend on f). Finally, we define

r(E, S,j) = lim sup .!. log rn(E, S,j) n~oo n

and

S(E, S,j) lim sup .!. log Sn(E, S,j). n---7 OO n

Then we set

hp(f, X) = lim S(E, S,f) = lim r(E, S,j) C---700 (---700

and (see Bowen [3] and Dinaburg [1])

h(f) = hp(f, X).

For Iv E (0, Iv *), every trajectory of the map x --* Ivx(l -x) is a periodic trajectory or its ffi-limit set is a cycle. It is not difficult to check that, in this case, h(f) = O. The following statement was proved by Bowen and Franks [1] and Misiurewicz [1]:

The topological entropy of a continuous mapping f: I --* I is equal to zero if and

only if the period of every cycle is apower of two.

For piecewise monotone maps (in particu1ar, for maps with a single extremum), there

exists a simple formula for topological entropy (Misiurewicz and Szlenk [1])

h(f) = lim .!.. log mn , n~oo n

where mn is the number of intervals of monotonicity of r. Consequently, for I.. = 4,

the topological entropy of the map x ~ A.x (1 - x) is equal to log 2.

10. I.. > 4. Finally, we consider the map h .. for I.. > 4 and x E 1R. In this case, we

have ft...(l/2) = 1../4> 1 and, consequently, h .. (1) <I: I (Fig.lO). In particular,

h .. (1/2) ~ I and f';:(1I2) ~ -00 as n ~ 00. The same behavior is exhibited by all tra

jectories starting at the points of the interval J = {x E 1R: h..c x) > I} (the ends of the

interval J are the roots of the equation A.x (l - x) = 1 and, consequently, are given by

the expression 1(1 ± .)1- 4/1.. )). The interval I contains two intervals Jo and JI

which are preimages of the interval J (i.e., f(Jo) = f(JI) = J). Thus, the interval I

also contains two preimages Joo and JIO of Jo and two preimages Jo I and JII of

J I , and so on. Obviously, all trajectories starting from the set

00

J* = U ri(J) i=O

(in particular, from the intervals Jo and JI, Joo, Jol , JIO' and JIl) eventually leave the interval I and approach - 00 as n ~ 00.

The set J* is open. Moreover, one can show that it is dense on I and its comple

ment K = I\J* is a perfect nowhere dense set. Consequently, it is homeomorphic to

the Cantor set. Furthermore, mes K = 0 (see Theorem 6.3 and Henry [1]).

y

x

Fig.l0 Fig.ll

Section 1 Trajectories oJ One-Dimensional Dynamical Systems 17

The dynamical system defined on the set K pos ses ses the same properties as the dy

namical system generated by the map JA with A = 4 on I. Namely, the periodic points

are dense in K. Moreover, in any neighborhood of any point K, one can find periodic

points with arbitrarily large periods. The set K contains an everywhere dense trajectory.

For any A > 4, the map h. is conjugate (on IR) to the map

g: x H g(x) j 3x, x :s; ~, 1

3(1- x), x> -2

(there exists a homeomorphism h;...: IR ~ IR such that ft .. = h:;:1 0 g 0 hA) (see Fig. 11).

The points that do not leave the interval [0, 1] under the action of g form a set

{XE [0, 1]lgn(x)E [0, 1], n~O} (= hAK).

This is actually the standard Cantor set. Indeed, let us use the ternary representation of

the points XE [0,1], i.e., we set x = 0.ClICl2Cl3'" Cli"" where CI.j E {O, 1, 2}. For

ternary rational points, x = O. ClI ... Clm 000 ... = O. ClI .•. Cl~ 222 ... (Clm *' 0, Cl~ = Clm -

1), we use the first representation if Clm = 2 and the second representation if Clm = 1.

Then

ra2a3 ... ai ... , if ClI = 0,

g(x) = > 1, if ClI = 1,

0, <X2<X3 ... <Xi ... , if ClI = 2,

where <Xi = 2 - Cli (cf. the similar representation for map (3)). Therefore, the points that

contain 1 in their ternary representation leave the interval [0, 1] under the action of g, while points of the Cantor set

do not leave the interval [0, 1]. These examples demonstrate how sets homeomorphic to the Cantor set appear in the

theory of dynamical systems, and it becomes clear why these sets play an important role in the dynamics of systems.

2. co-Limit and Statistically Limit Sets. Attractors and Quasiattractors

As indieated above, the asymptotie behavior of trajeetories is deseribed by O)-limit sets. The examples eonsidered in Seetion 1 demonstrate that, in the most simple eases, the 0)

limit sets are fixed points and eydes. In more eomplicated eases, they ean be Cantor sets (as in the ease where A > 4 or A = A * "" 3.57 for map (1)) or intervals (as for A = 4, i.e, for map (2)).

What other types of O)-limit sets ean be diseovered for one-dimensional dynamieal systems? Is it possible for an O)-limit set to eonsist of finitely many points but not to be a eyde, for example, to eonsist of two different eydes ?

For a dynamieal system on an arbitrary loeally eompaet spaee X, the following statement is true (Sharkovsky [5]):

If an O)-limit set consists of finitely many points, then these points form a cycle.

This is true due to the following property of a dynamieal system on the O)-limit set of any eompaet trajeetory (Sharkovsky[5]):

(*) If F is an O)-limit set, then f(U) ~ U for any set U C F (U:;j:. F) open with respect to F.

In this ease, we say that the dynamieal system possesses the property of weak incompressibility. If we assume that F eonsists of finitely many points and eontains a

eyde F that does not eoineide with F, then F should be a dosed invariant set and, at the same time, it should be open with respeet to F, i.e., we arrive at the inc1usion f (F) c F that eontradiets the property of ineompressibility.

Ey the same reason, we have the following assertion (Sharkovsky [5]):

Each cycle that lies in an O)-limit set but does not coincide with this set is not isolated in this O)-limit set; more precisely, each point of this cycle is not isolated.

This means that eaeh point of this eyde is limiting for the points of the O)-limit set. This situation ean be eneountered in the ease where an O)-limit set eonsists of infi

nitely many points. This O)-limit set ean be either eountable or eontinual. It is worth noting that, in the first ease, the O)-limit set F neeessarily eontains at least one eyde.

Indeed, a sequenee of dosed sets F1:J F2 :J ... :J Fa:J ... , where F 1 = F, F a+l = 0) (xa) (xa is an arbitrary point from Fa) and Fa' = na<a' Fa whenever a' is a limiting ordinal number, is always stabilized, i.e., there exists a finite or eountable ordinal

number a* such that Fa* = Fa*+l (Aleksandrov [1]). If Fa is eountable, then F a+l :;j:.

Fa beeause isolated points of Fa do not belong to F a+l unless these points are peri-

Section 2 ffi-Limit and Statistically Limit Sets. Attractors and Quasiattractors 19

odic and belong to the trajectory {r (xJ}. Therefore, F Cl consists of finitely many points forming a cycle.

Is it possible for an ffi-limit set of a one-dimensional dynamical system to be countable? The answer is positive. Let us present a simple example. For this purpose, we recall the definition of homoclinic trajectories. If time in a dynamical system is reversible, i.e., if the dynamical system under consideration is a group (but not a semigroup) of maps, a trajectory is called homoclinic if it approaches the same periodic trajectory both as time infinitely increases and infinitely decreases. In our case, this definition is not correct because, generally speaking, time is not reversible. One of the possibilities to

preserve this notion for semigroups is to consider bilateral trajectories {x):: ~:, where

xi+l =f(x;). However, one may arrive both at the situation where there are many nega

tive trajectories {xi}:: = ~ for the point Xo (if r' is an ambiguous function) and at the

situation where there are no negative trajectories at all (if f(l) *" I).

We can now apply the definition of homoclinic trajectories presented above to the

trajectory {xi}:: ~:. Abilateral trajectory is homoclinic to some periodic trajectory y

if ffi(XO) = Y and the set of limiting points of the sequence {x-;li= 0 coincides with y.

To present an example of a countable ffi-limit set, we consider map (3) once again. Assume that the point Xo E 1 is such that ffi (xo) = {O}. Since x = 0 is a repelling fixed point, this is possible only in the case where gm (xo) = 0 for some m > O. The set of

points {gi (xo), i = 0, 1, ... , m, Xo / 2i , i = 1,2, ... } forms a homoclinic trajectory. It is easy to show that this homoclinic trajectory is the ffi-limit set for other trajectories (there are many trajectories of this sort; the points of these trajectories form a set of the third Baire class (Sharkovsky [7]).

Property (*) implies the following statement, which is also valid in the general case (Sharkovsky [6]):

If an ffi-limit set F is different from a cycle, then any its open (with respect to

F) zero-dimensional subset (ifir exists) contains at least one nonperiodic point.

The requirement that an open set be zero-dimensional is essential. For example, for maps on the plane, an ffi-limit set may be an interval consisting only of fixed points.

The abovementioned property also implies that, for one-dimensional maps, the following stronger statement (Sharkovsky [6]) is true:

If X = I, then on any ffi-limit set that is not a cycle, nonperiodic points are dense

(i.e., nonperiodic points form a dense subset on any ffi-limit set of this type).

An ffi-limit set may contain a trajectory for which it is the ffi-limit set (this means that the trajectory is dense in this set). Then, similarly to the reasoning in Section 1 (see Proposition 1.2), we conclude that almost every trajectory is dense in the ffi-limit set (such trajectories form a Go-set in it). If not all trajectories are dense in this set (i.e., if the set is not minimal), then the set of points that generate nondense trajectories is also

relatively large: such points are dense in this ffi-limit set (Sharkovsky [6]).

If an ffi-limit set contains a trajectory that is dense in this set, then, on this set, the dynamical system possesses the following "mixing" property stronger than property (*):

(**) For any two open (with respect to F) subsets U l' U2 CF, there exists

m > 0 such that fm l1 n U2 '# 0.

Properties (*) and (**) completely describe the behavior of a dynamical system on ffi-limit sets (Sharkovsky [5], [9]) in the following sense:

Suppose that a continuous map f given on a closed set Fex satisfies the condition f F = F. Then

- if property (**) holds, there exists a point XE F such that ffi(X) = F;

- ifproperty (*) holds, then, provided that F is nowhere dense in X (i.e., it does

not contain open subsets of X), the map f can be extended to a closed set X', F C

X' <: X, such that the map f on the set X' is continuous and there exists a point x E X' for which ffi(X) = F.

Thus, the question about the admissible topological structure of ffi-limit sets can be

reduced to the following one: What topological structure should the closed set F have in order that one can define a continuous map on it that possesses either property (*) or the stronger property (**) ?

Since, on any connected set, the identity map (all trajectories of which are fixed

points) possesses property (*), any closed connected set can be an ffi -limit set of a dynamical system.

On the other hand, one can easily give examples of closed sets that cannot be ffi-limit sets. For example, it is not possible to define a continuous map with property (*) if the set F consists of

- finitely many connected components, at least one of which is a point and another one differs from a point;

- infinitely many connected components, only finitely many of which are not onepoint sets and at least one of the components is isolated from the others.

These statements are simple consequences of the fact that the components which are not one-point sets must form an invariant set. It follows from (Kolyada, Snoha [1]) that there are no exceptions for sets that can be imbedded into the real line: in this case, one can find a continuous map possessing property (*) if and only if the set is not a set of the form indicated above, i.e., if it is not the union of finitely many intervals and finitely or infinitely many points the closure of which has no common points with at least one of these intervals.

This means, that in the case of continuous maps on an interval, one can define a

Section 2 O)-Limit and Statistically Limit Sets. Attractors and Quasiattractors 21

continuous map with property (*) on a closed subset of the realline only if the set

(i) does not contain intervals (i.e., is nowhere dense on the realline);

(ii) consists of finitely many intervals;

(iii) consists of finitely many intervals and a countable set of points the closure of which intersects each of these intervals;

(iv) consists of finitely many intervals and an uncountable nowhere dense set;

(v) consists of countably many intervals.

As shown by Kolyada and Snoha [1], any set with the structure described above can be an O)-limit set for continuous maps on the plane.

However, for continuous maps on an interval, sets that contain intervals and separated points cannot be O)-limit sets. Thus,

for continuous maps on an interval, a closed set F can be an O)-limit set only in the following cases:

- F is an arbitrary nowhere dense set;

- F consists offinitely many intervals.

The second possibility can easily be realized, e.g., by maps similar to (2) and (3). The realizability of the first possibility was proved by Agronsky, Bruckner, Ceder, and Pearson [1].

For the O)-limit set of each trajectory, one can select its smallest closed subset such that, for any neighborhood of this subset, the trajectory stays in it almost all time. This is especially important in connection with the fact that it is often impossible to get a precise mathematical description of dynamical systems and just this subset (but not the entire 0)

limit set) is, as a rule, observed in experiments. For a map f: I ~ I, we define

1 n-1 p (x, U) = lim sup - L Xu(l(x)),

n-'l = n k=ü

where x E I, U is an arbitrary subset of the interval I, and X u is the indicator of U.

The trajectory of a point x is called statistically asymptotic with respect to the set M if the equality p (x, U) = 1 holds for any neighborhood U of M (Krylov and Bogolyubov [1]). It is clear that each trajectory is statistically asymptotic with respect to its own O)-limit set. The smallest closed set for which the trajectory of a given point x E I is statistically asymptotic is called the statisticallimit set or the o-limit set of the trajectory of a point x; it is denoted by 0f(x) or simply by o(x). As indicated above, we


havetheinclusion cr(X)Cffi(X) butthe situation where cr(X):;t:ffi(X) isalsopossible. Indeed, if ffi (x) is the closure of a trajectory homoclinic to a certain cycle, then cr (x)

coincides with this cycle and, hence, does not coincide with ffi(X). Unlike ffi(X), the set

cr(x) may consist of finitely many points being not a cycle, e.g., it may consist of a pair of cycles. This situation is observed when ffi(X) is a pair of cycles joined by heteroclin

ic trajectories (a trajectory is called heteroclinic to cycles Band B' if it approaches B

as time increases and B'as time decreases).

The dynamical system generated by the map ft., for A ~ 4 (see Section 1) possesses

an important property, which is known as mixing of trajectories. For A = 4, mixing takes place on I = [0, 1], while for A > 4, it takes place on an invariant Cantor subset

of I. The definition of mixing can be formulated as follows:

If {X,j} is a dynamical system and A C X is a compact invariant set but not a cy

cle, then we say that this dynamical system is mixing on A or, for the sake of brevity, that A is a mixing set if, for any open (in A) set V and any open finite covering

L = {crj} of the set A, there exists m = m (V, L) and r ~ 1 depending only on A and such that

for all j. This property can be characterized by the following physical analogy: Imagine

a "drop" (a set V open in A) that gets into A and, after a certain period of time, fills

the entire set A. The property ofmixing on A implies transitivity. Indeed, for any two open (in A)

sets VI and V2 (C A), there exists a number m such that fm (VI) n V2 :;t: 0. Transitivity is equivalent (if fA = A) to the existence of a dense trajectory in A and, in this

sense, any transitive set is indecomposable. We have already noted that, for the map ft." both for A = A * and A > 4, there exists a Cantor-type sub set of the interval 1= [0, 1]

that contains dense trajectories. For A = A *, it is easy to show that mixing is absent. At the same time, for A > 4, on this invariant set (denoted by K), the dynamical system pos ses ses not only the property of mixing but also a stronger property of expansion, i.e.,

for any set V C K open in K, there exists a number m depending on V and such that

r(V)=K. The map A with A = 4 pos ses ses the same property in the entire I (this is just the

assertion of Lemma 1.1) by virtue of the fact that the map ft., is expanding on K.

Hence, both for A = 4 and A> 4, we observe mixing; moreover, the number m in the

definition of this property can be chosen independently of the covering Land r can be chosen to be equal to 1.

If {X,j} is a dynamical system and A is a compact invariant set, then we say that

A is astrange or mixing attractor (Sharkovsky, Maistrenko, and Romanenko [2]) whenever

Section 2 (fj-Limit and Statistically Limit Sets. Attractors and Quasiattractors 23

(a) A is an attractor, i.e., there exists a neighborhood U of A such that U::>

f(U)::>f2(U)::> ... , U-::I=A, and ni<!ofi(U)=A;

(b) A is a mixing set.

For the map ft.., a c10sed interval land a Cantor-type subset K C I are Illlxmg sets

but not attractors for I.. = 4 and I.. > 4, respectively. Indeed, for I.. = 4, we have

r(x)~-oo as n~oo foranypoint XE 1R\I. For 1..>4, theset K=I\J*, where

J* = U r n (J), J = {x E I I fex) > I}, n=O

and fn (x) ~ - 00 as n ~ 00 for any point x E J*, is mixing.

It is not difficult to "improve" the map ft.. with I.. = 4 on the set 1R \ I to transform

the mixing set I into an attractor. Thus, one can set

_ {'Ax(1- x), ft..(x) =

0,

x ~ 0,

x< O.

In this case, the interval [0, 1] is a mixing attractor of the map A with I.. = 4. This proves that one-dimensional maps may have mixing attractors either in the form

of intervals containing a dense trajectory or in the form of a collection of intervals cyc1ically mapped into each other. The following statement is true:

If fE QJ (I, I) and I is an interval, then a mixing attractor consists of one or finitely many intervals cyclically mapped into each other.

Thus, a Cantor-type set cannot be a mixing attractor. In particular, the mixing

Cantor-type set K mentioned above for the map A with I.. > 4 does not possess

property (a) despite the fact that (fj(x) C K for any XE lR. Let us explain this in brief. Since any mixing set contains a trajectory dense in this

set, it is a perfect set and if it is not dense at least at one point of I, then it is nowhere dense on this interval. Hence, the mixing sets are either homeomorphic to the Cantor set or consist of finitely many intervals. Any neighborhood of a nowhere dense set which is dense in itself always contains points that are not attracted to this set. This result is due to Sharkovsky [2, 8]. In [2], Sharkovsky established the following fact:

Every nonisolated point of an arbitrary (fj-limit set is a limiting point of the set of periodic points.

Therefore, in order that a set be an attractor, it is necessary that the periodic points be dense in it (for this reason, the minimal set that exists for I.. = 1..* and is not a cyc1e can-

24 Fundamental Concepts oj the Theory oj Dynamical Systems Chapter 1

not be an attractor). At the same time, if the rn-limit set contains periodic points, then the dynamical system possesses on this set the property expansion of (relative) neighborhoods (Sharkovsky [8]). As a result, any sufficiently small neighborhood of this set contains points leaving this neighborhood after a certain period of time.

For some values of the parameter A., the mapping h., may have mixing attractors,

e.g., for A. = 3.678 ... when the point x = 1/2 hits the fixed point x = 1 - 1/A. after 3 steps (Fig. 12). In this case, the interval

J = [f2(1/2), j(1/2)],

wherej(l/2) = A./4 "" 0.92 andj2(1/2) = A.2 (1-A./4)/4 "" 0.27 isanattractor. In

deed, for any closed interval I' such that /' c (0, 1), one can indicate m such that

jm (I') C J. In the interval J, the mapping is mixing and, in particular, possesses aB properties exhibited by the map ft.. with A. = 4 on the interval / (the set of periodic points is dense, there are everywhere dense trajectories, there is an invariant measure

absolutely continuous with respect to the Lebesgue measure). In the interval J, the map Jt, is conjugate to the piecewise linear map

{(2/3) (1 + x),

x ~ g(x) = 2(1- x),

x:O; 1/2,

x:2:1/2,

defined on the interval [0, 1]. The interval [0, 1] is a mixing attractor of the map g (Fig. 13).

y

!I

Fig.12 Fig.13

It should be noted that the mapping h., possesses a mixing set whenever the value of

the parameter A. is chosen so that the point x = 1/2 (the point of extremum) hits some

Seetion 3 Return of Points and Sets 25

repelling periodic point of period m for finitely many steps. This set is an attractor and

consists of m intervals provided that the periodic point does not coincide with the ends

of one of the intervals (as for A. = 4, where x = 1/2 hits the fixed point x = 0 which is

one of the ends of the interval [0, 1]). In particular, if the point x = 1/2 hits the repelling cycle with period 2 (as already mentioned, it is formed by the points

A. + 1 ± ~ A.2 - 2A. - 3 ) 2A.

and the parameter A. takes the least possible value (A. "" 3.593), then the mixing attractor consists of 2 intervals.

A mixing set which is not an attractor and, in addition, does not belong to any larger

ü)-limit set is sometimes called a mixing repeller. We have already encountered such

sets in our presentation. The map iJ,,, possesses a mixing repeller for A. = 4 (the interval

I = [0, 1]) and for A. > 4 (a Cantor-type set on I). Repellers and attractors play an important role in the theory of difference equations and, especially, in the theory of equations with continuous argument.

As already mentioned, the minimal set K which exists for the map x --? A.X (1 - x)

with A. = A. * and differs from a cycle is not an attractor. However, the set K is, in a certain sense, a quasiattractor. (Moreover, the ü)-limit sets of almost all points in I co

incide with K.)

A set A C I is caBed a quasiattractor if

(i) for any neighborhood U of the set A, there exists a neighborhood V C U such

that fi (V) C U for all i ~ 0;

(ii) there exists a neighborhood U of the set A such that the ü)-limit sets of almost

all its points belong to A.

3. Return of Points and Sets

As already mentioned, the asymptotic behavior of the trajectories of a dynamical system may be fairly diverse. In order to understand a dynamical system as a whole, it is convenient to select in its phase space the sets which attract aB or almost aB trajectories. One of the most important properties of trajectories belonging to such sets is the property

of return. In the theory of dynamical systems, it is customary to distinguish between several

types of return. The simplest type is connected with the return of points to their initial location after a certain period of time. Points with this property are called periodic (in the previous sections, they have been studied in detail). The set of periodic points of a

map f is usuaBy denoted by Per Cf).

A more general type of return is connected with the return of a point into its own neighborhood (even after an arbitrarily large period of time): A point x E X is called recurrent if XE (0 (x), i.e., for any neighborhood U of x, there exists an integer m > 0 such that fm (x) E U and, consequently, one can find an infinite sequence of return times ml < m2 < .. , such that fmi(x) E U for i = 1, 2, .... Recurrent points can be, in turn, c1assified depending on the properties of the sequence {mi}' For example, if

{mi} is a relatively dense sequence, then x is called a regularly recurrent point; if, in

addition, mi = mi (m depends on U), then x is called an almost periodic point, and so on.

The set of recurrent points of a map f is denoted by R (f), the set of regularly recurrent points is denoted by RR (f), and the set of almost periodic points by AP (f). (It should be noted that some authors use the terms "Poisson stable", "almost periodic", and "isochronous" points instead of "recurrent", "regularly recurrent", and "almost periodic" points, respective1y).

It follows from the definitions introduced above that AP (f) !:; RR (f) !:; R (f). Note

that there exist maps such that R (f) \RR (f) # 0 (far example, it follows from Proposition 1.2 that map (2) has a trajectory everywhere dense in I whose points belong to

R (f) \RR (f) and maps such that RR (f) \AP (f) # 0 (e.g., the piecewise linear map

f in Fig. 14, where fha,bl is topologically conjugate to f~ [{(al, bl' For this map, the point

b belongs to AP (f) while its preimage b' belongs to RR (f) \ AP (f). For the proof of this property, see Section 4 in Chapter 4).

x

Fig.14 Fig.15

A weaker type of return is exhibited by the so-called nonwandering points. A point x E X is called nonwandering if, for any its neighborhood U, there exists an integer m > 0 such that fm (JJ) n U # 0, i.e., a subset of Ureturns into U after m steps. It is clear that the points exhibiting all types of return described above are nonwandering as weIl as the (O-limit points of the trajectories. The set of all nonwandering points of a dynamical system generated by a map f is denoted by NW (f).

It follows from the definition of NW (f) that NW (f) is always a c10sed set and if

Section 3 Return of Points and Sets 27

the dynarnical system is a group of maps, then NW (f) is invariant (i.e., f(NW (f)) = NW(f)).

The following assertion is weH known (the Birkhoff theorem):

Consider a dynamical system defined in aspace X. Assume that the space X is compact. Then,for any neighborhood U of NW (f), there exists an integer m (depending on U) such that the time of stay of any trajectory outside U does not exceed m, i.e., the following inequality is true for any x EX:

'tex, U) = L XX\U(/(x)) ~ m· ,

here, XA is the indicator of a set A.

a b

Fig.16

If a dynarnical system is generated by a continuous map (and is nothing more than a

semigroup of maps), then it is possible that f(NW (f)) =I- NW (f), although it is

obvious that the inclusion f(NW (f)) C NW (f) is always true. As an example (Sharkovsky [2]), we consider the map represented in Fig. 15. For this map, the point x = c is nonwandering but one can indicate no points XE NW (f) such that fex) = c. It is easy to see that the point x = b does not belong to NW (f). Note that c ~ Per(!) and c is not an ffi-limit point far any trajectory. Hence, far this map, we have

NW (f) =I- Per(j) and NW (f) =I- Ux ffif(x). It is not difficult to verify that, in this

case, Per(j) = U x ffi fex) and

NW (f) = Per(f) U {cl.

Note that, in any neighborhood of the point x = c, there exists a point x' such that

fm' (X) = c for some m' > O. It turns out that any nonwandering point possesses this property, provided that X = l.

Since the point x = c is nonwandering, one of the images of its neighborhood (c - lO,

c + lO) necessarily intersects this neighborhood in the course of time. However, for sufficiently small lO, the images of the left and right unilateral neighborhoods never intersect the corresponding unilateral neighborhoods. This type of behavior exhibited by dynamical systems indicates the necessity of distinguishing between the sub sets of unilateral

nonwandering points NW - (f) and NW + (f) in the set of nonwandering points.

Namely, a point x belongs to NW- (f) (NW + (f) if, for any open (in 1) interval

U whose right (Jeft) end coincides with the point x, we have fm (U) n U:f:; 0 for some m > 0 which depends on U. Thus, for the map displayed in Fig. 15, we can write

CE NW (f) \ (NW- (f) U NW+ (f).

Theorem 1.1. (i) (Sharkovsky [11])

Per(f) U NW-(f) U NW+(f) = U COf(x); XEI

(ii) (Blokh [4])

n i(NW(f)). i2:0

This theorem, in particular, implies that a point x E I is an co-limit point of a certain trajectory if and only if, for any neighborhood U of the point x, there exist x' E U and integer numbers 0< m\ < m2 such that fm;(x) E U, i = 1,2.

Denote the set Ux co fex) by n (f). This set is sm aller than NW (f) but satisfies

the following analog of the Birkhoff theorem:

Theorem 1.2 (Sharkovsky [11]). For any neighborhood U of the set n (f),

there exists an integer m = m(U) such that the time of stay of the trajectory of any point fram I outside U does not exceed m.

The set n (f) in Theorem 1.2 cannot be replaced by a smaller c10sed subset: Indeed, for any point x' E n (f), there exists a point x" such that CO f(x) .3 x' and,

hence, the trajectory of the point x" hits any neighborhood of the point x' infinitely many times.

Note that Per Um) = Per (f) for any m. Generally speaking, the set NW (f) does not possess this property. The example given in Fig. 16 (Coven and Nitecki [1]) is char-

Section 3 Return oJ Points and Sets 29

acterized by the property NW (p) =I- NW (f) (note that this example is a modification of the previous one). In this case, x = a is a nonwandering point of the map J but, for

the map p, this point is not nonwandering as can easily be seen from its graph. Never

theless, the equality NW (r) = NW (f) always holds for odd m (Coven and Nitecki [1]).

By definition, the set NW (f) consists of points at which one observes the return of domains of the space X. At the same time, the situation where relative regions (i.e., sub

sets of NW (f) open with respect to NW (f)) do not return is possible. Therefore, in the theory of dynamical systems, parallel with NW (f), it is reasonable to consider a smaller set C (f) called the center of a dynamical system and characterized by the return of relative domains.

If JE CO (X, X) and X is an arbitrary compact space, then we can define C (f) as follows: Let Cl = NW (f) and let, for a ~ 1, Ca+l be a set of the nonwandering

points of the space Ca' i.e., NW Ulc )' If a is the limiting ordinal number, then we IX

set

According to the Baire-Hausdorfftheorem, we have Cr = Cr+l = ... for some finite or

countable ordinal number r. Then C (f) = Cr This r is called the depth oJ the center,

provided that it is the least possible ordinal number of this sort. The center of a dynamical system can also be defined as follows: C (f) is the largest

c10sed invariant set characterized by the property of incompressibility of the regions, i.e.,

for any subset U C C (f) open in C (f), we have either J(U) = U or J(U) <t: u. It is well known (see, e.g., Birkhoff [1], Nemytsky and Stepanov [1]) that C (f) is

the c10sure of the set of recurrent points. For any trajectory, the probability of its stay in

any neighborhood of the center is equal to one, i.e., for any set U => C (f) open in X, we have

m-l

lim ~ L Xu(l(x)) m~oo m i=O

for any point x E X. In the case where X = I, some statements can be made more precise. Thus, in the

general case, the depth of the center can be equal to any finite or countab1e ordinal number but, for X = I, the depth of the center is not greater than 2.

Theorem 1.3 (Sharkovsky [2]). C (f) = NW UINW(f»)'

For the map whose graph is depicted in Fig. 16, the depth of the center is equal to 2.

30 Fundamental Concepts oJ the Theory oJ Dynamical Systems Chapter 1

As mentioned above, in the general case, the recurrent points are den se in C (f).

This does not mean that periodic points are also dense in C (f). Thus, for the circle SI

and J defined as a rotation of SI about an irrational angle, we have Per (f) = 0 but

C (f) = SI. At the same time, periodic points are everywhere dense in C (j) for X = l.

Theorem 1.4 (Sharkovsky [2]). C (f) = Per (f).

Note that there exist (nonsmooth!) mappings fE (!J(I, l) with co-limit points that are not limiting points for the set of periodic points (see Chapter 4). For these mappings, we have C (f) ;f: Q (f).

The weakest property of return that may take place for some points of dynamical systems is chain recurrence. A point x E I is called chain recurrent if, for any E > 0,

there exists a sequence {xJ;=o such that Xo = x = Xn and IJ(xD -Xi+tI < E for any

i < n (the points {Xir=o are called E-trajectories of the point xo).

The concept of chain recurrence is closely related to the notion of weak incompressibility (Vereikina and Sharkovsky [2]). We recall that a closed invariant set F

exhibits the property of weak incompressibility if, for any subset U C F open with

respect to F and not equal to F, one can write J(U) <I: U (Sharkovsky [15]).

We have already mentioned in the previous section that the property of weak incompressibility is observed for any co-limit set; it is also typical of cycles, the closures of homoclinic trajectories, etc. On the other hand, the set that consists of two cycles does not exhibit the property of weak incompressibility. In general, this property is not observed for any set that consists of two disjoint closed invariant subsets (one can choose U in the definition of incompressibility in the form of one of these invariant sub sets and, in

this case, J(U) = U).

A point that belongs to a set with the property of weak incompressibility can be called an almost returning point. One can easily show that every point of this sort can be made periodic by arbitrarily small perturbations of the dynamical system, i.e., this point will have the strongest property of return. This fact enables us to say that the points of sets with the property of weak incompressibility are almost returning points.

The set of almost returning points coincides with the set of chain recurrent points for any map J (Vereikina and Sharkovsky [2]).

This set is denoted by CR(f), i.e., XE CR(f) if there exists a set F 3 x with the property of weak incompressibility.

As a rule, there is no weak incompressibility in the entire set CR(f). For example, this is true if CR (f) consists of two fixed points-a sink and a source-as in the case of mapping (1) with 1< A ~ 3 (see Section 1).

Section 3 Return of Points and Sets

r-----I I I ,

Fig.17

31

x

We now recall some properties ofthe set CR(}), which can easily be derived from the definition. The set CR(f) is closed and invariant. Every nonwandering point is

chain recurrent, i.e., NW (f) C CR(f) but it is possible that NW (f) ::f. CR(}). For the map depicted in Fig. 17,

00

CR(f) \NW (f) = (a, b) U U ri([c, dl), i=O

while for the map presented in Fig. 15, we have

00

CR(f)\NW(f) = U ri(c). i=1

The definition of CR(f) immediately implies that CR(flcR(f) = CR(f). Unlike

the map fH NW (f) of the space cl(X, X) into 2x, which admits Q-explosions (i.e., is neither upper nor lower semicontinuous), the map fH CR (f) is upper semicontinuous.

Finally, in many cases (in particular, for X = I), the set C R (f) coincides with the set of points that return as a result of infinitesimal perturbations of dynamical systems themselves, i.e., it coincides with the set of weakly nonwandering points.

A point x E X is called a weakly nonwandering point of the map fE er (X, X) if, for any neighborhood U(x) of the point x and any neighborhood ~(f) of the map f (in er(X,X)), one can find JE ~(f) andinteger m>O suchthat Jm(U(x) n U(x)

::f.0.

As for as we know, the dependence of the property of weak nonwandering on r has not been studied yet.

The point x E I is called almost periodic in the sense of Bohr if, for any E > 0, one can find N> 0 such that, for any i > 0, there exists n > 0 such that i + 1 ::;; n ::;; i + N

and !fj + 1: x) - fj (x) ! < E for any j ~ O. The set of all points of a map f almost periodic

in the sense of Bohr is denoted by APB (j).

Theorem 1.5. Let fE Cl (I, 1). Then

Per(j) ~ APB(j) ~ AP(j) ~ RR(j) ~ R(j) ~ C(j) ~ n(j) ~ NW(j) ~ CR(j).

All inc1usions, except n (f) ::J C (f), follow from the definitions. For Cl (I, I), we

have APB (j) ~ AP(j) (Fedorenko [4]) and n (f) = n(f) (Sharkovsky [2]). This

enables us to conclude that n (f) ::J C (f).

Sometimes, it is possible to study the structure of sets indicated in Theorem 1.5 and represent these sets in the form of a finite (or countable) union of sub sets which are, in a certain sense, dynarnically indecomposable (e.g., contain a dense trajectory). Representations of this sort are usually called spectral decompositions. The spectral decomposition of the set of nonwandering points is the most popular object of investigations. As a rule, in terms of this decomposition, one can easily describe the typical behavior of the trajectories of the corresponding dynamical system.

To explain this in detail, we consider quadratic mappings from the family ft,,, de

scribed in Section 1. For these mappings, the sets Per (f), n (f), and NW (f) always coincide as follows from the results of Chapter 5.

The examples presented in Section 1 demonstrate that, for 0 < A ::;; 1, the set NW (f) consists of a single fixed point x = O. For 1 < A ::;; 3, it consists of two fixed

points, name1y, the repelling point x = 0 and the attracting point x = I - 1 JA. For 3 < A ::;; 1 + -J6, we observe the appearance of an attracting cyc1e of period 2 and NW (f) consists of three dynamically indecomposable components, namely, the repelling fixed

points x = 0 and x = 1 - 1 JA and the attracting cyc1e of period 2. One can check that the trajectories of all points on the interval I = [0, 1] (except countably many) are attracted by the cyc1e of period 2 (see Chapter 5 for detailed explanation). Further, if the value of A increases to A = A * , then the number of elements in the spectral decomposition of the set NW (f) increases to infinity. Thus, for A = A *, the set

NW (f) is a union of two repelling fixed points x = 0 and x = 1 - 1 JA, infinitely many cyc1es of periods 2i , i = 1,2, ... (with one cyc1e of each period), and the minimal Cantor set K. Note that, for A < A *, the generic behavior of trajectories on the interval I (i.e., the behavior of trajectories of almost aB (with respect to the Lebesgue measure) points) can be described as the asymptotic convergence to an attracting cyc1e. For A = A *, a typical trajectory on I is asymptotically approaching the set K, i.e., it is asymptotically almost periodic.

Section 3 Return of Points and Sets 33

For A = 4, the set NW (f) coincides with I = [0, 1] and can be regarded as dynamically indecomposable because I contains a dense trajectory. It has already been indicated that, in this case, the trajectories of almost all points from I are dense in I. We recall once again that the structure of the set NW Cf) is investigated in more details in Chapter 5.

From the practical point of view, it seems reasonable to select the properties typical of the trajectories not of all points of the phase space but of almost all points of this space. In this case, the term "almost all points" may denote either a collection of points forming a set of the second Baire category (i.e., almost all points in the topological sense) or almost all points with respect to a certain measure in the phase space (i.e., almost all points in the metric sense).

This point of view, in particular, leads to the notion of probabilistic limit sets (or

Milnor attractors, see Arnold, Afraimovich, Il'yashenko, and Shilnikov [1] and Milnor [2]), i.e., to the notion of the smallest c10sed set that contains the co-limit sets of trajec

tories of almost all points in the phase space (this set is denoted by :M(f). In a similar way, the notion of statistical limit set introduced in the previous section

leads to the notion of the minimal center of attraction of almost all trajectories of a dynamical system (or to the notion of statisticallimit set, as it is defined in Arnold, Afraimovich, Il'yashenko, and Shilnikov [1], i.e., to the smallest c10sed set that contains statistical limit sets of the trajectories of almost all points of the phase space; this set is de

noted by 5'1. (f). It follows from the definition that, as a rule, this is just the set observed in the experimental investigation of dynamical systems.

It is worth noting that if we replace the words "almost all" by "all" in this definition, then we arrive at the notion of the minimal center of attraction (of all trajectories), which is weIl known in the theory of dynamical systems since thirties; this set is defined as the smallest set in any neighborhood of which all trajectories stay almost always. As already mentioned, the trajectories stay almost always in the neighborhood of the center of the

dynamical system. Therefore, the minimal center of attraction is a subset of C (f).

It follows from the definition that 5'I.(f) C :M(f). There exist maps for which these sets do not coincide. An example of this sort is presented in Chapter 6 (Fig.44); for this

map, the set :M(f) is an interval with a dense trajectory and 5'I.(f) consists of a single repelling fixed point.

Consider a mapping almost all trajectories of which are attracted by a repelling fixed point. For XE [0, 1], we define

g(x) j3X,

1,

3(1- x),

o ~ x < 1/3,

1/3 ~ x ~ 2/3,

2/3 < x ~ 1.

By using the reasoning applied in Section 1 to the investigation of the family h.. for

A > 4, one can show that the trajectories of all points, except the points of the Cantor set K C [0, 1], hit the repelling fixed point x = 0 after finitely many steps. Moreover, K contains a dense trajectory. Therefore, both n (g) and the minimal center of attraction


of the map g coincide with the set K U {O} = K. At the same time, the sets :M (f)

and 5lL(f) consist of a single repelling fixed point x = O. The fact that there exists a mapping for which its generic trajectory "is attracted" by a

repelling cyde seems to be unexpected. However, the map g may be untypical or even, in a certain sense, exdusive. As an argument for this assertion, one can recall, e.g., the following fact: The repelling fixed point may lose its property to attract almost all trajectories as a result of infinitesimally small perturbations of the map g.

It is also interesting to study a more general question: What properties of a dynamical system generated by a map from a certain space IDC of maps can be regarded as typical? Any property can be regarded as generic (typical) if a collection of maps characterized by this property forms a set of the second Baire category in IDC Clearly, the answer to the posed question depends on the space IDC under consideration. Thus, as shown in Chapter 6, far a sufficiently broad dass of smooth mappings, almost all trajectories are attracted either by an attracting cyde, ar by a Cantor-type set, or by a set that consists of finitely many intervals cydically permutable by the map and contains an everywhere dense trajectory. At the same time, none of the indicated types of behaviar is observed

for typical mappings in Ql(I, 1). In particular, far these mappings, no cyde is attracting and no trajectory is dense on any interval.

Let us now formulate an assertion about the typical behavior of the trajectories of Co_ typical dynamical systems recently proved by Agronsky, Bruckner, and Laczkovich [1].

The space Ql(l, I) contains a set CI of the second category such that, for any map

fE CI, there are continuum many minimal Cantor-type sets Fa on each of which f is a homeomorphism and, moreover,

(a) P(Fa) = {x E 11 ffij(X) = Fa} is nowhere dense in I;

(b) U P(Fa) is a set ofthe second category. a

This means that almost all (on I) trajectories of a dynarnical system are asymptotic

ally regularly recurrent almost always in Ql(I, I). This result is, to a certain extent, unexpected. Actually, almost all mappings (in par

ticular, in CI) possess cydes with periods *- i and, consequently, Cantar-type quasiminimal sets that contain cydes on which almost all trajectories are recurrent but not regularly recurrent or asymptotically regularly recurrent. Although each quasiminimal set of this sort contains continuum many Cantor-type minimal sets, in the typical case, they attract not too many trajectories (which form a set of the first category in 1). Therefore, in the case of smooth mappings, almost all trajectories almost always are either recurrent or asymptotically approach recurrent trajectories. The above-mentioned

result by Agronsky, Bruckner, and Laczkovich [1] states that, for Ql-typical mappings, the situation is absolutely different: Due to the very complicated structure of typical

continuous Ql-maps, one observes the appearance of (continuum) many Cantor-type minimal sets, which "seize" almost all trajectaries.

2. ELEmEnTS OF svmBOLIC DvnamICS

1. Concepts of Symbolic Dynamics

Symbolic dynamics is a part of the general theory of dynamical systems dealing with cascades generated by shifts in various spaces of sequences

where 8n are letters of an alphabet JL = {8 1,8 2, ... ,Sm}. The methods of symbolic

dynamies are now widely applied to the investigation of various types of dynamical systems.

Let TI be the space of all unilateral sequences e = (8o, 81, 82, ... ) (or infinite words, if it is reasonable to omit commas) with the metric

p(e', eil) = L r(8~~Z), n~O m

where

We define a shift cr: TI ~ TI as folIows: If e = (8o, 81, 82, ..• ), then cre = (81, 82, ••• ). For the dynamical system (TI, cr), many standard problems of the theory of dynamical systems, in particular, those conceming periodic trajectories can be solved almost trivially.

Thus, for the dynamical system (TI, cr), every point e corresponding to the periodic sequence 81 ... 8k81 ••• 8k81 ••• with the least period k generates a k-periodic trajec

tory in the space TI (since crk e = e and crie "* e for 1:S; i < k). Hence, this dynamical system possesses periodic trajectories of all periods and these periodic trajectories are everywhere dense in TI. The last property folIo ws from the fact that, for any

e=(80,81,··.,8k_l,8k, ... )E TI and E>O, thepoint e'= (80, ... ,81<-1,80, ... ,81<-1'

35

36 Elements 01 Symbolic Dynamics Chapter 2

8o, ... ) belongs to the E-neighborhood of the point e and is periodic whenever k satis

fies the inequality 1/2k < E.

The spaee n also eontains dense trajeetories. Thus, the trajeetory that passes through the point

h r ili2···i!t-lis - ril···i!t-l 8 is - 2 3 r il - 8 il .. . - 1 . tr w ere - , s - , , ... , - ,I" '2' ... , 's - , ... , m, IS a a-

jeetory of this sort. The sequenee e* eonsists of all possible words written in the following sueeession: First, we write the words of length 1, then the words of length 2, and

so on. The trajeetory e*, 0- e*, 0-2 e*, ... is dense in n beeause the "eylinders"

Dil···is= {e E nie = (ril···is8s+,8s+2 ... )}, 1::; i" ... , is ::;m, s = 1,2, ... , form a

base of the spaee TI and, for any i;, ... , i;, one ean indicate an integer k sueh that o-k e* E D i{ ... i; .

Fig.18 Fig.19

Let us now analyze the possibility of applieation of symbolic dynamies to the investi

gation of individual dynamieal systems, e.g., on the real axis 1R. The dynamieal system on [0, 1] generated by the map (see Fig. 18)

I: x ~ mx (mod 1) (1)

is isomorphie to the dynamieal system of shifts with alphabet 8',... , 8 m. If we use the

m-digit representation of the points XE [0, 1], then, clearly, 8 i - i-I, where i = 1, ... ,

m. Henee, the dynamieal system generated by (1) possesses the properties of the dynam-

Section 1 Concepts of Symbolic Dynamics 37

ical system of shifts, i.e., there are periodic trajectories of arbitrarily large periods, periodic trajectories form an everywhere dense set in [0, 1], and there are trajectories dense in [0, 1], e.g.,

9* - x* =0.01 ... m-1 0001 ... m-1 m-1 000001 ... = 11m2 + 2Im3 + ....

The dynamical system generated by (1) does not belong to the c1ass of one-dimensional dynamical systems considered in the book because map (1) is not continuous. At the same time, the methods of symbolic dynamics can be successfully applied to the analysis of continuous maps. To illustrate this assertion, we present several examples.

Let f be an arbitrary continuous function IR ~ IR satisfying the condition

f(x) = 3x(mod 1) for XE J = [0, 1/3] U [2/3, 1] (2)

(Fig. 19). Clearly, one can use symbolic dynamics to investigate the dynamical system

generated by (2): If we use the temary representation of points in J, i.e., if we have x = 0.81828384 •.• , where 8i E {O, 1, 2} and 8 1 :F- 1 for XE J, then fex) = 0.8283 ..•.

Some points of J eventually leave J under the action of f The points leave J if and only iftheir temary representations 0.818283 ... contain at least one 8i equal to 1 (the

point 9 leaves J after 1= 1(9) steps, where 1(9) = min{il 8 i = I}). The points of

the set K = {O. 8182 .•. 8i ... I 8i E {O, 2}} (K is the standard Cantor set) and only these

points do not leave J and fK = K. The map facts on the set K as the dynamical sys

tem of shifts with alphabet {O, 2}. Hence, the map f: IR ~ IR has periodic points of

all periods on J (thus, the point 0.20202 ... =3/4 has period 2, the point 0.2002002 ...

= 9 113 has period 3, the point O. 20 ... 020 ... 02 ... = 2/3 r ,... has period m, '---v--' ~ 3 - 1

m m etc.).

The possibility of application of the methods of symbolic dynamics (with two-letter alphabets) to the investigation of map (2) is certainly explained not by the special form

of the map f on J but by the fact that J is the union of two intervals Jl and J2 such

that f Jl ~ Jl U J2 and f J2 ~ Jl U J2· This means that any word 8i 8i ... 8ik generat-1 2

ed by the two-letter alphabet 8 1,8 2 can be associated with a sequence of intervals :h, 1

!h2, ... ,:hk' where is = 1 or 2 and s= 1, ... ,k, which contains trajectories ofthe map f that pass through the intervals Jl and J2 in the indicated order. There are many trajectories ofthis sort. Moreover, these are the only trajectories passing through the points of

the set fl c!h successively constructed from the intervals !hk,:h , ... , :h as follows: 1 ~l 1

fs =!hs nf-l fS+l, s = k - 1, k - 2, ... , 1; fk = :hk. The set fl always contains a

nondegenerate interval. In the next chapter, a similar approach is used to study periodic trajectories of arbit

rary continuous maps IR ~ IR. More precisely, we consider the problem of coexistence of periodic trajectories of various periods and types.

Fig.20

It is important to mention the following fact: We have always assumed that the shift map is defined in the entire space rr. At the same time, in analyzing individual dynamical systems by using symbolic dynamics, we most often encounter the situation where the shift map (J is defined not in the entire TI but in a certain subspace TI' (of "admis

sible" sequences 8182 ",), Thus, for the map displayed in Fig. 20, there are intervals J I

and J2 such that I JI ~ JI U J2 and I J2 ~ JI · Therefore, if we pass to a symbolic

dynamical system with alphabet {8 I, 8 2} to study the map on the intervals J I and J2, then we arrive at the shift map defined only on the sequences SI82 ... Sn"" satisfying

the following condition: If 8n = 8 2, then Sm-I = SI. It is clear that restrictions of this

sort may significantly complicate the investigation of symbolic dynamical systems. Note that, in simple cases, the subspace TI' of admissible sequences may be determ

ined by a matrix of admissible transitions (of the mth order). In the last example, this is

the matrix (~ ~) (the only forbidden transition is J2 ~ J2 because I J2 ::p J2)'

In the set TI of sequences with alphabet (8 1, ..• , Sm) we introduce the following

naturallexicographic ordering: 8' = (8;82 ... ) < 8" = (S;'82 ... ) if, for some n, we

have 8i = 8i' for i< n and 8~ = 8s', 8~ = 8s" for some i< s". In the examples presented above, the map I is monotonically increasing on the inter

vals J I and J2' Thus, when we pass to symbolic dynamies, the correspondence be

tween the points XE UrJ,. and the sequences 8(x) = (8182 ... ) is monotone, i.e., if x',

x" E UrJ,. and x' < x", then 8(x') < 8(x") in the lexicographic order.

In studying one-dimensional dynamical systems, we mostly deal with piecewise monotone maps and, hence, both with intervals of increase and intervals of decrease of

the function f At the same time, if the intervals of decrease of the function I are involved in the construction of symbolic dynamies, the monotonicity of the correspondence between x and 8(x) is violated. Therefore, it is necessary to modify the method used to construct symbolic sequences.

Section 1 Coneepts 0/ Symbolie Dynamies 39

In this chapter, we analyze the possibilities of the method of symbolic dynamics in more details for fairly simple piecewise monotone maps, namely, for unimodal maps.

Let /: I ~ I be a unimodal map, let I = JI U J2' let / be a function monotonically

increasing on JI and monotonically decreasing on J2' and let e be the point of extremum.

Let us define the address of a point x EI:

{Js,

A(x) = e, if x = e.

The route is defined as a sequence of addresses

The operation of shift cr on the space ofunilateral sequences (;10' AI' A2, ..• ) is de

fined, as usual, by the equality cr (;10' AI' A2, ... ) = (A I' A2, ... ). The map / and the shift

cr are connected by the equality cr(Alx) = Ai/(x). In constructing symbolic dynamics, it turns out to be useful to take into account not

only the changes in the addresses A ({'(x) but also the changes in orientation. This idea was applied by Milnor and Thurston [1] to the theory of kneading invariants (see also Guckenheimer [1]).

We associate the intervals Js with the signs

and set t(e) = O.

{+ 1, if / / Js increases, t(J) =

s _ 1, if / / Js desreases,

Parallel with a route Alx) = (Ao, AI' A2, ... ), we consider a sequence 8 f (x) = (80,

81,82, ... ), where 80 = to, 8 1 = totl, ... , 8n= €otl ... t n ... , ti =t(Ai } The sign of

to' tl ..... t n corresponds to the local behavior of {' in the vicinity of the point x, i.e., it

is equal to + 1, - 1, or 0, respectively, if {' increases, decreases, or has an extremum at

the point x. Due to the fact that the phase space is one-dimensional, the lexicographic ordering is connected with the natural ordering of real numbers by the following assertion:

Lemma 2.1 (on monotonicity). The map x ~ 81x) is monotone.

Proof. Note that the map x ~ 8 f (x) is either nonincreasing or nondecreasing de

pending on the type of extremum (minimum or maximum) attained at the point e by the function f: I ~ I. Assume that e is the maximum point. If x' < x", then let n be the

least integer for which 8n(x') "* 8ix"). For n = 0, we have x'::; e ::; x" and 80(x');:::

40 Elements oJ Symbolic Dynamics Chapter 2

80(x"). For n;;:: 1, Jn is a homeomorphic map ofthe interval <x',x") onto the interval

<r(x'),r(x"» and the interval <r(x'),r(x"» contains the point c (here and below, <a, b) denotes the closed interval bounded by the points a and b). Assume that

8n-l(x') = 8n-l(X") = -1. Then the homeomorphism r: <x', x") ~ <r(x'),fn(x"»

changes orientation. Therefore, J'tx') E 52' J'tx") E 51> and 8Jx") S 8jx'). The

cases where 8Jx') = 8Jx") = 1 and 8n-l(x') = 8n-l(x") = 0 are analyzed similarly.

Let C = Ui~O J-i(c). The equality 8ix') = 8f (x") holds for x' '1= x" if and only if

the points x' and x" belong to the same component of the set 1\ C. In this case, Jn:

<x', x") ~ <r(x'),r(x") is a homeomorphism for an n > O. Note that 8 ix) = (80,

81,82> ... ) with eiE {-1,+1} for XE I\C.

The topology of coordinatewise convergence generates ametrie p in the set {8f (x)}.

Due to the lemma on monotonicity, the limits

exist for an x E I. Moreover, for any X E I, an elements of the sequences 8ix±) are nonzero.

Let 0" be a shift in the set L of unilateral sequences with alphabet {- 1, + I}.

For a = (aO,al,a2, ... )E Land ßOE {-I, I}, we set ßoa = (ßoao, ßOal'

ßOa2'···) anddefine lai bytheequality lal=aoa. Then 0"(8ix))=80(x)8/J(x)).

Let L' = {8f (x): x E 1\ C}. The lemma on monotonicity implies that the map x ~

8lx) is continuous at the points ofthe set L'. Consequently, the set ~' \L' consists of

at most countably many sequences ofthe form 8 f (z±), where Z E C. Moreover, the set

L' is invariant under the transformation cr': (80 81, 82> ... ) ~ 80(81, 82> 83, ... ), which coincides with the shift map multiplied by sign 80.

Thus, by neglecting a countable set of points and identifying points x and y such

that rl <x.y> is a homeomorphism for an n ;;:: 0, we reduce the investigation of the dy

namics of unimodal mappings to the investigation of the symbolic system (L', 0"').

2. Dynamical Coordinates and the Kneading Invariant

In this section, we give a description of the theory of kneading invariants (Milnor and Thurston [1]) for unimodal maps. It suffices to consider unimodal maps J: [-1, 1] ~

[-1,1] such that J(-I) = J(I) = -1. Let x ~ 8/(x) be the correspondence constructed in the previous section. A se

quence 8ix) = (80, 81, 82, ... ) can be associated with the formal power series

Seetion 2 Dynamical Coordinates and the Kneading Invariant

8(x) = L 8Jx)ti.

i=O

41

This power series is called the dynamical coordinate of the point x. The lexicographic

ordering and topology of coordinatewise convergence on {8f (x)} induce the lexico

graphie ordering and topology of coordinatewise convergence on the set {8(x)}. More

over, the correspondence x ~ 8(x) remains monotone and, for any x E I, there exist

8(x+) = lim8(y) and 8(x-) = lim8(y). y~x y~x

The series vf = 8(r) is called the kneading invariant of the map f We have chosen the series 8(r) but not 8(c) because 8(c) = O. However, if CE

Perf, then the map x ~ 8(x) is continuous at the point f(c) and we have 8(r) = 1 +

t8(f(c»). If c is a periodic point with period n, then the sequence 8tCr) is also peri

odic with period n or 2n (depending on the side on which j"(r) approaches the point

c). In both cases, 8(c-) = 1 + t8(f(c»)(mod tn:>. Hence, the series 8(f(c») contains

the same amount of information as the series 8(r) and, therefore, 8(f(c») can also be

chosen as kneading invariant. We also note that 8(r) = - 8(c+). The lemma presented below demonstrates that kneading invariants contain almost

complete information about the behavior of the orbits of maps.

The formal power series 8 is called vradrnissible if, for any n :2: 0, we have either

I cr"(8) I :2: vf or crn(8) = 0, where cr corresponds to the operation of shift. By virtue

of the lemma on monotonicity , the dynarnical coordinate of any point is a v radrnissible

power series.

Lemma 2.2. For any vf-admissible formal power se ries 8, there exists a point

x E I such that 8 is equal either to 8(x), or to 8(x-), or to 8(x+).

Proof. Let x = inf {y I 8(y) ~ 8}. Then 8(x-) :2: 8 :2: 8 (x+). If 8(y) is con

tinuous at the point x, we have 8( x) = 8. If 8(y) has a jump at the point x, then

f'(x)=c forsome n:2:0 and,consequently, cr'(8(x-») = -crn(8(x+») andisequalto

± vf" The series crnc8) is vradmissible and lies between cr'(8(x-») and cr n(8(x»).

Therefore, we have either crn(8) = ± v f or crn(8) = 0, and this implies the required

assertion.

Corollary 2.1. Let fand g be unimodal maps and let cf and c g be their maxi

mum points. lf vf = v g> then there exists an orientation preserving map

i;::O i;::O

such that hof = g 0 h.

42 Elements of Symbolic Dynamics Chapter 2

Proof. For x E Ui~O f-i(cf~ we set hex) = inf {y I G/y) :5: Gf(x)}. As in the

proof ofthe previous lemma, we show that hex) E Ui~O g-i(cg) and hof = go h.

The assertions established above demonstrate that vf contains all information on the

behavior of trajectories except the answer to the following question: Is the map x ~

Gfx) constant on some intervals ?

Lemma 2.3. If the map x ~ Gf(x) is constant on an interval J, then one of the

following possibilities is realized:

(i) there exists an integer n < 00 for which r(J) consists of a single point;

(ii) there exist n;::: 0, k> 0, and an interval L such that r(J) C L, teL) C L, and fkl L is a homeomorphism;

(iii) J is a wandering interval of f, i.e., J, J(J), f2 (J), ... are mutually disjoint intervals.

Proof. If (i) is excluded, then, for any n;::: 0, f(J) is an interval and r( x) =1= c for

XE int (J). In particular, rlJ is a homeomorphism for any n;::: o. Assurne that (iii) is

also not true. Then there are n;::: 0 and k> 0 such that r+k(J) n f'(J) =1= 0 and fk

is a homeomorphic mapping of the interval L = Ui~O r+ik (J) into itself.

Let us now return to the concept of vradmissible series. Let Gf be the sequence

that corresponds to the series vI" Then all elements of the sequence Gf differ from zero

and the inequality I anGfl ;::: Gf is true for all n ~ 0 because the series vf is also vradmissible.

Any sequence a = (ao' ai' a 2, ... ) is called admissible if ao = + 1, a i = ± 1 for

i E N, and I ana I ;::: a for any n;::: o. Thus, for any unimodal map f, the sequence Gf is admissible. On the other hand,

by the intermediate-value theorem (see Theorem 2.6 below), for any admissible sequence

a, there exists a map f such that Gf = a. The structure ofthe set of admissible sequences was investigated by Jonker and Rand

[1].

For a given periodic sequence ß = (ßi' ß2, ... , ßm' ßi' ß2, ... , ßm' ... ) with minimal

period m, we set

The sequence ß(l) is called an antiperiodic sequence ofperiod m. For n> 1, we suc

cessively dehne the sequence ß{n) = (ß{1I--1»(l) with period m2n.

Section 2 Dynamical Coordinates and the Kneading Invariant 43

Any periodic sequence a is admissible if and only if a(1) is admissible. If, for

some periodic sequence a, we have a< ß < a(1), then ß cannot be admissible.

Let 11' be the set of all adrnissible sequences, let P C 11' be its subset of periodic

sequences, and 1et P' C P be the subset ofperiodic sequences that are not antiperiodic. The structure of the set 11' is described by the following theorem:

Theorem 2.1. (Jonker and Rand [1]). Every sequence V E 11'\P is limiting both

for a> v and a< v. Everysequence v E P' is limiting for a> v and isolated for a< v. Every sequence V E P\P' is isolated in 11'. Moreover, anyantiperiodic

sequence is equal to v(k) for same V E P' and belangs to the sequence V > v(1) > V(2) > ... > v(~) generated by v.

Consider two arbitrary sequences a = (ao' a 1, a 2, ... ) and ß = (ßo' ß l' ß 2' ... ).

We say that a is ß-adrnissible if, for any n ~ 0, either I ana I ~ ß or ana = (0, 0, 0, ... ). Due to the existence of one-to-one correspondence between unilateral sequences and formal power series, one can apply the notation and notions introduced for se

quences to power series. Furthermore, by Lemma 2.2, if ß is a formal power series and,

for the map f, one can find a point x such that either 8(x), or 8(x-), or 8(xt) is

equal to ß, then, for any ß-admissible series a, there exists a point y such that either

8(y), or 8(y-), or 8(y+) is equal to a.

Lemma 2.4. If,for same point x, 8(x), 8 (x-), or 8(x+) iseitheranadmissible periodic or an admissible antiperiodic series of period n ~ 1, then f possesses a periodic point ß of period n. If f possesses a periodic orbit of period n, then there is a point x such that one of its se ries 8(x), 8 (x-), or 8(xt) is either an admissible periodic or an admissible antiperiodic series of period n.

Proof. First, we prove the second statement. Let ß be a periodic point of period n

and let B be the cyc1e that contains ß. If the point of extremum c belongs to B, then

we set x = c. If c ~ B, we consider a point ßo E B such that f(ßo) ~ ß1 for any ß1 E B

and assume that x coincides with a (unique) point ßü such that ßü < c and f(ßü) f(ßo)' In both cases, x is the required point.

Now assume that, for some point x, one ofthe series 8(x), 8(x-), or 8(x+) is ei

ther admissible periodic or adrnissible antiperiodic with period n ~ 1. Denote this series

by a. If x ~ Ui~O f-i(c), then a = 8(x) and one can easily show that ro(x) is a pe-

riodic trajectory ofperiod n. If x E Ui~O f-i(c), then the adrnissibility of a implies

that x = c, a = 8(x-), and c is a periodic point of period n. Let /-lek) be the maximal admissible series in the lexicographically ordered collec

tion of adrnissible periodic (but not antiperiodic) series ofperiod k. Thus, in particular, the following adrnissible sequences occupy the first positions in

the indicated lexicographically ordered collection (here, we write only the relevant signs

instead of ± 1):

J.l(20) = (+++ ... ),

J.l(2OO ) = (J.l(20))(00) = (+--+-++--++-+--+-+ ... ).

Lemma 2.5. The following sequence is lexicographically ordered:

> ... > J.l(2· 7) > J.l(2· 5) > J.l(2· 3) > ... > J.l(7) > J.l(5) > J.l(3).

To prove this assertion, it is necessary to determine the sequences J.l(2n . k). Let k =

2i + 3, i;::: 0. Then J.l(2n . k) is generated by the periodic replication of the finite chain

a(2n ·k) oflength 2n ·k. For n=O, wehave a(k)=a(2i+3)=(+--+-+- ... +-),

where the pair (+-) isrepeated i times. For n;::: 1, weget a(2n ·k) = a(2n-1k)·(+-),

where the chain on the right-hand side is obtained from the chain a(2n-1 k) by replacing

every sign with (+ -) or (- +) depending on the sign to be replaced. The statements established above imply the following assertion about the coexistence

of periods of cyc1es for unimodal mappings:

Corollary 2.2. Assume that the natural numbers are arranged in the following or

der: 1 <I 2 <I 4 <I ... <I 2·7 <I 2·5 <I 2·3 <I ... <I 7 <I 5 <I 3. If a unimodal map f has a cycle ofperiod n and k <I n, then it has a cycle ofperiod k.

In the next chapter, we prove this assertion for general continuous maps.

3. Periodic Points, ~-Function, and Topological Entropy

In this section, we establish the re1ationship between the kneading invariant and the

number of extrema Yn, the number of the intervals of monotonicity ln' and the number

lln offixed points of the map f".

Seetion 3 Periodic Points, s-Function, and TopoZogicaZ Entropy 45

We set 'Yn to be equal to eard {f-n(c)} and eonsider the power series 'Y f =

L~=o 'Yn tn . Similarly, let Zn be the number ofthe intervals of monotonieity of rand

let If = L~=o Intn . Sinee In = 1 + L~:~ 'Yk' we have If = (1 + t'Yf)/(l- t).

If fk has finitely many fixed points for any k:2: 1, then the funetion S ean be defined as the formal series

s = exp {L Tlk l}, f k~1 k

where Tlk = eard {xl fk(x) = x}.

The eontribution of every orbit of period p to Sf is L:o t ip = (1 - tl)-I. There

fore, Sjl = TIß (1 - t Aß), where the produet is taken over all periodie orbits.

The following definitions enable us to establish the relationship between the s-fune

tion and the kneading invariant: Points x and y are called monotonically equivalent

with respeet to fk whenever fk is a homeomorphic mapping of <x, y> onto <x, y>.

Let TJ k be the number of equivalenee classes of points with period k and let

~ = exp {L TJk l}. f k~1 k

Lemma 2.6. The following equalities hold:

where Q(t) == whenever v f is nonperiodic, and Q(t) = 1 - t P if v f is periodic

with period p.

Proof. We prove only the first equality. Consider

Let x be a point of the set f-{n-i)(c). Then f(n-i\x) = c and fn+-' possesses a loeal mi

nimum (maximum) at the point x if and only if Vi = 1 (or Vi = -1). This implies that

LO~i~n 'Y n-i Vi is equal to the differenee between the numbers of maxima and minima

of j'lf-l. Therefore, this sum ean be equal to + 1, 0, or -1. Thus, it is equal to zero if

and only if fnf-l either simultaneously inereases or simultaneously deereases at the ends of the interval.

Let J = (a, b) and let f1l+-1 : J -t IR. Then the differenee between the numbers of

maxima and minima of f1l+- 1 1 J is equal to t (8 n(a+) - 8n(b+»). Therefore, v JYr(J) =

tC8(a+) - 8(b-». Forthemap f: [-1, 1] -t [-1,1], wehave fe-I) =f(I) = -1,

and V/YJ= 1 +t+t2 + ... = (I-t)-I.

Consider some examples of finding v l' 11' and ~ J for maps from the family h.. (x) = /...x(I-x), XE [0, 1], A E (0,4], eorresponding to examples presented in Seetion 1 of Chapter 1.

1. 0< A ::; 1 (see Fig. 3). In this case, we have

8 h. (r) = (+ + + ... ),

A -I (~JJ =I-t,

2. 1 < A ::; 3 (see Fig. 4). It is neeessary to eonsider the following two subeases:

(i) 1 < A ::; 2. The map fA has two equivalent fixed points. Here, 8 A (r), v JA' 1 A'

and ~JA are as in the previous example but (~A)-I = (1- t)2 t: (~JTI.

(ii) 2 < A ::; 3. As A passes through the point 2, the sign of the multiplier of one of the fixed point ehanges and fixed points beeome nonequivalent. Here,

3. 3 < A ::; 1 + -J6 (see Fig. 5):

(i) 3 < A ::; 1 + {5. The map h.. has two nonequivalent fixed points and the eycle of period 2. Both points of this eycle are monotonieally equivalent to one of

Seetion 3 Periodic Points, s-Function, and Topological Entropy 47

the fixed points. Here, 8 h,.(c-), v fi. y 1-,; and ~/, are the same as in case 2(ii)

but

(ii) 1 +.J5 < A, ::; 1 + .J6. As A, passes through the value 1 + .J5, the multiplier of the 2-periodic cycle becomes negative. This results in the doubling of the period of kneading invariant and the points of 2-periodic cycle become nonequivalent to the fixed point. In this case,

8h,.(c-) = (+--++--+ ... ),

1-t v., =--J;" 1 + t 2 '

4. A, = 3.83 (see Fig. 8). For this value of A" the map h .. has two nonequiva1ent cycles of period 3 and the kneading invariant is antiperiodic with period 3. Moreover,

8/,(c-) = (+---+++-- ... ),

1 + t'

1 + t 2 y~, = --2 (S.,)-l = (Ö-1 = (l-t)(1-t-t 2)(1-t\

J',.. (l - t)' J',..

5. A, = 4. In this case (see Fig. 9),

1- 2t vI, = 1-t-? + ... = --,

~ 1- t

The importance of Lemma 2.6 is corroborated by the following assertion:

48 Elements oJ Symbolic Dynamics Chapter 2

Corollary 2.3. The power series V I is convergent Jor all t. The power se ries "tl

and II are meromorphic and convergent inside the circle I tl < r(f), where r(f)

is the smallest positive real zero oJ the Junction (1 - t) V /I).

Proof. Since vI = Li~O Vi ti, where Vi = ± 1, the series vI is convergent for all

with Itl<1. Theequalities vt'YI=(I-t)-1 and ll=(l +t"tI)/(I-t) implythat II

and "tl are meromorphic and their poles in the circle I tl < 1 coincide. It follows from

the positivity of the terms of the series "tl and Abel' s theorem that the radius of conver

gence r(f) of "tl isapoleofthefunction "tlt) if r(f» 0. If r(f)< 1, then r(f) is

the smallest positive (real) zero of the function V /I). Corollary 2.3 and the inequality l(f 0 g) ~ l(f)l(g) imply that

lim (l;)lIn = lim ("tJlIn = _1_ n--7= n--7= r(f)

The number s(f) = r(~) is called the growth exponent of the map fItis obvious that,

for unimodal maps, we have 1 ~ sU) ~ 2. Let h(f) be the topological entropy ofthe map f Misiurewicz and Szlenk [1] pro v

ed that h(f) = lim 1 log "t n" Hence, sU) = exp (h(f) and, consequently, h(f) E [0, n---7 00 n

log 2] for unimodal maps.

Theorem 2.2 (Milnor and Thurston [1]). The mapping h: J ~ h (f) is continu

ous in the space oJ unimodal Cl-maps endowed with Cl-topology.

Proof. First, we assume that the critical point of J is not periodic. Then the de

pendence ofthe series vI = Li~O Viti = 8 (c-) on J is continuous. Moreover, the Cau

chy integral theorem implies that the smallest zero of vI continuously depends on f, but

this means that entropy is continuous. Now 1et c E Per J and let the period of c be equal to n. In this case, any unimodal

map g sufficiently close to J in Cl possesses an attracting periodic orbit that contains

a (unique) critical point in its domain of immediate attraction. Let V g = Li~O Vi t i.

Then the sequence v g is either periodic or antiperiodic with period n. Thus, v g can be

written in the form

Inside the circle I tl < 1, the poles of these two functions coincide. Therefore, the entropy of the map g is equal to h(f). Theorem 2.2 is proved.

Section 4 Kneading Invariant and Dynamics oJ Maps 49

Consider the space ofunimodal maps with cl-topology. To define F: [0, 1] ~ [0,

1], weset F(x)=" x forxE [0, tl and F(x) = 3(I-x) for XE [t, 1]. LetA(x)=

AF(x), A~ 1.

By using the equality h(f'}) = lim ~ log 'Yn(A), we can show that heft) = 0 for A = 1. n ____ =

For A> 1, we consider the map J~ on the interval [ß', ß], where ß = ß(A) is the nonzero fixed point of the map JA! ß' t:- ß, and A(ß') = A(ß). In this interval, the map

g = J~ is unimodal and expanding (moreover, I (f~)'(x) I > 3 for all XE [ß', ß]).

Therefore, L (gn) ~ 2n and h (g) ~ log 2. Consequently, hUf) ~ ! log 2 for any A> 1.

At the same time, for unimodal maps with positive topological entropy, continuous changes in the map induce continuous changes in entropy. More precisely, Misiurewicz [5] proved the following statement:

Let V O be the space oJ unimodal maps endowed with CO-topoLogy. Then the map

h : J ~ h (f) is continuous at a point Jo oJ the space UO whenever h(fo) > O.

4. Kneading Invariant and Dynamics of Maps

For mappings J: I ~ I whose exponents of growth s(!) are greater than 1, one can

construct piecewise linear models. Consider a function Lf(5} / Lf' where

= L/J) = L I (rl J) tl>-I, Lf = lJI)·

n+1

It follows from the results of the previous seetion (see Corollary 2.3) that this function is

meromorphic in the circle I t I < 1 and satisfies the condition L f(J) / L f :s; 1 for t > O.

Hence, LlJ) / L f possesses a removable singularity at t = r (f). We define

A(J) = lim Lf(J). Hr(f) Lf

It is easy to show that

(i) if J1 and J2 have a common end, then

(ii) if J does not contain points of extrema, then

A(f}) = s(!)A(J);

(iii) lim A(J) = O. IJI~o

Let I = [0, 1]. We set A(X) = A([O, x]). Then there exists a unique map F: [0,

1] ~ [0, 1] that satisfies the condition F 0 A = A 0 f The map F is piecewise linear,

its derivative is equal to ± se!), and the number of the intervals of monotonicity of F

does not exceed the number of the intervals of monotonicity of f The function A(X) may be not strictly monotone. Thus, if 8(x) = 8(y) and y =1= x,

then A maps the interval <x, y> into a point. The map x ~ A (x) is called the semiconjugation of 1 to F.

The kneading invariants of 1 and the piecewise linear model

F - 2 !SX, XE [0, .!.], s - 1

s - SX, XE [2' 1],

where s = s (f), are connected as folIows: If A -I ({ t }) is a point, then v f = V Fs If

A -I ({ t }) = J is an interval, then v Fs is periodic with period n. In this case, r(J) C J and g = rlJ is a unimodal map for which it is possible to introduce the kneading in

variant v g-

Let 8 Fs = (808182 ", 8n-1' 8081 ", 8n-1' ... ) and 8 g = a = (ao' al' a 2, ... ). Then

8 f satisfies the equality

In terms of kneading invariants, we can write

In the general case where v(t) is an admissible periodic series of period n and a(t) is admissible, we define v * a(t) as the following admissible series:

v * a(t) = (1 - tll) v(t)a(tll).

For v E TI', we set s(v) = sC!), where 1 is an arbitrary map such that v f=v. We have

the following theorem on decomposition of v into a product of irreducible factors:

Seetion 4 Kneading Invariant and Dynamics 01 Maps 51

Theorem 2.3. Let V E n' and sI = s(v). lf SI = 1, then V = Jl(2 i ), 0 ~ i ~ 00.

lf sI> 1, then V admits one 01 the 10110wing decompositions:

(i) there exist S2' S3' ... , Sm* E (1,2] and a E n', s(a) = 1, such that

V = VFi * VF~ * ... * VFfn * a,

. d' fi 111< < * d I/P2 II Pm' (h VF~, areperlO IC ora _m_m an sI>s2 > ... > sm' ere,

m-I

Pm = rr nk k=I

and nk is the period 01 V F ), and a = 11(2i), i ~ 00; Sk

(ii) V = VF * ... * VF , where the series VF with 1 ~ m ~ m* -1 have the i·· • same properties as in (i) and VF is a periodic series;

'm,

(iii) V = lim VF * ... * VF, , where all V F. have the same properties as in (i) m*--700 SI 'm+ -m

d 1· II Pm' 1 an m*l~~ sm' =.

For a unimodal map 1, the decomposition of V f in Theorem 2.3 corresponds to the

spectral decomposition of the set NW (f) (see Theorem 2.5). The theory of kneading invariants takes the most perfect form for maps with negative

Schwarzian. For 1 E C\I, I) with l' (x) *- 0, the Schwarzian (or Schwarzian derivative) SI is defined by the equality

S x = 1'''(x) _ ~ (f"(X»)2. 'j() 1'(x) 2 1'(x)

Let SV denote the set of unimodal maps 1 such that Sf(x) < 0 for all x except c (in what follows, maps of this sort are studied in detail).

Theorem 2.4 (Jonker and Rand [1]). Let V f = vg 10r f, g E SV.

(i) 11 the series v f is not periodic, then 1 and gare topologically conjugate.

(ii) 11 V f is periodic with period n, then both 1 and g possess an attracting or

neutral periodic trajectory 01 period n 0 r n /2; moreover, 1 and gare topo-

logically conjugate whenever these trajectories are of the same type, i.e., either both attracting or both neutral, and the corresponding points of these trajectories have the same dynamical coordinates.

By applying Theorem 2.4 to maps with negative Schwarzian, we arrive at the follow

ing decomposition of the set NW (f).

Theorem 2.5 (Jonker and Rand [1]). For fE SV, the set NW (f) admits the following decomposition

NW(f) = U Qffl' m*::; 00,

l";m";m*

where Qm are closed mutually disjoint invariant sets such that

(i) for 1::; m < m*, the sets Qm are representable in the form Qm = Pm U Cm'

where Pm consists of finitely many periodic orbits and f I Cm is topologically

conjugate to a transitive topological Markov chain; moreover, Qm are hyper

bolic sets;

(ii) for m = m*, one ofthefollowing cases is realized: if m* < 00, then

(a) Qm* is the same as Qm with m < m* but Cm* contains a periodic trajec

tory whose derivative on the period is equal to + 1 (this corresponds to case (i) ofTheorem 2.3, a = (+ + + ... ));

(b) Qm* is the union offinitely many repelling periodic orbits ofperiods Pm*2i,

0::; i < n, and an attracting periodic orbit of period Pm*2" (case (i) of The

orem 2.3, a = 11(2"), n::; 0);

(c) Qm* is the union ofrepelling periodic orbits ofperiods Pm*2i, 0::; i < 00,

and a minimal invariant set equal to orb (c) (case (i) of Theorem 2.3, a =

11(2=)) ;

(d) Qm* is the union of finitely many repelling periodic orbits and finitely many

intervals (cyclically permuted by the map f) such that fPm. is topologically conjugate on these intervals to a piecewise linear map (case (ii) of Theorem 2.3).

If m* = 00, then Q= = orb (c) is a minimal invariant set (case (iii) of Theorem 2.3).

Seetion 4 Kneading Invariant and Dynamics of Maps 53

In all cases, the domain of attraction of the set n m*, Le., P (nm*' f) = {x E I I (f) f(x)

c n m *} is a set ofthe second category and mesP(nm"f) = mesI, where mes denotes the Lebesgue measure of the corresponding sets.

For the exponent of growth of the farnily of piecewise linear maps Fs' SE [1, 2], we

have s(Fs) = s. The kneading invariant monotonically changes from 1 + t + t 2 + ... = (l

- t)-I for s = 1 to 1 - t - t 2_ ... = (1 - 2t)(1 - t)-I for s = 2. This is a consequence of the following result of Jonker and Rand [1]:

If V > v in the sense of lexicographic ordering, then s(v):S; s(v). Moreover, V F s

does not take all admissible values, e.g., V F '# !!(2i ) for any s, i E N. s

At the same time, this is impossible for the farnilies of smooth maps. In the case of smooth maps, we have the following intermediate-value theorem for

kneading invariants:

Theorem 2.6 (Milnor and Thurston [1]). Let {fS}SE [0.1] be afamily of Cl-dass

maps which continuously depend on s in the CI-topology, let V fo > v fl' and let

a E n satisfy the inequalities v f o > a > v f 1' Then there exists So E [0, 1] such that

a=vfsa'

3. COEXISTEnCE OF PERIDDIC TRßJECTORIES

Dynamieal systems generated by eontinuous maps of an interval into itself are eharaeterized by the following important property: The data on the relative loeation of points of a single trajeetory on the interval I may eontain mueh information about the dynamieal system as a whole. Clearly, this is explained by the faet that the phase spaee (the interval 1) is one-dimensional. The points of a trajeetory define a deeomposition of the phase spaee, and information on the mutualloeation of these points often enables one to apply the methods of symbolie dynamies. These ideas are espeeially useful for the investigation of periodie trajeetories.

As already shown, the existenee of eycles of some periods implies the existenee of eycles of other periods. At present, the problem of eoexistenee of eycles is fairly weH studied and there are numerous papers dealing with this problem. Many important results on the eoexistenee of eycles were established for eontinuous maps of a eircle, of one-dimensional branehed manifolds, and some other classes of topologie al spaees. In this ehapter, we present the most important faets established for eontinuous maps of an interval into itself.

1. Coexistence of Periods of Periodic Trajectories

First, we present several simple assertions.

If a map fE CJ (/, 1) has a eycle of period m > 1, then it also possesses a fixed point. Indeed, if ß' and ß" are the smallest and the largest points of this eycle, respeetively, then f(ß') > ß' and f(ß") < ß" and it follows from the eontinuity of the fune

tion f on [ß', ß"] that f(ß) == ß forapoint ß E [ß', ß"]· In what foHows, (a, b) denotes a closed interval with ends at a, b E lR. This nota

tion is eonvenient in the ease where the relative loeation of the points a and b is unknown or inessential.

Lemma 3.1. A map f has a cycle of period 2 ~ there exists a point a E I

suchthat a*f(a) and aEf(a,f(a»).

55

56 Coexistenee of Periodie Trajeetories Chapter 3

Prooj. For any periodic point a of period 2, we have a"* f(a) and a E f «a,

f(a»). Therefore, it remains to prove the converse assertion.

For definiteness, we assume that f(a) > a. Then there exists a point a' E (a,/(a)]

such that I'(a) == a. If a' ==f(a), then a is a periodic point with period 2. FOLa' < f(a), we have only two possible cases:

(i) there are fixed points for x> a';

(ii) there are no fixed points for x> a'.

We consider each of these possibilities separately.

(i) Let b be the smallest fixed point in the interval {x> a'}. Since f([a, a']):::l [a,

a'], there are fixed points in the interval [a, a']. Let b' be one of these points. Since f([a', b]):::l [a, a'], one can find a point e E ra', b] such that fee) == b'. Hence,

f2 (a') > a' and f2( e) == b' < e and, consequently, there are periodic points of period 2

in the interval [a', e ].

(ii) Consider f2. Since the interval I is mapped into itself, there exists a point d:2:

a' such that f2( d) ~ d. Moreover, f2( a') > a' and there are no fixed points of f in the interval [a', d]. Therefore, the interval [a', d] must contain a periodic point of period 2.

Lemma 3.2. If a map has a eycle of period m > 2, then this map has a eycle of

period 2.

Prooj. Let B be a cyde of the map f of period m and let ßo == max {ß E B I f(ß) > ß}· It is dearthat ßo E f([ßo,f(ßo)]) and it remains to apply Lemma 3.1.

Corollary 3.1. If a map f has a eycle of period 21 for 1:2: 0, then f has ey

cles of periods 2i, i == 0, 1, ... , 1- 1.

Corollary 3.2. If a map f has a eycle of period "* 2i, i == 0, 1, 2, ... , then f also

has eycles of periods i, i == 0, 1,2, ....

In order to prove that f has a cyde of period 2n, it suffices to apply Lemma 3.2 to

the map g = f2n -I. Thus, in the case of Corollary 3.2, the map f has a periodic point of

period 21m with odd m and 1:2: 1. For the map g, the period of this periodic point is

greater than 2 (namely, it is equal to 21- n+ 1 m for n ~ land to m whenever n> I).

According to Lemma 3.2, the map g possesses a periodic point of period 2 which is ob

viously a periodic point of period 2n for f

Section 1 Coexistence of Periods of Periodic Trajectories 57

Actually, we have the following theorem (Sharkovsky [1]):

Theorem 3.1 (on coexistence of cycles). lf a continuous map of the interval onto it

self has a cycle of period m, then it also has cycles of any period m' such that

m' <l m, where

<l 2·7 <l 2·5 <l 2·3 <l ... <l 9 <l 7 <l 5 <l 3.

Moreover, for any m, there exists a map with cycle of period m and no cycles

ofperiods m' if m <l m'.

(Maps of this sort are studied in Section 2 of this chapter).

Apart of the statement of Theorem 3.1 (concerning the existence of cycles of periods

2i, i = 0, 1,2, ... ) is contained in Corollaries 3.1 and 3.2.

At present, there are several known versions of the proof of Theorem 3.1 (Arneodo, Ferrero, and Tresser [I], Block, Guckenheimer, Misiurewicz, and Young [1], Burkart [1], Guckenheimer [1], Ho and Morris [1], Jonker [1] Shapiro and Luppov [1], Sharkov

sky [1]). Here, we present a proof based on the use of symbolic dynamics and properties of cyclic permutations (a cyclic permutation of length m is defined as a map of the set { 1, 2, ... , m} onto itself which has no invariant sub sets other than {I, 2, ... , m} ).

Every cycle can be associated with a cyclic permutation 1t, a transition matrix, and (or) an oriented transition graph. The investigation of these objects gives vast information on properties of a dynamical system as a whole.

Ifa cycle B consists of the points ß1 < ß2 < ... < ßm and f(ßi) = ßS? 1 $ Si $m,

i = 1, 2, ... , m, then

( 1 2 ... m) 1t = sl s2 ... sm .

Due to the continuity of the map f on the intervals Ji = [ßi' ßi + 1]' i = 1, ... , m - 1, we can write

fC!;) ~' ,+1 lJs. U ... U Js -1'

JSi+1 U ... U JSi_ l ' if Si > si+1.

In this case, we say that Ji covers (or f-covers) the corresponding intervals JS ( This

cycle can be associated with a matrix { J.1 is} of admissible transitions (of points of the

intervals :J;), where

58 Coexistence of Periodic Trajectories Chapter 3

jO,

Ilis = 1, if f(Ji ) => J s '

and with an oriented transition graph with vertices J1, ••• , Jm- I and oriented edges that

connect J; and Js if f(J;) => Js · For convenience, we write J; --+ Js if fU;) => Js (i.e.,

in the case where J; f-covers 1). In what follows, the transition graph is called the B

graph of a cycle. Thus, the map displayed in Fig. 21 has a 3-periodic cycle formed by

the points ßI' ß2, and ß3· For this cycle, we have 1t = (~ ~ 1)' f(Jd => J2, and

f(J2) => J1 U J2· Hence, the transition matrix has the form (? l) and the B-graph is

depicted in Fig. 22.

Fig.21 Fig.22

By analyzing the transition matrix or the B-graph of a map, one can show that the map possesses periodic trajectories of various periods. Thus, if, e.g., we use symbols aj,

a2, ... , am-I as an alphabet, then any symbolic sequence ar1 ar2 ... arprjtl ... Cl ~ rj ~ m - 1) admitted by the transition matrix (Ilrr 1 = 1 for all j = 1, 2, ... ) corresponds to

1J+

(at least one) trajectory of the system which passes through the intervals Jl'···' Jm - I in

the following order: Jr --+ Jr --+ ... --+ Jr. --+ Jr. --+ .... In particular, if the symbolic 1 2 1 J+ 1

sequence is periodic with the smallest period n, then the system has at least one periodic

trajectory of period m (which passes through the intervals JI, J2' ... , Jm- I in the indicated order).

This fact is a consequence of the following simple geometric lemma, which allows us to pass from the intervals covering each other to periodic points:

Lemma 3.3. 1. If there exists a closed path Jr ~ Jr ~ ... ~ Jr ~ Jr (1:5 o 1 n -I 0

ri :5 m - 1), then there exists a periodic point 13 such that

i(ß) E Jr .. i = O,l, ... ,n-l, r(ß) = ß.

2. Furthermore, if n is the smallest period of the sequence ro, rl,'" , r n--I, ro, rl,'" and 13 ~ B, then the period of 13 is equal to n.

Proof. There exists a c10sed interval I' C Jr such that i (l') C J" i = 0, 1, ... , o I

n - 1, and rU') = Jr . Therefore, there exists a point 13 E I' for which r(ß) = ß. o

We prove the second assertion. Assurne that the period of 13 is n' and n' < n (dear-

ly, in this case, n' is a divisor of n). The condition ß ~ B means that the points 13, f (ß), f2 ( 13 ), ... lie in the interior of the intervals JI , J2, ... , Jm -I' Since the interiors of

these intervals are mutually disjoint and i(ß) = i+h(ß) for i = 0, 1,2, ... , the inter

vals Jr and Jr • coincide whenever 1 i' - i 1 = n'. Hence, the sequence ro, r I' ... , I I

r n-I' ... is periodic with period n' < n but this contradicts the assumption of the lemma.

Remark 1. It is obvious that the condition 13 ~ B is not necessary in the case where

m is not a divisor of n. If m is a divisor of n, this condition is essential. For any m > 2 and any n divisible by m, which is regarded as the (smallest) period of a sequence ro,

r I, ... , r n- I' r 0' r I' ... , one can always indicate a map such that the periodic point b

from the condition ofLemma 3.3 belongs to the cyde B (and, hence, is m-periodic).

Lemmas 3.4 and 3.5 presented below deal with the properties of cyclic permutations.

Let 1 i l , i2 1 denote the segment ofthe sequence ofnatural numbers lying between i l

and i2, i.e., the set {i E N 1 i l :5 i:5 i2 }. Segments 1 i l , i2 1 with i2 = il + 1 are denot

ed by 1 i I' * I. Any permutation 1t of length n generates an operator Alt acting on the

segments 1 i l , i2 1 eil, n 1 as follows:

Alt 1 i l , i2 1 = 1 min 1t(i), i E lil • i21

max 1t(i)l. i E lil • ;21

Thus, for a map fE cO U, I) with a periodic trajectory 131 < 132 < ... < ßn of type 1t,

the action of the operator Alt has the following sense: If Alt 1 i l , i2 1 = 1 i{, i~ I. then

f([ ßi ' ßi ]):::> [ ßi', 13,"]' The operator Alt pos ses ses the following obvious properties: 1 2 1 2

Property 1 follows from the fact that n is a one-to-one map. Property (ii) is a conse

quence of the absence of proper n-invariant subsets of the segment 11, n I. In what follows, we use the notation I iJ, i2 1 ~ I i{, i2 i, which means that Alt I i I'

i2 1 :::l I i{, i2 I· Also let ~ be the operator Alt applied k times, i.e., Arr 0 ••• 0 Arr.

Lemma 3.4. Let n be a cyclic permutation of length n > 2. Then

'------v---' k times

(i)onecanindicate iOE II,n-II and kE {1, ... ,n-2} such that lio,*lc

Alt I io, * I c ... c A~ I io, * I = 11, n I;

(ii) for any i J Eil, n - 11 other than io, there exists a set of distinct elements ij ,

2'5,. j '5,. r '5,. k, of the set 11, n - 11 such that I io, * I ~ I i" * I ~ I ir-I' * I ~ ...

~ I i j , * I~ ... ~ I i l , * I·

Proof (i). Let iOE max{iE II,nlln(i»i}. Itisclearthat ioE II,n-II be

cause n(l) > land n(n) < n. Since n(io+I) < io+I '5,. n(io), wehave lio,*1 ~

Alt I io, * land, consequently, I io, * I ~ Alt I io, * I ~ ... ~ Aj{1 io, * I ~ ... , j = 1,2, ....

Let k = min {j I Aj{ I io, * I = Aj{+ J I io, * I}. Property (ii) of the operator Alt implies that

~lio,*1 = II,nl. Since lio,*1 *II,nl, wehave k<::l, cardlio,*1 = 2, and,the

refore, card Aj{ I iQ' * I <:: 2 + j, 0 '5,.j '5,. k. Hence, k'5,. n - 2.

(ii) Since io * iJ and I il> * I eil, n I, one can find h such that Aj{1-11 io, * 11:J

I i J , * I and A~I I io, * I :::l I iJ, * I· This implies the existence of an element i2 such that

I i2 , * I c Aj{1-JI io, * I and I i2, * I ~ I iJ, * I. Given i2, by applying the same procedure,

we choose an element i3. Since Aj{+ll io, * I :::l Aj{ I io, * I, by repeating the same argu

ments r times, r< k, we arrive at the element io. The fact that the elements i j, 0 '5,.

j'5,. r, are distinct is a consequence of the fact that, in each step, we choose an element of

the set Aj{+l I io, * I \ A~ I io, * I.

Let 2r be the set of cyclic permutations n with the following property: The lengths

n of all n are larger than 2 and one can indicate an element i* Eil, n I such that i* and n (i*) simultaneously belong either to the segment 11, io I or to the segment

lio+l,nl, where ia = max{iE Il,nlln(i»i}.

Lemma 3.5. Let n be a permutation of length n from the set 21. Then

(i) there exists a collection of distinct elements i j of the set 11, n - 1 i, 0 '5,.j '5,. r,

1 '5,. r '5,. k, such that the diagram depicted in Fig. 23 is realized;

(ii) there exist elements i\ < i2 < i3 of the set 11, nl such that the diagram

depicted in Fig. 24 holds for the operator ~.

Fig.23 Fig.24

Proof. (i). Let io = max {i E 11, n Iln(i) > i}. As follows from statement (i) of

Lemma 3.4, 1 io, * 1 ~ 1 io, * I. It is c1ear that at least one of the sets {i Eil, io Iln(i) E

11, io I} or {i E 1 io + 1, n " n(i) E 1 io + 1, nl} is nonempty. For definiteness, we assume that this is true for the first set. We set i\ = max{iE 11, iolln(i)E 11, iol}.

Then n(i\):S; io < io + 1 :s; n(i\ + 1), i.e., 1 i\, * 1 ~ 1 io, * I. To complete the proof, it suffices to apply statement (ii) of Lemma 3.4.

(ii). Let {i E 11, io Iln( i) E 11, io I}:;/: 0 (the case where this set is empty is inves

tigated analogously). As in (i), we assurne that i\ is the maximal element of the indi

cated set and i2 is such that

n (i2) max n (i), i E {i!. i2 }

and i3 = io + 1. Obviously, i J < i2 < i3.

There exists an element i E 1 io + 1, n (i2) 1 such that n (i) :s; i J' Indeed, due to the

choice of io, iJ , and i2, we have An 1 i j + 1, io 1 Cl io + 1, n (i2) 1 and n (i) < i for all

i E 1 io + 1, n(i2) I. Therefore, ifthere are no such l, then An 1 i J + 1, n(i2) 1 ~ 1 i\ + 1,

n (i2) I. But this contradicts Property 2 of the operator An.

Let us now check the required inc1usions. It follows from the inequalities n (i j ) :s;

io < i :s; n(i2) that Anl i j , i2 1 ~ {io, I}. Since n(i):s; i j < i3 :S; n(i2), we have ~ 1 i j ,

i21 ~ 1 i j, i3 I. Similarly, the inequalities n (i3) :s; i2 < 1 :s; n (i2) imply the inc1usion

~ I i2, i3 1 ~ I i j , i3 1·

Remark 2. The set U contains permutations such that it is impossible to add any edge to the graph depicted in Fig. 23 (e.g.,

62 Coexistence of Periodic Trajectories

(1 2 3 4 5 6 7 8 9) 1t= 469875321'

Chapter 3

where io = 5, il = 1, i2 = 8, i3 = 3, and i4 = 6).

Lemma 3.6. If a map possesses a cycle of odd period m, m > 1, then it has cycles

of all odd periods greater than m and cycles of all even periods.

Proof. If the period of a cycle is odd and greater than one, then the corresponding cyclic permutation belongs to ~. Hence, the assertion ofLemma 3.6 follows from Lemmas 3.5 and 3.3.

Lemma 3.6 yields the remaining part of the proof of Theorem 3.1: If a map possesses

a cycle of period 21 (2k + 1), k ~ 1, then it has cycles of periods 2 1 (2r + 1) and 21 + I s

with r > k and s ~ 1. Indeed, if the map f has a cycle of period 21 (2k + 1), then the

map f21 has a cycle of period 2k + 1. It follows from Lemma 3.6 that f21 has a cycle

of period 2s for any s ~ 1 and, consequently, f has a cycle of period 21 + I s. Moreover,

Lemma 3.6 implies that the map f21 has a cycle of period 2r + 1 whose points are peri

odic with period 21(2r + 1) or 21' (2r + 1), l' < 1, for f In the latter case, the existence

of a cycle with period 21 (2r + 1) follows from the already proved part of the theorem. This completes the proof of Theorem 3.1.

The theorem on coexistence of cycles guarantees the existence of cyc1es of any peri

od m' <] m when the map has a cycle of period m, but this theorem contains no infor

mation about the number ofthese cycles. For m = 21, there are maps (e.g., Iv * x(1 - x))

which have a single cycle of period m' <] m. However, this is not true for m * 21, I = 0, 1,2, .... Numerous papers (e.g., Bowen and Franks [1] and Du [1]) are devoted to the

estimation of the lower bound of the number of cyc1es of period m' <] m.

We now present another formulation of Theorem 3.1. Let Pm = {fE CJ (I, I) I f has a cyc1e of period m}. Obviously, CJ (I, I) = PI'

Theorem 3.2 (on the stratification of the space of maps). If m' <] m, then P m ~

Pm" i.e., PI ~ P2 ~ P4 ~ ... ~ PlO~ P6 ~ ... ~ P9 ~ P7 ~ Ps ~ P3 and all inclu

sions are strict.

Note that there are maps fE Pm \ Pm' of arbitrarily high smoothness (inc1uding analytic maps). The corresponding examples can be constructed by analogy with maps from

the c1ass Cl (I, I) indicated above.

The theorem on coexistence of cycles deterrnines the periods of cyc1es of the map f in the case where f has a cyc1e of period m. Is it possible to say anything about cyc1es

of maps that are close to f? Is it possible to establish any (lower and upper) bounds for

the periods of cycles of maps that are Cr -c1ose to f, r ~ 0 ?

If a map f has a cycle of period m, then it is possible that a map arbitrarily Cr-close to f (for any r ~ 0) has no cycles of period m (for any m). This may happen, e.g., in the case where the existence of a cycle ofperiod m is guaranteed, say, by the tangency

of the curves y = fn(x) and y = x (the corresponding examples can be constructed quite easily). Nevertheless, the following theorem establishes a "lower" bound for the periods of cycles of maps close to f (Block [3]).

Theorem 3.3. If a map f has a cycle of period m, then there exists a neighbor

hood U c ~ (l, I) ofthe map f such that U C Pm' for any m<l m.

At the same time, it is impossible to establish an "upper" bound of the periods of cy

cles in ~ (l, I). Actually, the maps with cycles of all periods are dense in ~ (l, I). In

deed, if I is a bounded closed interval, then any map f: I ~ I has a fixed point. If ß

is a fixed point, then, for any E > 0, one can find Ö > ° such that If(x) - ß I < E for

I x - ß I < Ö. Hence, it remains to replace f(x) on the interval (ß - ö, ß + Ö) by any Eclose map with cycles of all periods, preserving continuity. Thus, one can choose Ö' <

min{E,Ö} andset j(x) = ß+Ö / -2Ix-ßI for Ix-ßI<ö /.

For any map j, there exists a neighborhood in ~ (l, I) that consists of maps with cycles of all periods. Therefore, the collection of maps that have cycles of all periods

(i.e., P3) has an open dense subset contained in ~ (I, I).

Hence, for any m, the set Pm contains an open dense subset of CJ (l, /). It follows from Theorem 3.3 that the set ofmaps with cycles ofperiods "* 2i, i = 0,1,

2, ... (i.e., Um,d Pm) is open in ~ (l, I). Therefore, this set is open and dense in

~(l, I). It should be noted that maps with cycles of all periods cannot be constructed in a

similar way by using Cl-perturbations because these maps are not dense in Cl (l, I).

Theorem 3.4 (Sharkovsky [17]). If f ~ Cl (l, I) and fE Pm' then there exists

a neighborhood U C Cl (l, I) ofthe map f such that

(i) U n Pm = 0 if m =F- 2i , i = 0, 1,2, ... ,

In particular, it follows from Theorem 3.4 that, for any m"* 2i, the set Pm is closed

In Cl (l, I).

2. Types of Periodic Trajectories

The eyclie permutation assoeiated with a eycle is ealled the type of this eycle. Sinee the type of a eycle depends not only on its period (this is the length of the permutation) but also on relative positions of the points of this eyde, the classifieation of eycles by types is more eomprehensive than the classifieation by periods. Thus, eycles of period 3 may have only one type, namely,

(21 2 3) 3 1

(to within the inverse permutation). At the same time, eycles of period 4 may have several different types, e.g.,

and

= (1 2 3 4) 2 3 4 1

(1 2 3 4). 3 4 2 1

If a map f has a eycle of type 1t~I), then it is easy to show that f has a eycle of type 1t3

and, eonsequently, it has eycles of all periods (in this ease, Theorem 3.1 implies only the existenee of eycles of periods 2 and 1).

Let us study the problem of "unimprovability" of Theorem 3.1. This means that we indieate maps whieh have a eycle of period m (m > 1) and have no eycles of period m'

if m' <l m. Any eyclie permutation

(1 ... m)

1t = sI. .. sm' m > 1,

and an arbitrary eolleetion of points ßI < ... < ßm are assoeiated with a eontinuous

pieeewise linear map fn: [ßI' ßm] ~ [ßI' ßm] linear in the intervals Ii = [ßi' ßi+ d and such that fn(ßi) = ßs' i = 1, ... , m. The map fn does not depend on the ehoiee of

I

the points ßi in the sense that the maps fn and In eonstrueted for two eolleetions of

points {ßi} and {ßi} are topologieally equivalent; the eorresponding eonjugating ho

meomorphism h is an arbitrary homeomorphism such that h (ßi) = ßi.

Section 2 Types of Periodic Trajectories 65

We say that apermutation 1t of length m, m > 1, is minimal if the map frr intro

duced above has no cycles of period m' for m' <I m (the permutation C) is also

called minimal). A cycle is called a cycle of minimal type if the corresponding permutation is minimal. Minimal permutations can be described in the following way:

Fig.25

1. If m = 2k + 1, k;:: 1, then the following permutations are minimal:

for k = 1 and

k + 1 k + 2 . . . 2k 2k + IJ k+2 k ... 2 1

for k> 1 (and the inverse permutations). The B-graph of a 1t2k+ I-type cycle for the

map F is displayed in Fig. 25. Jrr2k +\

2. For m = 2k, k;:: 1, permutations 1t are minimal if they possess the following

property: The sets {I, ... , k} and {k + 1, ... , 2k} are invariant under 1t2 and the re

striction of 1t2 to each of these sets is a minimal permutation (the same is true for the inverse permutations).

In order to prove that all these permutations 1t of length m are minimal, one must

directly check that the B-graph of frr has no closed 100ps of length m' for m' <I m. For

66 Coexistence 01 Periodic Trajectories Chapter 3

the first time, cyc1es of minimal type were described by Sharkovsky [1]. For more details concerning this problem, see Alseda, Llibre, and Misiurewicz [1], Alseda, Llibre, and Serra [1], Block [2], Block and Coppel [2], Coppel [1], Snoha [1], and Stefan [1].

Note that if m is odd, then the minimal permutation of length m is unique (to within the inverse permutation). At the same time, for even m, m ~ 6, there are several minimal permutations of length m. Thus, there are two minimal permutations for m = 6:

(1 2 3 4 5 6J.

4 6 5 2 3 1

Theorem 3.5. [I a map I E CJ (I, I) has a cycle 01 period m, t he n I also has a cycle 01 minimal type ollength m.

The proof of this theorem is similar to the proof of Lemma 3.5.

If a map has a cycle of period 2' k, where k is odd and greater than 1, then this map has cycles that are not of minimal type. The following theorem is true:

Theorem 3.6. A map I E CJ (I, I) has cycles 01 minimal type if and only if the period 01 any cycle is apower 01 two.

Assume that a map I has a cycle of period 2' k, where k is odd and greater than

one. It follows from Lemma 3.3 and 3.5 that the map g = li has a cyc1e of type

(to within the inverse permutation). The cyc1e of type 1ts is not minimal. Therefore,

the map I has a cyc1e of period 2' + 3, which is not of minimal type. Consider a cycle of any map with cyc1es whose periods are equal only to powers of

two. Let 1t2/' I> 1, be the type of this cyc1e. By virtue to Lemma 3.5, we have 1t2/ E'

m:. Therefore, max {i Eil, 2'1 : ~/(i) > i} = 2'-1 and each set 11, 21- 1 I, 12' + 1, i 1 is invariant under 1t~/. Hence, the cyc1e under consideration is of minimal type if I = 2.

For I > 2, the argument presented above must be repeated for 1t~/, and so on.

Parallel with the problem of coexistence of periods of cycles, it is natural to consider the problem of coexistence of their types. To do this, we equip the set of cyc1ic permutations with relation of ordering ( -<) as follows: we say that 1t -< 1t' if, for any I E

CJ (I, I), the condition that the map I has a cycle of type 1t implies that it has a cycle of type 1t'. The relation -< is not linear. For example, the map 11 in Fig. 26 has a cyc1e oftype

Section 2 Types 0/ Periodic Trajectories 67

1t~) = G 2 3

~) 4

but has no cycles of type

1t~2) = G 2 3

~} 3 4

On the contrary, the map h has a cycle of type 1t~2) but has no cycles of type 1t~). One can formulate general theorems on the coexistence of cycles of various types but, unfortunately, these theorems are cumbersome (Fedorenko [1]) and we do not present them here. We restrict ourselves to the investigation of cycles of some special types.

y y

)( X

Fig.26

Let

1t = C ... 2) SI Si ...

be a cyclic permutation such that sI = n and let i* be an element of the set {2, ... , n}

such that 1t (i*) = n. The permutation 1t is called unimodal if si> Si + I for 1:::; i < i* and Si< Si+ I for i* < i:::; n. The set of permutations of this sort is denoted by L.

It turns out that the relation of ordering -< induces the relation of linear ardering in the set L. Let us prove this assertion. Far any permutation 1t E L, we define a se

quence B(1t) = (So(1t), So(1t), ... ) by setting

68

for k = 0, 1, 2, ... and

Coexistence of Periodic Trajectories

r

er (1t) = rr Ck(1t)· k=ü

Chapter 3

For any 1t E L, the sequenee e(1t) is admissible in the sense of the definition intro

duced in Section 2 of Chapter 2. Recall that the set {e (1t), 1t E L} is lexicographically ordered and this relation is denoted by <.

Theorem 3.7. If a map fE CÜ (I, I) has a cycle of type 1t E L, then f has a

cycle of type 1t' E L whenever e (1t) < e (1t').

Proof. Assurne that f has a cycle of type 1t E L formed by points ßl < ß2 < ... < ßi < ... < ßn and i* is such that f(ßi') = ßn. Given 1, we eonstruct a continuous fune-

tion J nondecreasing on [ßl' ßi'] and noninereasing on [ßi"~] as folIows: If XE

[ßi' ßi + d, then

_ jmin {f(ßl+J)' fex) =

max{f(ßi)' min f(y)} ß,,,y,,x

for 1 ::; i < i*,

for i* ::; i < n.

It is clear that J has a cycle of type 1t. It follows from Seetion 2 of Chapter 2 that if a unimodal map has a cycle of type 1t, then this map has a eycle of type 1t' whenever

e (1t) < e (1t'). This result remains true for the map J. Therefore, to eomplete the proof

it remains to note that every cycle of the map J is, at the same time, a eycle of f. In eonclusion, it should be noted that the relation of ordering --< in the set of uni

modal permutations is closely related to the loeation of periodie points of the map

!2X, 0 ::; x < !, x ~ fex) = 1 2

2 - 2x, - ::; x < 1. 2

Any 1t E L eorresponds to a eycle of type 1t of this map, and viee versa. Denote the

minimal point of a cycle of type 1t of the map f by xmin ( 1t). As ean be proved by di

reet eomputation, for any 1t, 1t' E L, we have xmin(1t) < Xmin(1t') if and only if 1t --< 1t'.

i. SImPLE nvnarmcm. SYSTEmS

As shown in previous chapters, maps of the interval onto itself exhibit fairly diversedynamical behavior. Therefore, in studying dynamical systems of this sort, it is naturalto decompose the entire set of maps into classes exhibiting "similar" dynamical behavior.

At present, dynamical systems are usually regarded as simple or complex, dependingon whether their topological entropy is equal to zero or not. The dynamics of complexmaps is characterized by the following property: There exists a subset of the interval onwhich some iteration of a map is semiconjugate to the Bernoulli shift on the set of allunilateral sequences with two-symbol alphabet. It was shown in Chapter 3 that complex

maps form an open dense set in CO (I , 1) and, in this sense, they are generic. Moreover

these maps with infinite topological entropy are generic in CO (1,1).

Simple maps, in turn, form a closed nowhere dense subset in CO (1, 1). However ,

they are not exceptional in the space C"(1, I), r > 0, and form a closed set that containsan open subset.

The dynamics of simple maps is described by using the notion of splitting (see Section 2). Then they are classified according to the criterion of coincidence of differenttypes of return and various criteria for a map to belong to a certain class are applied .Note that, by establishing criteria of simplicity for dynamical systems, we, in fact, also

established criteria of their complexity (i.e., generic properties of maps from CO (I , 1)),which can be obtained by "converting" the simplicity criteria.

1. Maps without Periodic Points

Let :Fm denote the set {fE CO (1,1) IPer (f) =Fix Um)}. The set :Fm consists of maps

such that the period of any their cycle does not exceed m . In this case, m =2k, k < 00 ,

and, consequently, any set :Fm (m = I, 2, 22 , • • • ) consists of maps generating simplesystems .

First , we consider the case where a map has only fixed points but no periodic points,

i.e., f E :!J . The fact that Per (f) =Fix (f) can easily be verified. Indeed, it suffices to

69

70 Simple Dynamical Systems Chapter4

show that Fix (f2) = Fix (f). Note that, for maps from :Fi, any closed set can playa

role of their set of fixed points. Indeed, for any closed set F C IR, we have

Per (f) = Fix (f) = F for the map x ~ x + P (x, F), where p (x, F) denotes the

distance from x to F, because fex) ;::: x for x E IR and fex) = x only for XE F.

What kinds of behavior can be demonstrated by the trajectories xl' x2' ... ,xm' ... ,

xm+ 1= f(xm) of maps from :Fi? How to describe the relative arrangement of points of a

single trajectory in I?

If a function f is monotonically increasing, i.e., if fex') ;::: f(x") for any x';::: x",

then fm (x') ;::: fm (x") for any m > O. Any trajectory is monotone: If xl ;::: x2 = f(xI)'

then xl;::: x2 ;::: ... ;::: xm ;::: ... ; if xI ~ x2' then x I ~ x2 ~ ... ~ xm ~ .... Hence, any

trajectory is attracted by a fixed point of f, i.e., the co-limit set of any trajectory is a fixed point.

The following theorems describe the behavior of trajectories of arbitrary maps in ~:

Theorem 4.1 (Sharkovsky [4]). fE :Fi {::> for any XE 1 and m > 0, there are

no points xi with i < m between xm and xm+ Iprovided that x m and x m+ I are

distinct or, equivalently, for i > m, all points xi lie on the same side of the point xm

Theorem 4.2 (Sharkovsky [4]). fE :Fi {::> the co-limit set of any trajectory is a

fixed point.

Let us prove these theorems. In Section 1 of Chapter 3, we proved the following assertion:

A map f has a cycle of period 2 {::> there exists a point x E 1 such that f(x) * x

and x E f «x,J(x)), where (x,J (x) is the closed interval with ends at x and f (x) .

Below, we present another formulation of this lemma.

Lemma 4.1. fE :Fi {::> forany XE I, either f(x)=x or x~f«x,J(x)).

Since the interval f«x,J(x)) contains points fex) and f2(x), one can replace this

interval in Lemma 4.1 by the interval (f(x),J2(x) contained in it.

Lemma 4.2. Let fE CÜU, 1). 11, for x E I, the relations fm'(X)E (fm(x),

fm+Xx) and fm(x) *fm+Xx) are truefor some m and m', m'<m, then the

collection of points f i (x), i = m', m' + 1, ... ,m, contains a point x' such that x' E

f( (x',J(x')).

Proof Let m" be such that fi(;:) ;:::fm(x) for i = m', m' + 1, ... , m" and fm"+I(x) <

Section 1 Maps without Periodic Points 71

f m (x). Denoteby F theset {t(x),i==m', ... ,m"}. If f(y)<y forevery YEF,

then x'==fm(x). If f(y') > y' forsomey'E F, then x'==max{YE Flf(y»Y}.

Theorem 4.1 follows from Lemmas 4.1 and 4.2. For any x E I, we define

=

== U (f(x), f+\x)), i=m

m == 0,1,2, ....

It follows from this definition that To(x) d Tl (x) d T2 (x) d ... ::> ro(x). Let T =(x) ==

nm20 Tm (x). Clearly, T=(x) iseitheraclosedintervalorapointand T=(x) d ro(x).

Theorem 4.1 immediately implies the following assertion:

Lemma 4.3. fE 1] {:::} f i (x)1l Ti+l(x)forany XE land iE N providedthat

fi+l(x) 7= fi(x).

We now prove Theorem 4.2. Assurne that fE 1] and X is an arbitrary point of I.

Letusshowthat ro(x) isafixedpoint. If T=(x) isapoint,then ro(x) isalsoapoint.

Consider another possibility: T = (x) is an interval. It follows from Lemma 4.3 that

int T = (x) does not contain points of the trajectory of x. Hence, ro (x) == fJT = (x), i.e.,

ro(x) consists of two points. We know that ro(x) is an invariant set which cannot be formed by two fixed points. Therefore, ro (x) must be a cycle of period 2 but this is impossible.

Assurne that a trajectory approaches a fixed point ß. We are interested in the behavior of this trajectory near the point ß. If f is a monotonically increasing function, then all trajectories are monotonically convergent (increasing or decreasing) as mentioned

above. If the function f is differentiable in ß, then the trajectories convergent to ß approach this point monotonically ß provided that l' (ß) > O. At the same time, if l' (ß ) < 0, then the trajectories approach ß spasmodically so that the points with odd

numbers are located on the same side of the point ß, while the points with even numbers are located on the other side. In the case where 1'(ß) == 0 or the derivative does not exist at the point ß, the trajectory may exhibit extremely irregular behavior approaching this point.

Consider an example: Let

fex) == e sm l/x, x { -lIx2 .

0, x 7= 0,

o.

In this case, If(x) I < I x I as x 7= 0 and, consequently, the map has a single fixed point x == 0 and all trajectories approach this point. At the same time, for any partition

{NJ , N2 } of the set of natural numbers, one can indicate a point x such that fn(x) > 0

for nE N1 and rex) < 0 for nE N2.

This property of f follows from the fact that, in any neighborhood V of x = 0, there

are intervals VJ, U2, ~ C {x< O} n V, U2 C {x> O} n V such that f(V1 ) and

f (U 2) are neighborhoods of the point x = 0 (in our example, we can choose the inter

vals 2krr.< lI/xi< 2(k+ I)rr., O<k<=, i.e., (-Y"'-Yk+I) and· (Y"'Yk+I)' where

Y k = 1/ 2krr.). Due to the possibility of passing from one half neighborhood of the point

x = 0 into the other under the action of f, we can apply aversion of symbolic dynamics (with symbols +, -) that admits sequences generated by trajectories {xJ with arbitrary

collections of sign xi (the matrix of admissib1e transitions is (: :}.

Let us now dweIl upon the rate of convergence of various trajectories. It is weIl known that typical trajectories of smooth maps converge as geometric progressions,

namely,if xi~ß and If'(ß)1 = b<I, then Ixi+1-ßI/lxi -ßI '" b. However,if

If'(ß) I = 1, then the rate of convergence is substantiaIly smaller. Thus, for fex) = x

a(x-ß)2m+I(1+o(I») as x~ß and a>O, wehave IX i2 -ßI/lxi]-ßI '" b onlyif

i2-i1 '" (blx i]-ßD- 2m /2am.

In general, for fE :Fi, the rate of convergence may vary from arbitrarily high to ar

bitrarily low. In particular, for the map presented above with 1'(0) = 0, there are trajec

tories that converge arbitrarily rapidly (among trajectories that do not hit the point x = 0 after finitely many steps). Let us now formulate the corresponding general assertions.

Let ß be a fixed point of f with an invariant half neighborhood, i.e., there exists a

neighborhood U of ß such that fU- C U-, where U- = U n {x ~ ß}. For the sake of simplicity, the required assertions are formulated only for this case.

We say that f has trajectories that approach ß arbitrarily rapidly if, for any se

quence <XI < <X2 < ... ~ ß, one can indicate a trajectory xl < x2 < ... ~ ß, xi * ß, such

that xi> <Xi for i = 1, 2, .... If we replace the last inequality by the inverse one: xi< <Xi beginning with some i;:: 1, then we say that f has trajectories that converge arbitrarily

slowly.

The foIlowing assertions are tme:

A. A map f has trajectories that converge to a fixed point ß arbitrarily rapidly

if and only if there exists a sequence of points y I < y{ ~ Y2 < Yl ~ Y3 < Y3 ~ ... ~ ß such that f(Yi) = f(yj) = ß and fex) < ß for XE (Yi' y[), i = 1,2, ....

B. A map f has trajectories that converge to a fixed point ß arbitrarily slowly if and only if there exists a sequence of fixed points y I < Yl ~ Y2 < Yl ~ Y3 < Y3 ~ ...

~ ß such that Yi+ I E intf([Yi' y;]), i = 1,2, ....

Section 1 Maps without Periodic Points 73

Thus, for the existence of trajectories whose convergence is arbitrarily slow, it is necessary that any neighborhood of the point ß contain trajectories that do not converge to this point (e.g., fixed points). For the existence of trajectories whose convergence to ß is arbitrarily rapid, it is necessary that any neighborhood of the point ß contain trajectories whose points immediately hit the point ß (e.g., the trajectories starting at the points

Yi' yj, i=1,2, ... ).

The proof of assertions A and B is quite simple and we do not present it here.

As an example that illustrates assertion B, we consider a CO-map f: [-1, 0] -; [-1,0] generatedby f(x)=x+ap(x,F), where F={-1,-1I2,-1I3, ... ,0} isthe set of fixed points, p is the distance from a point x to the set F, and 1 < a < 3.

The properties of maps from :Ji established in this sec ti on can be summarized as

follows: For any map fE CO (J, 1), the following statements are equivalent:

(i) Per (f) = Fix (f) (i.e., fE Ji);

(ii) Fix (f2) = Fix (f);

(iii) 'v' x E I, m(x) is a fixed point;

(iv) 'v' x E land m> 1, the points f(x) with i > mare located on the same side

of the point fm (x),

(v) 'v'XE I, either xll.f(x,f(x») or x =f(x).

The following two statements established in Section 6 of this chapter are equivalent to properties (i)-(v):

(vi) NW (f) = Fix (f);

(vii) CR (f) = Fix (f),

where Nt\' (f) and CR (f) denote the sets of nonwandering and chain recurrent points, respectively.

If fE :hk, k> 0, then g = f2k E Ji. Therefore, all statements formulated above are tme for g and one can reformulate them for f. In particular, the following statements are equivalent:

(i) Per (f) = Fix (f2k ) (i.e., fE '!2k);


(iii) 'v' x E I, Cü(x) is a eycle of period 2i , 0 ~ i ~ k;

(iv) 'v'xE/and'v'sE{O,l, ... ,m-l},thepointsJ i 2'+s(x) with i>m areloe

ated on the same side of the point Jm 2'+ s (x);

(vii) CR (f) = Fix (f2k).

Eaeh of the sets J2k is closed in CO (I, 1) (if fi E 1) and fi ~ J*, then J* E 1) because it follows from xi E Fix (f;) and xi ~X* that X* E Fix (f*».

The set of all simple systems is not exhausted by the sets J2k, k = 0, 1, 2, .... The

maps from the set :fr \ U;=O J2k, where

J2~ = 0 J2k = {JE C°(l,I)IPer(f) = 0 Fix (f2k) }, k=O k=O

i.e., simple maps with cycles of arbitrarily large periods have not been studied yet. These maps are investigated in the remaining part of this chapter.

2. Simple Invariant Sets

Let us now study the structure of Cü-limit sets of dynamical systems with eycles whose

periods are necessarily equal to 2i , i ~ O. As we have already explained, for maps from

::r 2k, every Cü-limit set is a cycle. At the same time, in the ease where simple maps have cycles of arbitrary large periods, one may encounter much more complicated situations. Thus, there are maps such that each their Cü-limit set is also a cycle. However, the maps whose Cü-limit sets are Cantor sets are more typical. In addition, there are maps whose Cü-limit sets are composed of a Cantor set and a countable set of points. It ean be shown

that infinite Cü-limit sets of maps from the set J2~ are eharacterized by the properties similar to the properties of cycles.

The maps from ::r rare usually called simple maps. It is also convenient to say that the cycles of simple maps are simple cycles. (Thus, the topologie al entropy of a map is equal to zero if and only if all cycles of this map are simple.) As stated in Theorem 3.5, any simple cycle is a cycle of the minimal type, which means that simple maps may have

Section 2 Simple Invariant Sets 75

only cycles of the minimal type with periods 2i . Therefore, simple cycles possess the following property: A cycle

of a map f is simple if and only if either B is a fixed point or the sets {ßI ,···, ß2k -I}

and {ß2k-1 + I' ... , ß2k} are invariant under the action of f2 and the restriction of f2

to each of these sets is a simple cycle. Hence, each simple cycle B which is not a fixed point can be decomposed into two

subsets BI and B2 such that f(B I ) = B2 and f(B2) = BI. By generalizing this property to the case of arbitrary invariant sets, we arrive at the following notion of splitting:

We say that a closed invariant set M C I admits splitting under the map f2 if it can be decomposed into sets MI and M2 such that

(i) MI and M2 belong to two different closed disjoint intervals;

We say that a closed invariant set M admits njold splitting (n> 1) under the map

f2 if it can be split under f2 and each of the sets MI and M2 admits (n - 1)-fold split

ting under the map g2, where g =f2 (the terms I-fold splitting and simply splitting are synonyms).

In the case where M admits splitting, the sets MI and M2 are determined unique

ly; therefore, in the case where M admits n-fold splitting, the set M is decomposed, in

a unique manner, into subsets Mi(n), i = 1, 2, ... , 2 n, such that

(a) there exist 2n mutually disjoint intervals JI' ... , J2" ordered on 1R by increas

ing of their subscripts and such that Min) = Ji n M;

2n

(b) M = U Min);

i=J

Note that the permutation

1t

76 Simple Dynamical Systems Chapter 4

defined by eondition (e) (where si = j if f(M) = M) determines a simple eycle, i.e., it

is a minimal permutation of length 2~

Any deeomposition of M into sub sets {Mi(n), i = 1, ... , 2n} is ealled a simple

decomposition ofrank n provided that this deeomposition satisfies eonditions (a)-(e). Propositions 4.1 and 4.2 presented below are obvious eonsequenees of eondition (e).

Proposition 4.1. If {Min), i = 1,2, ... ,2n} and {M;n+l l , i = 1,2, ... ,2n+ l }

are simple decompositions of the set M of ranks n and n + 1, respectively, then M (n l = M(Hl) U M(n+1) . = 1 2n

I 2l-1 2l' l , .. " •

Proposition 4.2. An invariant set M admits n-fold splitting if and only if, for any

n' :<:; n, there exists a simple decomposition of the set M of rank n'.

A eycle of period 2n is simple if and only if it admits n-fold splitting. It ean be

shown that infinite (ü-limit sets of maps from :::r zoo also possess the property of splitting

and, moreover, that these sets admit n-fold splitting for any n.

A closed invariant set M is called simple either if it is a fixed point or if it admits

n-fold splitting for any n:<:; log2 card M (Fedorenko [3]). Simple sets have the following properties (some of these properties are obvious, and we give them without proofs):

(i) Any simple set cannot be decomposed (i.e., eannot be represented as a union of two closed disjoint invariant subsets).

(ii) Any finite simple set is a simple eycle.

(iii) Any infinite simple set has the cardinality of continuum.

(iv) Any infinite simple set eontains no periodic points.

Proof Let {M;nl, i = 1, ... ,2n} be a simple deeomposition of rank n of an infi

nite simple set M of a map f Then fk( Mfnl) n Mfnl = 0 for k = 1, ... , 2n - 1. For

any j> 0, there exists a simple deeomposition of rank n such that 2 n > j. Hence,

fj (x) ::f. x for all x E M and j > 0. Therefore, x ~ Per Cf).

(v) Any simple set that contains a periodic point is a simple cycle.

(vi) If a simple set eontains an open interval, then eaeh point of this interval is a wandering point.

Proof Any simple set M that contains an open interval U is infinite. Moreover,

the interval U contained in M neeessarily belongs to a single element of the deeompo-

Section 2 Simple Invariant Sets 77

sition {Mi(n), i = 1, ... , 2n }. Therefore, by using the same argument as in the proof of

(iv), we conclude that fi(U) n U = 0 for j> 0, i.e., each point of U is wandering.

(vii) Any simple invariant set contains an almost periodic point.

ProoJ. If M is a finite simple set, i.e., if it is a periodic trajectory, then property

(vii) is evident. Let M be an infinite simple set of the map fand let {Min ), i = l, ... ,

2n } be a simple decomposition of the set M of rank n. It follows from Proposition 4.1

h M (n) M(n+l) U M(n+l) . 1 2 2n L M- (n) d h 1 f h t at i = 2i-1 2i' I = , , ... , . et enote tee ement 0 t e

decomposition {Min ), i = 1, ... , 2n} with the smallest diameter. Since M (I) and M (2)

belong to two disjoint closed intervals, we have diam M(l) < (1/2) diamM. Then, for

any n> 1 (beginning with n = 2), we successively choose a single element M(n) in a

simple decomposition of rank n as folIows: Assurne that the element M (n) is chosen.

Then M(n) contains exactly two elements of the simple decomposition of rank n + 1.

In this pair of elements, the element with smaller diameter is denoted by M (n+ 1). Obvi

ously, M(n) ::::> M(n+l) for any n> 1 and diam M(n) < (l/2n)diamM. Therefore,

nn>j M (n) must be a point, and we denote it by x.

Let E > O. We choose n such that diam M(n) < E. Since f2" (M(n») = M(n), we

have If2"i(X)-xl < E forall i>O. Therefore, XE AP(!).

(viii) Each point of a simple invariant set is a chain recurrent point.

ProoJ. Let M be a simple invariant set of the map f If M is a finite set, then M

is a simple periodic trajectory. Therefore, ME CR (f).

Now let M be an infinite simple invariant set. By virtue of Proposition 4.2, for any

n ;e: 0, there exists a simple decomposition of the set M of rank n, namely, {Mfn), i = 1, ... ,2n }. Since the elements of a simple decomposition of rank n belong to 2n mu

tually disjoint intervals, we have min diam Mfn) ~ 0 as n ~ 00. Let M(n) denote an i

element of the decomposition {Mi(n), i = 1, ... , 2n } such that

diam M(n) = min diam Min). i

We fix E > 0 and choose no such that diam M(no) < E. Let X be an arbitrary point of

the set M. It follows from the invariance of the set M, that there exist k j ;e: 0, k2 > 0,

and a point Y E M(no) such that k l + k2 = 2no, /I(X) E M(no), and /2 (X) = x. Since

diam M(no) < E, the points X o = x, !'(xo), and r (y), r = 1, ... , kl' S = 1, ... , k2, are an

E-trajectory of x. Hence, X E CR (f).

(ix) Any recurrent point of a simple invariant set is regularly recurrent.

Proof. Let M be a simple invariant set of the map f. If M is finite, then property (ix) is obvious.

Let M be infinite and let x E Mn R (f). Also let M(n) (x) be the element of the simple decomposition of the set M of rank n which contains the point x. Denote

M(x)= nn<:1 M(n)(x). Since M(n)(x), n = 1,2, ... , are c10sed sets, M(x) is a non

empty c10sed set. Denote by a and ß the maximal and minimal points of M(x), re

spectively. If a = ß = x, then XE AP(f). Indeed, let c> O. Since nn<:1 M(n)(x) is

a point, one can find n l such that diamM("I lex) < c. This implies that XE AP(f)

because f 2n1 (M(nj )Cx)) = M(nj)Cx).

Now assume that a =1= ß. It foBows from property (vi) that (a, ß)n NW (f) = 0. Therefore, x coincides either with a or with ß.

For definiteness, we assume that x = a. Let c be an arbitrary number satisfying the

inequality O<c<ß-a. Since M(x)= nn<:IM(n)(x), thereexists n2 such that the last

point ofthe set M(":!)(x) lies to the right ofthe point x - c, i.e., infM(":!)(x) > x - c.

Since XE R (f), f 2n\M("2)(x)) = M("2)(x), and (x, x + c] n R (f) = 0, the interval

[x - c, x] contains infinitely many points of the trajectory fx' Therefore, there exists n3

for which some element M~n3) of the simple decomposition of M of rank n 3 lies in

the interval [x - c, x). Since f 2n'(Mt3») = M~n3), one can indicate k :5: 2n3 such that

fk+i t''(x) E (x - c, x) for aB i ~ O. Thus, any segment of the trajectory of the point x of

length 2n3 contains a point from the c-neighborhood of x, i.e., XE RR (f). There are many other dynamical properties of simple invariant sets. We restrict our

selves to properties (i)-(ix) necessary for what foBows and proceed to the principal theorem of this chapter.

3. Separation of All Maps into Simple and Complicated

The foBowing theorem c1arifies common features in the dynamical behavior of all simple maps and common features in the dynamical behavior of aB complicated maps:

Theorem 4.3. Any map fE CO (I, I) possesses exactly one of the following

properties:

(i) the set of all chain recurrent points of the map f coincides with the union of

all simple sets ofthis map;

Section 3 Separation of All Maps into Simple and Complicated 79

(ii) there exist n ~ 0 and closed intervals I1 and 12 with intIl n intI2 = 0

such that f 2n (l1) ~ 11 U 12 and f 2n (l2) ~ I 1 U 12,

Property (i) is a general property of all simple maps, and property (ii) is a general property of all complicated maps.

Here, we present the proof of Theorem 4.3 suggested by Fedorenko [5]; this proof is based on the c1assification of trajectories of maps of an interval and on the properties of the set of chain recurrent points. To describe this c1assification, we consider the simplest dass of dynamical systems, namely, cyc1ic permutations, as an example.

Recall that the set {i E NI i l :s: i:S: i2 } is denoted by I i l , i2 1.

Lemma 4.4. Each cyclic permutation 1t of length i > 1 possesses exactly one of

the following properties:

(i) there are elements i l , i2 E 11,nl suchthateither il <1t(i1):S:i2 <1t(i2) or

i 1 > 1t(iI) ~ i2 > 1t(i2 ),

(U) n is even, 1t 11, nl21 = I nl2 + 1, n I, and 1t I nl2 + 1, n I = 11, n121·

Proof. Since 1t is a cydic permutation, we have 1t (1) > 1 and 1t (n) < n. Hence,

thesets N+ = {iE 11,nll1t(i»i} and N- = {iE II,nll1t(i)<i} arenonempty.

We denote io = maxN+ and ia = minN-.

There are two possible cases, namely, ia < io and ia > io' We consider each of

these cases separately.

Suppose that ia < io' Let i be the preimage of ia. We have either i < ia or

i> ia. In the first case, we have i < 1tU) < io < 1t(io)' In the second case, the ele

ments ia and i satisfy the inequality i > 1t U) ~ ia > 1t (ia). Suppose that ia > io' Due to the choice of the elements ia and io' we have i < 1t (i)

if i E 11, io land i> 1t(i) whenever i E I ia, n I, where ia = io + 1. Consider two

possible cases:

(i) there exists an element i such that i and 1tel) simultaneously belong either to

11, io I or to I ia, n I;

(ii) 1t (i) E I ia, n I if i E 11, io land 1t ( i) E IUo I if i E I ia, n I·

In the first case, either i < 1t ci) < 1t2 ci) or i > 1t (i) > 1t2 ci) and, therefore, we arrive at property (i) of Lemma 4.4. In the second case, in view of the fact that 1t is a

one-to-one map, we conclude that n is even, io = n / 2, and, hence, the permutation 1t

possesses property (ii). It follows from Lemma 4.4 that any cyclic permutation 1t is either a minimal permu

tation of length 2n or possesses the following property: There exists k = 0 such that one

of the orbits of the permutation 1t2k has property (i) of Lemma 4.4. Similar classification is also applicable to trajectories of maps of an interval.

Lemma 4.5. Let JE C°(J,1), a E I. The trajectory orb(a) has exactly one oJ

the Jollowing properties:

(i)J(a) = a, i.e., a isafixedpoint;

(ii) one can indicate points a', a" E orb(a) such that either a' <J(a') $ a" $

( " , (' " (" Ja) or a >Ja ) ~ a ~J a );

(iii) the trajectory orb(a) can be decomposed into sets orb'(a) and orb"(a) such that

(a) orb' (a) and orb" (a) belong to closed intervals I' and 1" such that

. I' n· I" - 0 f' C I' d f" C I"· Int Int - , a ' an a '

(b) J(orb'(a)) ~ orb"(a) and J(orb"(a)) ~ orb'(a).

Proof. Suppose that J( a) "* a. Let us show that, in this case, we have either (ii) or (iii).

Ifthe inequalities Ji' -I (a) "* i' (a) and i' (a) = i' + I (a) hold for some (> 0,

then Ja possesses property (ii). In this case, a' = Ji' -I (a) and a" = i' (a).

Now suppose that i(a)"* i+l(a) for all i > O. Denote orb-(a)= {XE

orb(a)lJ(x)<x} and orb+(a) = {XE orb(a)IJ(x»x}. Ifoneofthesesetsisempty,

then the trajectory orb (a) possesses property (ii).

It remains to consider the case where orb- (a) "* 0 and orb+ (a) "# 0.

Jfthere are points xI E orb-(a) and x2 E orb+(a) such that xI< x2' then the

trajectory orb(a) hasproperty(ii). Indeed,since xI' X2 E orb(a), thepreimageofat

least one of these points (XI or x2 ) belongs to orb (a). For definiteness, we assume

that this is the preimage of the point xI. Denote it by x3. Thus, x3 E orb (a), J (x3) =

xI' and J(x l ) < xI < x2 <J(x2). If x3 < xI' one should take a' = x3 and a" = x2· If xI

< x3, then a' = x3 and a" = xI. Now assume that the inequality xI > x2 holds for any xI and x2 such that X I E

orb-(a) and x2 E orb+(a). If there exists a point XE orb-(a) such that J(X)E

orb-(a), then the trajectory orb(a) has property (ii). The trajectory orb(a) also possesses the same property in the case where both x and f (x) belong to the set

orb+(a).

Consider the last possibility: f(orb+ (a)) \;::; orb- (a) and f(orb- (a» \;::; orb+ (a). In

this case, the trajectory orb (a) has property (iii). Indeed, one can choose orb' (a) =

orb+(a) and orb"(a)=orb-(a) and, hence, I' = [inforb(a), sup orb+ (a)] and

I" = [inforb-(a), sup orb (a)]. To complete the proof, it remains to note that the trajectory cannot have properties

(ii) and (iii) simultaneously. Lemmas 4.6 and 4.7 describe the properties of chain recurrent points. Lemma 4.6 can

be regarded as a consequence of the incompressibility of the set CR (f).

Lemma 4.6. Let a E CR (f) \ Fix (f). Then one ean indieate points b, e E I

sueh that f(b) = a, fee) = b, and aSe f(a).

ProoJ. Assume that a <f(a) (the prooffor the case a > f(a) is similar). Denote

p = minf(x). x:::::a

Assumethat p>a. Let E = (p-a)/3. Then f(X)E [p,l] foral! XE [a, 1].

Hence, any sequence of points {xi E I, i = 0, I, ... } such that Xo = a and If(x i )

xi + II < E belongs to the interval [p - E, 1] and, consequently, I xi - a I > E for all i >

0, i.e., a ~ CR (f).

Thus, pS a whenever a E CR (f) \ Fix (f). Therefore, the set {x ~ a If(x) = a}

is nonempty. Let b be the least point in this set. If f(a) > b, then f([a, b]) :::> [a, b].

Hence, there exists a point CE [a, b] such that f(e) = b. If f(a) = b, then e = a.

Consider the case where f(a) < b. Let q = max fex). We now prove that the ina5,x5,b

equality q a for all XE [a, b). Consequently, PI > O. Let 0 = (PI -

a)/4. Sincef[q,b]::::> [a,p], theset {XE [q,b]lf(x)=PI-O} isnonempty. Let d

be the least point in this set. Denote E] = min {O, d - q}. Then

Hence,anysequence {xJ;:o with xo=a suchthat IfCx)-xi+]1 < EI satisfiesthe

inequality I xi - a I > E] for all i > O.

Thus, q ~ band, consequently, there exists a point e E (a, b) such that f (e) = b.

Lemma 4.7. Let a E CR (f). If the trajeetory orb (a) of the point a eontains

points a' and a" such that either a' <f(a') S a" sfCa") or a' > f(a') ~ a" ~

f( a"), then one ean indieate closed intervals 11 an d 12 sueh that intl] n

82 Simple Dynamieal Systems Chapter4

intl2 = 0, f2" (I]) ~ I] U 12, and f2" (12) ~ I] U 12, where n is equal to either 0

or I.

Proof. We consider only the case where a' < f(a /) :s; a" :s; f(a") (for the second case, the proof is sirnilar). There are four different possibilities:

(I) I

< f(a ' ) " :s; f(a"); a = a

(11) I

< f(a ' ) " < f(a ''); a < a

(III) I

< f(a /) = " = f(a''); a a

(IV) I

< f(a /) < " = f(a"). a a

We prove Lemma 4.7 in each of these cases.

I. a' < f(a /) = a" :s; f(a"). By virtue ofLemma 4.6, one can find a point b such

that a' <b andf(b)=a' . If b<a", then I] =[a',b] and 12 =[b,a"] arejustthe

required intervals and n = O.

Now let a" < b. By Lemma 4.6, there exists a point e such that a' < c < band fee) = b. Since f(b) < e a". Denotetheintervals [a',e], [e,a], and [a,b] by 11' 12, and 13, re

spectively. Bythechoiceofthepoints b, e, and a, we get f(l]):::> 13 , f(l2) ~ 13 ,

and f(3 ) ~ I] U 12, This means that f2(1]) ~ I] U 12 and f2(12) ~ I] U 12,

11. a' <f(a/) < a" <f(a"). If f(a / ) < f2(a'}, we proceed as in case I. Assurne

that f( a/) > f2( a'). In this case, the set of fixed points from the interval [f(a'}, a"] is

nonempty. We denote by ß] and ß2 the least and the greatest points of this sort, re

spectively, i.e., ß] = min [f(a'}, a"] n Fix (f) and ß2 = max [f(a'}, a"] n Fix (f).

Since the points a' and a" belong to the same trajectory, one can indicate either i]

suchthati1(a") = a' or i2 suchthat fi 2(a' )=a". Supposethat i1(a") = a' (the

case where i 2 (a / ) = a" can be investigated in exactly the same way). Since i1(a") =

a' , there are points ß' = min {x E I I a" < x and f (x) = ß2} and ß" such that ß2 < ß" < ß' and f(ß") = ß/· By the choice of the points ß2, ß/, and ß", the intervals

I] = [ß2,ß"] and 12 = [ß",ß/] satisfyLemma4.7with n=O.

III. a' < f(a /) = a" = f(a"). It follows from Lemma 4.6 that f(b) = a' for I " ["]) ['''] ([ I ]) [' "] some b > a . If b < a , then f ( b, a ~ a, a and f a, b ~ a, a . If

a" > b, then Lemma 4.6 implies the existence of a point c such that e E (a', b) and

f(e) = b. If e > a", then the intervals [a", e] and [e, b] satisfy Lemma 4.7 with " 2 I ['''] 2(["]) [' "] n = O. If e> a , then f ([a , eD ~ a, a and f e, a ~ a, a .

IV. a' < f(a') < a" = f(a"). Since the points a' and a" belong to the same tra

jectory fa' one can indicate a point a'" E f a such that a"'"* a" and f(a"') = a". By

repeating the reasoning used in case 111 for the points a" and a'" , we complete the proof of Lemma 4.7.

Remark 1. It is clear that Lemma 4.7 is not true without the condition a E eR (J).

Actually, let f o E CO (I, I) be such that fo(x) < x for XE intl, fo(O) = 0, and fo(1) = 1.

Then, for any point x from intI, we have x > fo(x) > f 02(x). At the same time,

eR (Jo) = Fix (Jo) = {O, I}. Let I] and 12 be some intervals such that int I] () int 12 =

0. For definiteness, we assume that 12 is located to the right of I]. Then the intervals

r(l2) are located to the right of I] for all n > 0, i.e., r(I2) n I] = 0.

Theorem 4.4. F or fE CO (I, I), the following assertions are equivalent:

I. eR (J) does not coincide with the union of all simple sets of the map f

II. There exists a chain recurrent point which does not belong to a simple invariant

set.

In There exists a chain recurrent point a whose trajectory contains points a'

and a" such that either a' < f2k (a') :::; a" :::; f2k (a") or a' > f2k (a') ?

a" ? f2k (a") for some k? 0.

IV. There exists I? ° and closed intervals I] and 12 in I such that intl] n intl2 = 0, fi(I]) ;;:;? I] U 12, and fi (I2) ;;:;? I] U 12 for some I? 0.

V. There exists a cycle whose period is not apower of2, i.e., f ~ ::r 200.

VI. There exists a cycle which is not simple.

VII. There exists an (f)-limit set which is not a simple invariant set.

Proof. I ~ 11. This is a consequence of property (viii) of simple invariant sets which can be formulated as follows: eR (J) ;;:;? Ua Ma , where Ua Ma is the union of

all simple sets of the map f.

n => III. Let a E eR (f) \ Ua Ma . It is clear that a ~ Fix (J) because Fix(f) C

UaMa . Since eR (f) is an invariant set, for any i > 0, one can find a point a_i E

eR (J) such that fi(a_;} = a. Let M = U:o {fi(a), a_J, where ao = a. By defini-

84 Simple Dynamical Systems Chapter 4

tion, M is an invariant set and M C CR (f). If, for some i > 0, the trajectory orb (a _;)

has property (ii) of Lemma 4.5, then a_; is a required chain recurrent point for statement

III with k = 0. If property (ii) of Lemma 4.5 does not hold for any i> 0, then, by virtue

of Lemma 4.5, for every i> 0, the trajectory orb (a_;) admits a decomposition

{ orb' (a _;), orb" (a _;)} such that

(1) orb' (a _) and orb" (a _;) belong to closed intervals ( and (' such that

. I' n· " 0 mt; lllt I; = .

Therefore, the set M admits a decomposition {MI' M 2 } such that

(1) MI and M2 belong to closed intervals l' and 1" such that int l' n int 1" = 0,

MI C 1', and M2 C 1";

M b' ( ) M b" ( ) M I' I' d I" C I" J' 11· ° oreover, or a_; CI' or a_; C 2' ; C ,an; lor a I> .

Consider j2. The map j2 deeomposes the set M into two trajectories MI and M 2.

Since Fix (f2) C Ua Ma , every trajectory possesses either property (ii) or property

(iii) in Lemma 4.5. If at least one of these trajectories has property (ii), then assertion III

holds with k = 1. Assurne that both MI and M 2 possess property (iii) in Lemma 4.5.

In this case, M admits a decomposition into the sets MI and M 2 . For the sets MI

and M 2, we repeat the same argument as for the set M. Then the entire procedure is re

peated once again, and so on. After finitely many (k) steps, we arrive at a simple set. This yields III.

II1::::} IV. It suffices to apply Lemma 4.7 to the map j2k •

IV::::} V. The map j2k possesses a cycle of period 3 (see, e.g., Lemma 3.3 in Sec

tion 3.1). Therefore, j possesses a cycle of period 2/ 3, 150k.

V ::::} VI. Note that any cycle whose period is not apower of two is not simple by definition.

VI ::::} VII. Note that any cycle is an ü)-limit set.

VII ::::} I. It follows from VII that CR (f) contains a closed invariant indecompos

able set which is not simple. Hence, CR (f) 7:- Ua Ma .

Seetion 3 Separation of All Maps into Simple and Complicated 85

Note that Theorem 4.3 is equivalent to the statement "I ~ IV". Moreover, in order to prove "I ~ IV", it suffices to show that V ~ I (by analogy with the proof of the equivalence VII ~ I). However, we have added assertions VI and VII because these are quite useful and their proofs are very simple. It should also be noted that the equivalence "V ~ IV" was proved by Sharkovsky [3]; the equivalence "V ~ VI" was established by Block [2], and the assertions similar to "VII ~ V" can be found in Barkovsky and Levin [1], Blokh [1], Fedorenko [3], Li, Misiurewicz, and Yorke [1], Misiurewicz [2], and Smital [1].

At the end of this seetion, we present a lemma used in what follows. Recall that fE

~ 2~ <=> CR (f) = Ua Ma .

Lemma 4.8. Let M be a simple invariant set of a map fE :J' 2~ and let card M >

2n, n > O. Then there exist closed mutually dis joint intervals li' i = 1, ... ,2~ each of

which contains an element of a simple decomposition of the set M of rank n, f\/i ) n li = 0 for k = 1, ... , 2n - 1, and f2" (I) C li' i = 1, ... , 2n.

Prooj. For n > 1, the proof of the lemma is a simple consequence of its assertion

with n = 1. Thus, it suffices to prove Lemma 4.8 for n = 1.

Since card M > 2, the set M admits a simple decomposition {M?l, i = 1, ... , 22 }

ofrank 2. Denote a'i = min {x I x E M?l} and a'; = max {x I x E M?l}, i = 1, ... ,

22. Consider the map g = f2. It is clear that the map g has fixed points in the interval

[a'~, a'4]' Let ~ be one of these points. Moreover, since ~ E g([a'~, a'3])' there

existsapoint Y= max{xE [a'~,a'3]lg(x)=~}.

Let us now indicate a closed interval 12 that contains the element M~'l = Mj 2l U

Mi2l of the simple decomposition of rank 1. If the point y has no preimages to the

right of the point y, then 12 = {x EIl x ~ y}. Now suppose that there exists y, =

min{xE Ilx~y and g(x)=y}. If y, E [a,~], then g([y, 11]) ;;?[y,~] andg([y"

~ ]) ;;? [y" ~]. Hence, by virtue of Theorem 4.3, we have f (I': :J' T' If y, E [ß, a;J,

then there exists a point x, E [y, a;J such that g2 (x,) = Yl (the existence of x, is a

consequence of the inclusion g ([ y, ~]) :::> [ß, a;]). Therefore, g2 ([ y,x,]) ;;? [y, ~],

g2(xi'~);;?[y,~J, andf(l': ~2~'

Consequently, y, > a;. If the point y, has apreimage, then, by the same argument,

one can also prove that f (I': ~ 2~' Hence, g (x) < y, for xl E [y, y]] and g (/2) C 12

for the interval 12 = [y, y, ].

Similarly, one can prove the existence of a closed interval I] C [inf I, y] such that

I] :::> M[2l U M~2l, g(I,) C I" and I] n 12 = 0.


4. Return for Simple Maps

As shown in the previous section, a map is simple if and only if its set of almost returning points is the union of all simple sets of this map. However, the set of almost return

ing points of any map in ~ 2~ always contains points with the stronger property of return than chain recurrence.

Recall (see Theorem 1.5) that the following chain of inclusions is valid for fE

C°(I,l):

Per (f) ~ APB (f) ~ AP(J) ~ RR (f) ~ R (J) ~ cef) ~ Q (f) ~ NW (J) ~ CR (J).

What types of return appearing in this chain may simple maps have? This problem is completely solved by the following two theorems:

Theorem 4.5. The set of all chain recurrent points of the map f coincides with the union of alt simple sets of this map (i.e., f is a simple map) if and only ifRR (J) = R(f).

ProoJ. Let CR (J) = Ua Ma . Since any recurrent point of a simple invariant set

is regularly recurrent (see property (ix) in Section 2) and CR (f);;;2 R (J), we have RR (f) = R (f).

Now suppose that CR (J) * UaMa . By virtue of Theorem 4.3, in this case, there

exist n?:: 0 and closed intervals 11 and 12 such that int 11 n int 12 = 0, f2 n (11) ;;;2

I] U 12, and f 2n (l2);;;2 I] U 12 .

By using standard methods of symbolic dynamics (see the proofs of Propositions 1.1 and 1.2), one can show that f possesses an infinite closed invariant set F that contains an everywhere dense trajectory and an everywhere dense subset of periodic points. This dense trajectory consists of recurrent points. However, F is not a minimal set. Hence, by virtue of the Birkhoff theorem (Birkhoff [1]), the points of this trajectory are not regularly recurrent, i.e., RR (f) * R (J).

Theorem 4.6. (Sharkovsky and Fedorenko [1]). There exists a mapfo E CO (I, I) such that

Per (Jo) * APB (Jo) * AP(Jo) * RR (Jo)

= R(Jo) * C(Jo) * Q(fo) * NW(Jo) * CR(Jo)· (4.1)

ProoJ. To prove the theorem, we use the following considerations: Let ~, i = 1, ... ,

k, be closed mutually disjoint intervals on I. Assurne that maps !; E CO (Ii' li)' i =

Seetion 4 Return for Simple Maps 87

1, ... , k possess properties Ai invariant under topologie al conjugation. In addition, we

suppose that the 1eft end of the interval I, eoineides with the left end of the interval I,

i.e., inf I, = inf I, and that the right end of Ik coincides with the right end of I, i.e.,

sup Ik = sup I. Furthermore, let the map fE CO (I, I) be such that

(i) fl I is topologieally eonjugate to 1;, i = 1, ... , k; I

(ii) fis a linear funetion in eaeh eomponent of l\ U:=/i'

In this ease, the map f possesses all the properties Ai' i = 1, ... , k. Henee, in order

to prove the theorem, it suffiees to eonstruet aseries of maps with the following eommon property: For eaeh of these maps, the set of reeurrent points eoincides with the set of al

most periodie points (i.e., eaeh of these maps belongs to n:-2~)' Note that, for any two maps in this series, some pair of neighboring sets in (4.1) does not eoineide (these pairs are different for different maps).

Any map in the series eonstrueted below is a modifieation of two fixed maps. The first of these maps is

{2X'

fex) = -2x + 2,

XE [0, 1/2],

XE [1/2, 1],

and the seeond one was introdueed by Sharkovsky in [7].

I. Consider the map f given by (4.2).

(4.2)

Here, we use the binary representation of points in 1 = [0, 1] instead of their deei

mal representation. Let O. a, ... a i ... , where ai is either ° or 1, be the eoordinate of a

point x E I. Then

if (4.3)

if

where ai = 1 - ai"

The following property is a generalization of this representation for the n th power of

the map f

Property 1.

{O.:n+' ... ~ ... , O.an+, ... ai ... ,

if an = 0,

if an = 1. (4.4)

Proof. Since ai = ai, it follows from (4.3) that fn--l (O.a l a 2 ... ai ... ) is equal ei

ther to O.anan+ l ... ai ... or to O.anan+l ... ai .... In the first case, by substituting

O. anan+I ... ai ... in (4.3), we obtain (4.4). In the second case, we also arrive at (4.4).

Indeed, if an = 0, then an = land (4.3) implies that

rCO.al a2 ..• ai ... ) = O.an+1 ..• ai ....

Further, if an = I, then an = 0 and, thus,

rc o. a l a2 ..• ai ... ) = o. an+l ... ai ....

We now introduce several definitions and notation necessary for what folIows. Any

finite ordered sequence that consists of 0 and I is called a block. Let B be a block that

consists of elements a 1 •.. an' Then B denotes the block formed by the elements

al ... an' The infinite sequence ofblocks B is denoted by (B), i.e., (B) = BBB ....

Any positive integer i can be represented in the form i = Lj~O sj 2j , where Sj E

{O, I}. Weset

pU) = L Sj' qU) = rnin {j I sj';t: O}, j~O

and R(i) = p(i) + q(i).

The block formed by a single I is denoted by B2o. Beginning with B2o, we con

struct blocks for any k > 0 according to the formula

(4.5)

Let a = O. a l ... ai ... be a point of the interval I such that a l ••• a2k = B2k for any

k'?O.

Property 2. For any fixed k '? 0, a = O. cf ... ct ... , where

d = {B2k, I -

B2k,

if RU) is odd, (4.6)

if R(i) is even.

Proof. It follows from (4.5) that, for any k> 0 and k' < k, B2k is representable as - k k k an ordered sequence of blocks Bzk' and Bzk'. Therefore, Cl ... Ci ... , where Ci is

either B2k or B 2k for any fixed k. Let us deterrnine the block occupying the ith posi

tion. Let i = Lj~Osj2j, where SjE {O, I}. Denote k l = max{jlsj"* O} + 1.

Section 4 Returnfar Simple Maps 89

Then it follows from the definition of a and (4.5) that

k -k and, consequently, Ci = C i _ 2kl .

We set

By repeating the same argument for the block occupying the (i - 2k ] )th position, we ob

tain Cf = \~2kl _2k,· Iterating this procedure R(i) times, we arrive at the equality

:} I limes (4.7)

Cf = Cjk , where 1= R(i).

Since ~k = ct = B2k, relation (4.7) yields (4.6). By Property 2 with k = 0, we ob

tain, in particular, a = O.a j ••• ai ... , where

{O,

1,

if R(i) is odd,

if R(i) is even.

Property 3. a E RR (f).

Proof Let U(a) be an arbitrary neighborhood of the point a. It follows from the

definition of a that there exists an odd number k such that each point of the form

0.B2k a2k+ j ... ai ... belongs to the neighborhood U(a). We fix an arbitrary number i

and prove that there exists io E {i, i + 1, ... , i + i+ 2 - I} such that fiO( a) = 0.B2k ....

This, in fact, means that a E RR (f).

Indeed, let a = 0. C Jk+2 •.. Ct+2 .•.. Then f2k+2i (a), j = 1, 2, ... , is either the point

0.B2k+2 ••• or the point O.B 2k+2 •••• Formula (4.5) implies that

and

Since k is odd, R (2 k ) is even and, therefore, the last element of the block B2k is 0.

Hence, j2k+2i +2k+! (a) = 0.B2k ... for any j. Consider the representation i = 2k+2i l +

h, where 0 ~ i 2 < 2k +2. If h ~ 2k + I, then we take io = 2k+2i l + 2k+ I. If i 2 > 2k + I,

then io = 2k+2U1 + 1) + 2k+ I. Here, the inequality i ~ io ~ i + 2k+2 - 1 holds in both

cases.

Property 4. a g: AP (f).

Proof. Let a = O. c? ... Cp ... , and let N be an arbitrary fixed positive integer.

Since a = O. 1 ... , it suffices to show that there exists i> 0 such that j ~ (a) = 0.0 ...

We set .i = 2q(N)+ I. By the definition of the function R(j), we can write R (N) =

p(N) + 1 + 2q(N) and R(Nj + 1) = p(N) + 1. This means that the numbers R(N)

and R(N. + 1) are either both even or both odd. Thus, in the first case, C~ = 0 and } j

C~ = 0, while in the second case, C~ = 1 and C~. = 1. However, in any case J+! J J+!

j~(a) = 0.0 ....

Property 5. If k is odd, then x = O. (B2k ) is a periodic point with period 2k- 1 .

Proof. For odd k, the last element of the block B2k _! is 1 and the last element of

the block B2k- 2 is O. Consequently,

and

The first equality implies that the point O. (B2k ) is periodic and the second one means

that its period is equal to 2k- 1 because the number 2k-2 and, hence, any number 2k!

with kl < k - 1 cannot be aperiod of this point.

Proof. Let a = O. cf ... ct ... and let a l ... ak be apart of the block B2k. Then

j2k+l(a) = 0.a2,·.a2kB2kB2k ... andj(0.(B2k» = 0.a2 ... a2dB2d. Thismeansthat

Section 4 Return for Simple Maps 91

the first 2 k+ I - 1 elements of the points under consideration are equal, while the ('1- 1) th

element ofthe points f2k+ I(a) and f(0.(B2k » is equal to 1 orO, respectively. This,

in fact, proves Property 7.

,'ted«' CI( 'loel

Fig.27 Fig.28

Let a' be the point of 1 such that a' = 1 - a. Consider a map J E CO (I, I) (Fig. 27) such that

_ {f(a), if x E [a', a], fex) =

fex), if XE 1 \ [a', a].

The map J pos ses ses the invariant interval I 1 = [f2(a),f(a)]. Consider Pk It follows from Property 7 with k = 1 that f3(a) > O. (B2!). This condition implies the

followingpropertyofthemap J2: Theinterval I I containstwointervals 12 = [J2(a),

f4(a)] and 13 = [f3(a),f(a)] invariant under P and such that J(/2) = 13 and

J(/3) = 12, This enables us to conclude that

(i) the map J has no periodic trajectories with odd periods;

(ii) theinterval/3 contains points ßI and ß2 such that P[ßI' ß2 ] = [a', a] (Fig.28).

In view of the fact that Property 7 holds for any odd k, one can repeat this reasoning for k = 3 and the interval 13 , etc. As a result, we obtain the following properties of the

map J:

(1) J possesses a periodic trajectory of period 2k, k = 0, 1, ... ;

(2) j has no periodic trajectories whose periods are not powers of 2;

(3) j (a) E AP(f) \ Per (f) (this property is a consequence of the facts that O. (B2d

~ a as k ~ 00 and that, for any neighborhood U(j(a)), there exists an inter

val ofthe form [f2k + \a), a] invariant under j2k );

(4) foranyneighborhoodofthepoint j(a), onecanindicatepoints ß1 and ß2

suchthat ß1 <ß2 <a and jk[ßI'ß2 ] = [a',a] forsome k.

This enables us to conclude that the map j possesses a unique infinite simple invari

ant set M lying in the interval [0, a'] U [a, j(a)]. By construction, M~ [a', a] and all points of the set M \ int M are limit points of the set of periodic points. Conse

quently, M is a minimal set and fl M is Lyapunov unstable (see Property 4 of the map

j). Hence,

RR (j) ::> AP(j) ::> APB (]) = Per (]).

Let f l be a continuous function defined on the interval [a', a] and such that

(i) f 1 (a') = f 1 (a) = Jea);

(ii) fl(x) > j(a) forany XE (a',a) and m~x iJ(x) = y. xe[a ,al

Consider the map

fi (x), if XE [a',a],

j(x), if XE [O,j(a)] \ [a',a], fex) =

p(a), XE [j(a), y], if

fz (x), if x>y,

where f 2 (x) is a continuous function such that

(ii) f 2 (x) > j2(a) for x>y (Fig.29).

Since the maps J and j coincide on the set [0, a'] U [a, j(a)], the set M is

invariant both under J and j. Moreover, Property 4 of the map j implies that each

Section 4 Return for Simple Maps 93

pointoftheinterval (J(a), y) belongs tothe set CR(f)\NW(f) and YE NW(f). Finally, since any sufficiently small neighborhood of the point y contains at most two

points of each trajectory of the map j, we have Y II n (f). Hence,

I

I", r Fig.29

11. The second series of maps is formed by modifications of the map introduced by Sharkovsky in [7]. First, we describe this map.

Let hex) = (x + 2) /3. In the segment I, we choose two sequences {an} and

{bn}, n = 0,1,2, ... , such that an+! = h(an), bn+! = h(bn), ao = 0, and bo = 1/3. It

is c1ear that an< bn < an+! for n = 0, 1,2, ... , and an ~ 1, bn ~ 1 as n ~ 00.

Let f be a continuous function piecewise linear in I, linear in each of the segments [an' bn] and [bn, an+!], and satisfying the relations

(4.8)

f(x)l[lIQ.Lt!l = x + 2/3, and f(l) = 0 (Fig.30).

It follows from the definition of the function f that the equality

f(h(x» = ~ fex) (4.9)

holds for all x E I.


Fig.30 Fig.31

Since fex) = x+2/3 for XE [ao,bo] andf([al' 1]) = [ao,boL wehave

2 f(f(x)=f(x)+- for XE [al,I],

3

f(f(x) = f( X +~) for XE [ao, bo].

(4.10)

It follows from (4.9) and (4.10) that the maps fl[o,l] and f 21[ao,bo] UI[o,l] and

f 2 1 [al, I]) are topologically conjugate; furthermore,

h(f(x) = f 2(h(x),

(4.11)

g(f(x) = f 2(g(x),

where g(x) = x13. It follows from (4.11) that the following relations hold for any n = 1,2, ... :

f2 n (x) = hn(f(h-n(x))) for x E [an' 1], (4.12)

f 2n (x) = gn(f(g-n(x))) for xE[O,n/3].

Let us establish some properties of the map f

(1) fE ~ 2~' Indeed, we have f([ ao, bo]) = [al' 1], f([al' 1]) = [ao, bol, and

there are no periodic points in the interval [bo, al] except a fixed point.

Section 4 Returnfar Simple Maps 95

Hence, f pos ses ses a periodic trajectory of period 2 and has no periodic trajec

tories with odd periods. Therefore, the fact that fand f2 are topologically con

jugate implies that fE :y 2~'

(2) CR (f) = AP (f). The fact that f2n and f are topologically conjugate implies

that the intervals [all' 1] and [0, n/3] are periodic with period 2n• Furthermore,

Hence,

2n _l

CR (f) C U i[an ,1] U Fix (J2n-

1), n = 1,2, ....

i=O

2n -l

CR(f)C n U i[an , 1] U Per(J) n~1 i=O

2n 1 . The set nn~1 Ui=o f' [an' 1] is the standard Cantor set. This means that each its point

is almost periodic in the sense of Bohr. Relations presented below are necessary for what follows. It follows from (4.12) that

f 2n (x) = X + 3n2+1 for XE [0, 3}+1 1 Therefore, f 2n (X)E [0, 1/3n ] if XE [0, 1/3n+ l l Thisyields

f2n -l(X) = x+ 1- 3~ for XE [0, 3~ 1

(4.13)

(4.14)

We also note that if XE [0, 1 /3 n+ I], then fi(x) E U~=O [an' bn ] for = 1, 2, ... ,

2n+ I _1 and

By using the map constructed above, we can now present an example of a map f such

that CR (f) * NW (f) * Q (f) = APB (f) * Per (f).

1. Let b~ = bn + 1I3n +3. Clearly, b: < an+l' Consider a piecewise linear map f l

defined on [0, 1] and such that

fex) if x E [an' bn],

3(x + bn) + f(bn) if XE [bn, b:], f 1 (x) = f( ) - f(an+l) - f(bn) - 3(b: - bn) ( _) if XE [b:, an+l], an+l * an+l x

an+l - bn 0 if x:2: 1,

for any n = 0, 1,2, ... (see Fig. 31). By using (4.14), one can easily show that

2n * fi (bn ) = a, n = 1,2, ... ,

where a = 3(b~ - bo) + f(bo). Thus, for any n = 0,1,2, ... , the interval [an' a] is

periodic with period 2n. Moreover, each interval k([an, aD, i = 0, 1, ... , 2n_ 1, can

be split into three intervals so that fi2n is a homeomorphism defined on the central in

terval with a single fixed point and the other two intervals form a cycle of intervals with

period 2. Therefore, f 1 is a continuous map from !J' 2~ and

2"-1

.NWCf1)C n U /[an,a] U Per(fi)· n<:1 i=O

2" 1 . We also note that nn<:1 Ui=o P [an' a] is the standard Cantor set with a system of

closed intervals attached to each unilateral point of the Cantor set which is a left limit

point for points of this set (these intervals are preimages of the interval [1, a]). Since

each point of the interval [1, a] is wandering, we have NW Cf1) = APB Cf1 ).

Consider a map

{fi(X), if XE [0, a];

f 2 (x) = l'f X - a, x> a.

The point a is nonwandering for f 2. This follows from the fact that there exists a se

quence ofpoints (an };=1 such that

(ii) an ~ a as n ~ 00;

(iii) f 2 (an) = f 2 (b:), i.e., a is the limit point for its own preimages.

Section 4 Retumlor Simple Maps 97

2. By using 12, we can construct an example of a simple map 13 such that C(13 ) "#

R(3 )·

Let h(x)lu [ bOl = 1\ (x). We fix a sequence of points a* = a - l/3n+2 and n~O an' n n n

setl\x) =1\(x)+(1/3)(x-an) for XE [a:,an], n~O. Wealsoconsiderasequence

ofpoints

b' b 1 n = n + 3n+2' n~O.

Obviously,

'* b b* < b' an < an < an < n < n n < cn+ I < an+ \.

Fig.32

The function 13 can be extended by continuity to 1R + as follows (see Fig. 32):

{

X - a~ + h(an), for X E [a~, a~] U [b~ ,b~],

13(x) = 1- x, for XE{Cn } U {xlx~l},

linear, for xE[b~,cn]U[cn,a~], n=O,1,2, ....

Let us now describe the properties of the map 13, By virtue of the definition of 13

and (4.13), we can write

2" * [*' * *] (d) 13 (x) = an -x for XE ao - bn+! + bn+!, ao ;

2 * * [* '] (e) 13 (x) = ao-x+bo for XE bo,bo ' n~O.

2n+1 , I

It follows from (e)-(e) that h (bn) = an. Then (a) and (b) imply that the points

a~, b~, and < belong to the same periodic trajeetory with period 2n+ 2, n ~ O. More-

, 2"-1 ( ') 2"+1! ( , ) over, it follows from (e) and (d) that ao < 13 bn < 13 - bn+! ' n ~ 0, and

It-I (b~) ~ a~ as n ~ 00, i.e., a~ is a limit point of the set of periodie points. It is

known (see Sharkovsky [1]) that C(/3 ) = Per(f3). Therefore, a~ E C(3 ). However,

a~ ~ R ( 3 ), because the co-limit set belongs to the interval [ao' 1]. By using the same

reasoning as in the previous subsection, we can prove that 1 E :T 2~.

Fig.33

3. Let us now construct a map 14 with Q(f4) -:f. C(f4).

Assume that 14 (x) is a continuous piecewise linear funetion such that

Seetion 5 Classification of Simple Maps According to the Types of Return 99

for

for x = fr-I(b~),

for 2n I' 2n+1 I ' XE [13 - (bn ), 13 - (bn+l )].

The map f 4 belongs to ::r 2= and the point a; belongs to the co-lirnit set of the point

a;. However, the point a; does not belong to the center of the map f 4 because the in

terval [a;, a;l contains no periodic points ofthe map f4.

The graph of the function appearing in the theorem is displayed in Fig. 33.

5. Classification of Simple Maps According to the Types of Return

According to Theorem 4.6, simple maps may have a large variety of types of "returning" points. This is why it is reasonable to construct a c1assification of one-dimensional dynarnical systems based on the coincidence of different types of "returning" points from the following chain of inc1usions:

Per (f) k APB (f) k AP (f) k RR (f) k R (f)

k C(f) k o'(f) ~ NW(f) ~ CR(f) (4.15)

A c1ass of maps {J E C(I, I) I AI (f) = A2 (f)}, where AI (f) and A 2 (f) are two arbitrary sets from (4.15), is denoted by AI(f) = A2(f). As follows from (4.15), there are 36 c1asses of maps of the type AI (f) = A2 (f). All these c1asses are depicted in

Diagram 1, where "~" denotes (replaces) the sign of inc1usion "e". Recall that a map is simple if and only if it belongs to the c1ass RR (f) = R (f) (see Theorem 4.5). This and Diagram 1 imply that the problem of c1assification of all simple maps is reduced to the problem of selection of all c1asses of maps in Diagram 1 that belong to the c1ass RR (f) = R (f) followed by the identification of coinciding c1asses in the group thus selected. This problem is solved in Theorems 4.7-4.10 and corollaries to these theorems.

Theorem 4.7. The class of maps RR (f) = R (f) contains the following classes

ofmaps:

(i) R (f) = C(f);

(ii) AP(f) = RR (f);

...., ~ U

11

-=: I i 0 ...., pe: I u ~

-=: -=: Cf a i i

;~ ...., er

~ 11

-=: -=: u u

....,

~~ ....,

~ ~ er u ~ ~ 11

S S pe: pe: .... i i i i ~

.~ Q

~ ~ ~

...., 0 I c:

11 ~ 11 ~ ~ ~

S C; ~ P:" <C <C

...., ...., ...., ...., ...., ...., 0 ~ r a u ~ pe: ~ u pe: <C

11 ~ 11 ~ ~ ~ ~ 11 ~

S S S S S S C;

~ ~ ~ ~ ~ ~ ~

~ ~

...., ...., 0 0 ...., ...., ...., ...., ~ I c: u ~ ~ ~ f u pe: <C

11 ~ 11 ~ 11 ~ ~ ~ 11 ~ 11 ~

S S S S S S S -=: il ~ ~ ~ ~ ~ ~ ~ ~

Section 5 Classification of Simple Maps According to the Types of Return 101

(iii) APB(f) = AP(f).

Prooj. Instead of proving the theorem itself, we shall prove the equivalent assertion, i.e., the fact that C (f):::) R (f), RR (f):::) A P (f), and AP (f):::) APB (f)

whenever RR (f) *" R (f). Since RR (f) *" R (f), Theorems 4.5 and 4.4 imply the existence of I;::: 0 and

closed intervals I) and 12 such that intI) (") intl2 = 0, j2I(1));;;2 I) U 12, and f21(12)

;;;2 I) U 12. By using standard methods of symbolic dynamics, we can now establish the existence of a periodic point whose preimage is not a periodic point but belongs to the closure of the set of periodic points. Therefore, there exists a point x E I such that x ~

ffi(X) and XE Per(f). By virtue of the fact that C (f) = Per(f) (Theorem 1.4), we conclude that x E C (f) \ R (f).

Actually, one can complete the proof ofTheorem 4.7 by the methods of symbolic dynamics but, for diversity, we present another version of the proof.

According to Theorems 4.5 and 4.4, the fact that RR (f) t:: R (f) is equivalent to the existence of a periodic trajectory which is not a simple cyclic permutation. Denote this trajectory, its type, and period by B, 1t, and n, respectively. Since 1t is not a simple cyc1ic permutation, there exists k, 1::; k < log2 n, such that one of the orbits of

the permutation 1tk possesses Property (i) ofLemma 4.4.

Assume that k = 1 (for k> 1, the proof is similar). Let 51. denote the set of cyclic permutations with Property (i) of Lemma 4.4.

If 1t E 5'l, then the map f possesses a periodic trajectory of the type

(1 2 3 4 5 6)

1t) = 4 6 5 3 2 1 .

The points of this trajectory are denoted by a i , i = 1, ... ,6, according to the natural

ordering on IR. Denote a~ = max {x E [al' a 2] If(x) = a 6}. It follows from the form

of the cyc1ic permutation 1t) that f([ al' a 3]) :::) [a4, a 6] and f([ a4, a 6]) :::) [a),

a 3], where [a~,a2]:::) [al'a3]. Thismeansthattheinterval [al'a6] containsa c10sed invariant set which admits splitting (e.g., a periodic trajectory of period 2).

Consider the map g = f2. A periodic trajectory of the type 1t) admits splitting into two periodic trajectories of period 3. The type of 3-periodic trajectories be10ngs to the

set 51.. Therefore, in each of the intervals [a), a 3] and [a4, a 6], the map g pos

ses ses a periodic trajectory of the type 1t). As above, this means that the segment [a),

a 3 ] contains two intervals I) and 12 whose ends are points of a periodic trajectory of

type 1t) of the map g. Moreover, these intervals are such that g(ll) :::) 12, g (12) :::) 11'

and I) :::) [a~, a 2]. Similarly, the segment [a4, a6] contains two c10sed disjoint inter

vals 13 and 14 such that g(l3):::) 14 and g(l4):::) 13. Consequently, the interval [al'

a 6 ] contains a c10sed invariant set which can be split 2 times.

Considerthe map h = J22. Each periodic trajectory of type 1t 1 of the map g can be

split into two periodic trajectories (of period 3) of the map h, etc. This reasoning can

be repeated infinitely many times. Hence, the interval [al' a 6 ] contains an infinite

simple invariant set, which is denoted by M.

Let {Mfn), i = 1, ... , 2n} be a simple decomposition ofthe set M. It follows from

the construction that this decomposition indudes an element M(n) such that the interval

whose ends coincide with the minimal and maximal points of the set M(n) contains the

interval [a;, az]. Hence, the set nn~O M(n) consists of two points. This enables us to

condude that

(i) fl M is Lyapunov unstable (because for any element Mfn), one can find k < 2n

such that fk(Mfn» = M(n»;

(ii) there exists k> 0 such that fk( nn~O M(n») is a point (because M is a mini

mal set).

It folIo ws from (i) (see Sibirsky [1]) that M contains no almost periodic in the sense

of Bohr points. Moreover, (ii) implies that fl M is not a homeomorphism. Hence, f possesses an almost periodic point which is not almost periodic in the sense of Bohr.

Furthermore, since fl A IV) is a homeomorphism and M is a minimal set, the map f possesses a regularly recurrent point which is not almost periodic.

Corollary 4.1. Let A (f) be an arbitrary set from (4.15) other than RR (J) 0 r

R (f). Then the following classes of maps coincide:

(i) RR(f) = A(f);

(ii) R (f) = A(f).

Proof. We split the proof into two parts. First, we consider the case where A (f) is

equal to Per (f), APB (f), or AP (f) and then the case where A (f) is equal to C (f),

0. (f), NW (f), or CR (f).

Let A(f) be either Per (f), or APB (f), or AP (J). As follows from (4.15), the dass of maps R (f) = A(f) is contained in the dass RR(f) = A(f) (see Diagram 1). Furthermore, it follows from (4.15) that the dass of maps RR(f) = A(f) is contained

in the dass RR (f) = AP(f). This and Theorem 4.7 together imply that the dass of

maps RR(f) = A(f) is contained in the dass RR (f) = R (f). Hence, the dass of maps RR(f) = A(f) belongs to the dass R (f) = A(f). This means that the first part

of the proof of Corollary 4.1 is completed. Now suppose that A (f) is either C (f), or 0. (f), or NW (f), or CR (f). According

to Diagram 1, the dass R (f) = A(f) contains the dass RR(f) = A(f). Moreover,

Section 5 Classification of Simple Maps According to the Types of Return 103

the dass R (f) = A(f) is contained in the dass R (f) = C(f). Therefore, by virtue of

Theorem 4.7, the dass of maps R (f) = A(f) is contained in the dass

RR (f) = R (f) and, hence, in the dass RR(f) = A(f). The proof of Corollary 4.1 is

completed.

Theorem 4.8. Thefollowing classes ofmaps coincide:

(i) Per (f) = CR(f);

(ii) Per (f) = NW (f);

( iii) Per (f) = Q(f);

(iv) Per (f) = C(f);

(v) Per (f) = R (f);

(vi) Per (f) = RR (f) ;

(vii) Per (f) = AP(f).

Proof. According to Diagram 1, to prove Theorem 4.8, it suffices to show that the dass of maps Per (f) = AP(f) is contained in the c1ass Per (f) = CR(f). To do this,

we now establish the fact that the map f such that Per (f) = AP (f) possesses the prop

erty Per (f) = CR (f).

Suppose that the map f is such that Per (f) = AP (f). Then it follows from Dia

gram 1 and Theorem 4.7 that fE 3" 2~. By virtue of Theorem 4.4, each chain recurrent

point of the map f belongs to a simple invariant set. It follows from Properties 4 and 7 of simple invariant sets that each infinite simple invariant set of the map f contains an almost periodic point which is not periodic.

Since Per (f) = AP (f), we conc1ude that the map f has no infinite simple invari

ant sets. Therefore, each simple invariant set of the map f is a periodic trajectory. This

yie1ds Per (f) = CR (f).

Theorem 4.9. The following classes of maps coincide:

(i) APB(f) = AP(f);

(ii) AP(f) = RR(f).

Proof. Theorems 4.7, 4.5, and 4.4 imply that any map f from one of the c1asses in

the formulation of Theorem 4.9 belongs to 3" 2~. Let M be a minimal infinite set of the

map f (if M is a finite set, then the proof is evident). Since fE 3" 2~' the set M is a simple infinite invariant set of the map f

Let {M(n), i = 1, ... ,21 be a simple decomposition of the set M of rank n. As

sume that j is such that APB (j) = AP (f). Since M contains an almost periodic point (Property 7), it also contains aalmost periodic in the sense of Bohr point. Taking into account the fact that the dosure of the trajectory of any almost periodic in the sense of

Bohr point is Lyapunov stable (see Sibirsky [1]), we find that the map jlM is Lyapunov

stable. Hence,

max diam M~n) ~ 0 as n ~ 00.

i Eil, 2n l !

Therefore, each point of any minimal set is almost periodic, i.e., AP (f) = RR (f). Thus, the dass of maps APB(f) = AP(f) contains the dass AP(f) = RR (f).

Assurne that j is such that AP (f) = RR (f). Let x be an arbitrary point from M

and let M(n)(x) be an element of the simple decomposition of the set M of rank n

which contains the point x. Suppose that the set nn>OM(n)(x) is not a point. Denote

by xI and x2 the minimal and the maximal points ofthis set, respectively. Since M is

a minimal set, and AP (f) = RR (f), the points xI and x 2 are almost periodic. We

choose € O and k = 1,2. Consequently, jMN2(XI ) < xI<

x2 <jNIN2(X2 ). Hence, the interval [xl' x 2 ] contains a periodic point but this is

impossible by Lemma 4.8 and Property 4 of simple sets. Therefore, for any point

XE M, the set nn>OM(n)(x) consists of a single point and this means that each almost

periodic point is almost periodic in the sense of Bohr, i.e., AP (f) = APB (f).

Corollary 4.2. Let A (j) be an arbitrary set jram (4.15) but not Per (f), APB (f),

or AP(f). Then thejollowing classes ojmaps coincide:

(i) APB(f) = A(f);

(ii) AP(f) = A(f).

Proof According to Diagram 1, the dass of maps APB(f) = A(f) is contained in the dass AP(f) = A(f). Moreover, the dass of maps AP(f) = A(f) is contained in the dass APB(f) = RR (f) which, in turn, coincides with the dass APB(f) = AP(f)

(by virtue of Theorem 4.9). Hence, the dass AP(f) = A(f) is contained in the dass APB(f) = A(f). Therefore, the dasses APB(f) = A(f) and AP(f) = A(f) coineide.

Theorem 4.10. Thejollowing classes ojmaps coincide:

(i) AP(f) = CR(f);

Section 5 Classification 0/ Simple Maps According to the Types 0/ Return 105

(ii) RR (f) = CR(f).

Prooj. To prove the theorem, it suffices to show that the c1ass of maps RR (f) =

CR(f) is contained in the c1ass AP(f) = CR(f).

Actually, let / be such that RR (j) = CR (f). Then / E !Y 2~' Let x E CR (f). For

x E Per (f), the proof is obvious. Assume that x E CR (f) \ Per (f). Let M be a simple invariant set (maximal by inc1usion) that contains the point x, let

{Mfn), i = 1, ... , 2n} be a simple decomposition of this set of rank n, and let M(n)(x)

be the element of this decomposition that contains the point x. Consider the set

nn>oM(n)(x). For / E :y 2~' this set may be a segment. Every interior point of the set

nn>OM(n>(x) is chain recurrent; it is also a wandering point. Therefore, for any point

XE M, the set nn>OM(n)(x) is a point provided that RR (f) = CR (f). This means

that each point of a simple set of the map / is almost periodic. The results of this section are displayed in Diagram 2. This diagram inc1udes all

c1asses presented in Diagram 1 that belong to the c1ass of maps RR (f) = R (f) . All c1asses uni ted in a single block coincide.

Per (J) = CR(J) Per (f) =NW(f Per (J) = Q(J) Per(f) = C(f)

Per(f) = R f Per(f) = RR f) Per(f) = AP( f)

AP(J) = CR(J) APB(f) = CR(f

~ RR(J)= CR(J) ~ R (f) - CR(f

AP(f) =NW(J) RR(f)=NW(f)

APB(f)=NW(J) ~ R(f) = NW(J)

J., J.,

AP(f) = n(J)

APB(f) = n(J)

J.,

RR(f) = Q(J)

R(f) = Q(J)

J.,

RR(f) = C(J)

R (J) = C(J) J.,

R(J)I

Diagram2

The examples constructed to prove Theorem 4.6 demonstrate that the c1asses of maps that are not united in a single block in Diagram 2 do not coincide. Therefore, the space

/ E CO (I, I) admits a decomposition into the following c1asses of maps:

(i) Per (f) = AP (f);


(ii) Per (f) *- AP (f) = CR (f);

(iii) AP (f) = NW (f) *- CR (f);

(iv) AP (f) = Q(f) *- NW(f);

(v) AP (f) = C (f) *- Q(f);

(vi) AP (f) = R (f) *- C (f);

(vii) AP (f) *- RR (f) *- NW (f);

(viii) AP (f) *- RR (f) *- 0. (f) *- NW (f);

(ix) AP (f) *- RR (f) *- C (f) *- Q(f);

(x) AP (f) *- RR (f) = R (f) *- C (f);

(xi) RR (f) *- R (f).

AP(f) = CR(f) AP f -NWU) AP f - Q(f)

Per (f) = CR(f) PerU)-NWU

AP f-C(f) AP(f) - R(f) APU) - RR(f) RR(f) = Q(f)

PerU) - QU PerU) - CU RR (f) = CR(f)

R f - CR(f) RR(f)=NW(f) RR (f) = C (f)

~ R(f) =NW(f) ~ RRU) - RU RlfJ - QU) RU)-CU)

!PerU) - RU PerU) - RRU) Per (f) = AP(f)

3klf E Fxk

APB (f) - CR(f) APB U) -NW( f) APB (f) - Q(f) APB f) = C(f)

APBU - RU, APBUJ - RR(f) APB(f) = AP(f)

Diagram3

Note that for smooth or piecewise monotone maps, the classification displayed in Diagram 2 takes the form presented in Diagram 3.

Section 6 Properties of Individual Classes 107

6. Properties of Individual Classes

The dassifieation deseribed above is more or less eomplete from the following point of view: Topologieal dynamies deals not only with the property of return but also with many other important eoneepts, namely, with topologie al entropy, Lyapunov stability, homodinie trajeetories, etc. Many of these coneepts ean be regarded as eriteria that enable one to attribute a given map to a eertain dass in Diagram 2. Therefore, there is no need in more detailed dassifieations of simple maps. Theorems 4.11--4.19 presented below darify this observation. For simplieity, in Diagram 4, we depict a single representative of eaeh dass of equivalenee from Diagram 2 and present the numbers of the eorresponding theorems.

Th.11 I Per (f) = CR(f)1 ~ Th.12IAP(f) = CR(f)1

t

Th.13 IAP(f) =NW(f) I --t IRR(f)=NW(f)1

t t Th.14IAP(f) = Q(f)1 --t IRR (f) = Q(f)1 Th.17

t t Th.15IAP(f) = C(f)1 --t IRR (f) = c(f)1 Th.18

t t Th.16IAP(f) = R (f)I~ IRR (f) = R (f)1 Th.19

Diagram4

Theorem 4.11. Let fE CO (1,1). Then the following statements are equivalent:

( i) Per (f) = CR (f);

(U) Per (f) = NW (f);

( iii) Per (f) = Q. (f);

(iv) Per (f) = C (f);

(v) Per (f) = R (f);

(vi) Per (f) = RR (f);

(vii) Per(f) = AP(f);

(viii) Per (f) = PerU);

(ix) for any x EI, ffilx) is a cycle;

(x) any invariant ergodie measure is concentrated on a cycle;

(xi) for any x EI, ffilx) is a simple cycle;

(xii) CR (f) = {x EI 13n(x) 1!2n(x) = x};

(xiii) NW(f) = {XE 113n(x)lf2n (x)=x};

(xiv) Q(f) = {XE 113n(x)lf2n (x)=x};

(xv) C(f) = {XE 113n(x)1!2n(x)=x};

(xvi) R(f) = {XE 113n(x)lj2n(x)=x};

(xvii) RR(f) = {XE 113n(x)lf2n (x)=x};

(xviii) AP (f) = {x EI 13n(x) If2n (x) = x};

(xix) CR (f) is a union of alt simple cycles of the map f

The equivalence (i) {:::} (viii) was established by Block and Franke [1], the fact that (ii) {:::} (viii) was proved by Blokh [1] and Fedorenko and Sharkovsky [2], and the facts that (viii) {:::} (ix) and (viii) {:::} (iv) were established by Sharkovsky [3] and Blokh [1], respectively.

Let Ac I be such that f(A) ~ A. We say that fl A is Lyapunov stahle if, for any

E > 0, there exists Ö such that the inequality Ir(x) - f n(y) I < E holds for any I x - y I < 0, x, Y E A, and aB n > O.

Theorem 4.12. Let fE CO (I, I). Then the foltowing statements are equivalent:

(i) APB (f) = eR (f);

(U) AP (f) = CR (f);

(iii) RR (f) = CR (f);

(iv) R (f) = CR (f);

Section 6 Properties of Individual Classes 109

(v) fl cR (f) is Lyapunov stahle;

(vi) eR(f) = {XE 11 lim f2"(x)=x}; n---700

Theorem 4.13. Let fE CO (I, I). Then the following assertions are equivalent:

(i) APB (f) = NW (f);

(ii) AP(f) = NW(f);

(iii) flcR(f) isLyapunovstableand f(NW(f)) = NW(f);

(iv) NW(f) = {XE Illimf2n (x)=x}; n---7 00

We say that flA is chaotic if

lim suplr(x) -r(y)1 > 0 n---7 OO

and

lim inflr(x) - r(y)1 = 0 n---7 00

for some X, y E A. For maps of an interval, the definition of chaotic maps is equivalent (see Jankova and

Smital [1], Kuchta and Smital [1], and Smital [1]) to the definition of "Li-Yorke chaotic" maps (see Li and Yorke [1]).

We say that the trajectory of a point X E I is approximated by periodic trajectories

if,forany E>O, thereexists pE Per(f) and n>O suchthat Ifi(x) _fi(p)1 < E for all i > n.

An interval J ~ I is called periodic if there exists m > 0 such that fm (1) ~ J and

int (1) n intfi(J) = 0 for i = I, ... , rn-I. We say that a c10sed invariant indecomposable set Ac I admits cyclic decomposi

tion if it can be represented as a union of c10sed mutually disjoint sets Ai' i = 1, ... , n,

such that r(A i ) = Ai·

We say that a decomposition Jt of the set A improves a decomposition 'E of the

same set (and write Jt ~ 'B) if each element of the decomposition Jt is contained in a

single element of the decomposition 'E. We say that a c10sed invariant indecomposable set Apossesses an exhausting se

quence of cyclic decompositions if there exists a sequence of cyclic decompositions

{~} of the set A such that ~+1 ~ JI,. for all n and the maximum diameter of the el

ements of ~ tends to zero as n ~ 00.

110 Simple Dynamieal Systems Chapter4

Theorem 4.14. LetfE CO (I, 1). Then thefollowing assertions are equivalent:

(i) APB (f) = n(f);

(ii) AP (f) = n(f);

(iii) f is not ehaotie;

(iv) fln(n is Lyapunov stable;

(v) fINW(f) is Lyapunov stable;

(vi) n(f) = {XE Illim f2n(x)=x}; n~~

(vii) any trajeetory ean be approximated by periodie trajeetories;

(viii) for any two distinct points of an infinite (f)-/imit set, one ean find two disjoint periodie intervals eaeh ofwhieh eontains one ofthese points;

(ix) every (f)-limit set whieh is not a eycle possesses an exhausting sequenee of eyclie deeompositions.

The relations (iii) ~ (vii) ~ (viii) were established by Fedorenko and Sharkovsky [2] and Srnital [1]. The facts that (iii) ~ (iv) and (iv) ~ (v) were proved by Fedorenko Sharkovsky, and Srnital [1] and Fedorenko and Srnital [1], respectively.

Theorem 4.15. Let fE CO (I, 1). Then the following statements are equivalent:

(i) APB (f) = C (f);

(ii) AP (f) = C (f);

(iii) APB (f) = APB(f)

(iv) AP (f) = AP(!);

(v) c(f) = {XE Illim f 2n (x)=x}; n~~

(vi) flC(f) is Lyapunov stable;

(vi) flR(f) is Lyapunov stable;

Seetion 6 Properties of Individual Classes 111

(viii) fIRR(f) is Lyapunov stable;

(ix) fl AP (f) is Lyapunov stable;

(x) fl APB (f) is Lyapunov stable;

(xi) fIPer(f) is Lyapunov stable.

A c10sed invariant set is called minimal if it does not contain any proper c10sed invariant subset.

Theorem 4.16. Let fE CO (1,1). Then the following assertions are equivalent:

(i) APB (f) = R (f);

(ii) AP (f) = R (f);

( iii) APB (f) = RR (f);

(iv) AP (f) = RR (f);

(v) APB (f) = AP (f);

(vi) R(f) = {XE Illim f 2n (x) = x}; n-7~

(vii) RR(f) = {xEIllimf2n (x)=x}; n-7~

(viii) f is Lyapunov stable on every minimal set;

(ix) each trajectory of an arbitrary minimal set can be approximated by periodic trajectories;

(x) for any two distinct point of an infinite minimal set, one can indicate two disjoint periodic intervals each ofwhich contains one ofthese points;

(xi) any minimal set which is not a cycle possesses an exhausting sequence of cyclic decompositions.

The equivalence (i) <::::} (viii) is an analog of the Markov theorem on the relationship between the type of return on a minimal set and Lyapunov stability on this set (see Sibirsky [1]); the other equivalences are established in (Fedorenko [4]).

Theorem 4.17. Letf E CO(I,I). Then thefollowing assertions are equivalent:

(i) RR(f) = Q(f);

(ii) R (f) = Q(f);

(iii) for any x E I, co/x) is a minimal set;

(iv) co/x) = a/x) for any x E I;

(v) the map x ~ coj(x) regarded as a function I -7 2/ (with the Hausdorff met

ric) is not of the ft.rst Baire class.

The equivalence (i) <=> (iii) is an analog of the Birlchoff theorem on the type of return on minimal sets.

Theorem 4.18. LetfE CO (I, 1). Then thefollowing assertions are equivalent:

(i) RR (f) = C (f);

(ii) R (f) = C (f);

(iii) RR(f) = RR(f);

(iv) R (f) = R(f).

Theorem 4.19. LetfE CO(I,I). Then thefollowing assertions are equivalent:

(i) the period of every cycle is apower oftwo;

(ii) each cycle is simple;

(Ui) there are no homoclinic trajectories;

(iv) h(f) = 0;

(v) h(fICR(f)) = 0;

(vi) h(fINW(f») = 0;

(vii) h(fIQ(f)) = 0;

Seetion 6 Properties 01 Individual Classes 113

(viii) h (fIC(f) = 0;

(ix) h(/IR(f» = 0;

(x) h(fIRR(f» = 0;

(xi) h (11 AP (f» = 0;

(xii) h(fIAPB(f» = 0;

(xiii) h(fIPer(f» = 0;

(xiv) IICR(f) is not chaotic;

(xv) IINW(f) is not chaotic;

(xvi) Iln(f) is not chaotic;

(xvii) IIC(f) is not chaotic;

(xviii) IIR(f) is not chaotic;

(xix) IIRR(f) is not chaotic;

(xx) RR (f) = R (f);

(xxi) Per (I) is a Go-set;

(xxii) R (f) is a F cr -set;

(xxiii) AP (f) = {x E I I lim 12n (x) = x}; n---t~

(xxiv) APB(f) = {XE lilim 12n (x)=x}; n---t~

(xxv) every minimal set is simple;

(xxvi) there are no minimal sets with positive topological entropy;

(xxvii) eR (I) is a union 01 all simple invariant sets olthe map I;


(xxviii) every co-limit set contains a simple minimal set;

(xxviv) every co-limit set is simple;

(xxx) if co/x)=coj2(x1 thenco/x) isafixedpoint;

(xxxi) every co-limit set which is not a cycle does not contain any cycle;

(xxxii) there are no countable co-limit sets;

(xxxiii) trajectories of any two points are correlated, namely, for the two-dimensional

{X H fex)

map g: , the inequality CO «x, y»);f. co'f(x)xco'f(y) holds for any YHf(x) g

two points x, y E 1 provided that co/x) and co/y) are not flXed points;

(xxxiv) for any x E I, ö/x) is a minimal set;

(xxxv) any finite ö-limit set is a cycle;

(xxxvi) there are no countable ö-limit sets;

(xxxvii) every ö-limit set is a simple set;

(xxxviii) if ö/x) = Öj2(x1 then ö/x) is afixed point;

(xxxiv) any ö-limit set which is not a cycle contains no cycles;

(xl) for any x, y EI, the inequality ö/(x, y») ;f. ö/x) X ö/y) holds for the

{X H fex)

map g: provided that ö/x) and ö/y) are not fixed points; y H fex)

(xli) any trajectory can be approximated by trajectories ofperiodic intervals;

(xlii) for any closed intervals I) and 12 such that intl) n intl2 = 0 and any

m > 0, either ru)) ::t> I) U 12 or f m( 2 ) ::t> I) U 12;

(xliii) for any closed invariant set Fand any m > 0, the map fm IF cannot be

topologically semiconjugate to a shift in the space of unilateral sequences of two symbols;

(xliv) there are no m ~ 0 with the following property: for any k > 0, one can

Seetion 6 Properties of Individual Classes 115

2i, if is even and 1 :=:; i :=:; 2k - 1,

2i-l, if is odd and 1 :=:; i :5: 2k - 1,

Ttk (i) 2k+1 - 2i + 2, 2k- 1 :5: i :5: 2k , if is even and

2k+1 - 2i + 1, if is odd and 2k- 1 :5: i :5: 2k ;

(xlv) for any m-limit set F, the set {x E I I mix) = F} is at most of the second

dass according to the Baire-de la Vallee Poussin classification.

The equivalence (i) <=> (ii) follows from (Block [2]); Ci) <=> (iii) follows from (Shar

kovsky [13]); (i) <=> (iv) follows from (Misiurewicz [1]); (iv) <=> ... <=> (viii) is a general fact; (viii) <=> ... <=> (xix) follows from (Fedorenko, Sharkovsky, and Srnital [1]); (i) <=> (xx) follows from (Xiong [1]); (i) <=> (xxi) <=> (xlv) <=> (xliii) follows from (Sharkovsky [3]); (i) <=> (xxxi) <=> (xxv) <=> (xxviii) follows from (Sharkovsky [7, 10-12]).

For the first time, different statements of this type were put together by Sharkovsky in [17]. Note that the major part of the equivalences given in Theorem 4.19 can be proved on the basis of Theorem 4.3. Different proofs of the equivalence of certain statements from Theorem 4.3 can also be found in (Alseda, Llibre, and Misiurewicz [1]) and (Block and Coppel [2]).

s. TOPOLOGIC8L DVn8mICS OF unImODRL mRPS

1. Phase Diagrams of Unimodal Maps

Let f:I--'d beaunimodalmap (U-map). Wesaythatafinitefamily 5'1.. ={Jo,II"'" I n- I} of subintervals of the interval I fonns a cyc1e of intervals of period n if the inte

riors of I i are mutually disjoint and f(Ji)C I(i+I)modn for all i E {O, 1, ... ,n - I}.

Denote by YJ.,. = YJ.,. (f) the set of cyc1es of intervals of period n of the map f which

contain the critical point c. Suppose that, for some n ~ 1, the set YJ.,.(f) is not empty

(it is c1ear that 5'1..1 is not empty because f(I) C 1). The set YJ.,. contains an element

maximal by inc1usion. Indeed, let ~CJ.) = {IbCJ.), ACJ.), ... , I~~?I} and A~ß) = { Ibß) ,

I}ß), ... , I~~I} be cyc1es of intervals from YJ.,.. We say that A~CJ.) is bounded from

above by the cyc1e of intervals A~ß) if IjCJ.) C Ijß) for all i E {O, 1, ... , n - 1 }. If

:y = { ~ CJ.), a E ~} is a completely ordered (in the indicated sense) subfamily of the

set YJ.,., then the elements of ::r are bounded from above by the cyc1e of intervals

A - { U I(CJ.) U I(CJ.) U J(CJ.)} n - 0' I' ... , n-I' CJ.E~ CJ.E~ CJ.E~

Consequently, by the Zorn lemma, the partially ordered set YJ.,. contains a maximal

I A* { I* * I*} * .. e ement n = n,O' In,l' ... , n,n-I . We can assume that CE In,O' Therelore, the

cyc1e of intervals A~ is defined unambiguously. Clearly, A; = {I}. For n ~ 2, it follows from the unimodality of the function fand maximality of the cyc1e of intervals

A~ that

(a) for any i E {O, 1, ... ,n - I}, I;' is a c10sed interval;

(b) for any i E {I, 2, ... ,n - I}, the mapping of the interval I;' onto I~+I)modn is bijective;

117

118 Topological Dynamics of Unimodal Maps Chapter 5

(c) f(d l~) C dl;; therefore, if l~ = [y, y'], then f(y) = f(y') and either r(y) = y

or r(y') = y';

(d) if m > n and .91". "" 0, then m = kn for some k ~ 2 and A,: C A.: in the following sense:

{XE lllE A,:} C {XE lllE A.:}.

Let {Pm }::, be an increasing sequence that consists of all positive integers such

that .9/pm (f) "" 0. In this case, m* ~ 00. Let ~ = {x E II 1 E A;m }. It is clear that

f(~) C ~ and the sequence of closed sets {~}::, forms a kind of filtration,

which can be used to decompose the set of all trajectories of a given map into finitely or countably many natural classes and study some problems of the dynamics of one-dimensional maps in detail.

Dynamics ofMaps fl~. for m < m*. Consider a U-map g = fRnIl* . The m Pm,O

point c is the critical point of g. For definiteness, we assume that c is its maximum point. Define the sets

Rm = ~ \ U r i ( U intJ;m+l,j)' ~m = intRm, and 9{m =Rm \~m' i;:'O o<;,j<Pm+l

The sets Rm and 9{m are closed and the set ~m is open (note that ~m can be emp

ty but Rm "" 0 because fixed points of g must belong to Rm and, therefore, 9{ m "" 0 ).

Obviously, f(Rm)C Rm. It follows from the strict monotonicity of f on any interval

which does not contain the point of extremum that f( ~m) C ~m and the components of

the set ~m are bijectively mapped onto each other by the map f. Hence, f( 9{m) C 9{ m .

It follows from the definition that 9{m is a nowhere dense set.

Thus, the set ~ is decomposed into three sub sets characterized by different types

of dynamics, namely, trajectories that hit the interior of ~+, after finitely many steps,

trajectories that belong to the set ~m' and trajectories that belong to the set !R..m. We investigate the dynamics ofthe map f on ~m' Components U and V of this

set are called equivalent if there exist i, j ~ 0 such that fi(U) () P(V) "" 0 (and, conse

quently, fi(U) = P(V)). It is clear that the dass of components equivalent to a compo

nent U is formed by the family of components of the set {U . Ji (U)} lying in -00<1<+00

 ~. The set of all classes of equivalent components is at most countable because ~m is

open.

The components of ~m may exhibit the following two types of dynamics:

Section 1 Phase Diagrams of Unimodal Maps 119

(i) the trajectory of a component eventually forms a cycle of intervals;

(ii) the trajectory of a component consists of infinitely many intervals, i.e., this com

ponent is a wandering interval: fi (U) n Ji (V) = 0 whenever i*, j.

a

Fig.34

A* p",.,

-----6' lT1

b

Let {B~)} {,::! be the family of all different cycles of intervals formed by compo

nents of 13m. The class of components equivalent to the components of a cycle of inter

vals B~) is also denoted by B~). Symbols r~) denote classes of components formed

by wandering intervals of 13m ; these classes form the set {r~)} 7:!. If there are no sets

B~) or r~), then we assurne that jm = 0 or km = 0, respectively. In Fig. 35b, we pre

sent a formal illustration of the dynamics of the map on the set <1>:. Arrows mean that

there are trajectories of Ap* that hit intervals of the set Ap* and there are sets B m(i) , m m+l

1 ::; i ::;jm, and r~), 1::; i::; km; dotted lines mean that the sets of types B~) and r~) may be absent for the map f

Below, we consider two cases different from the dynamical point of view.

A. Pm+!/Pm=2. Let J;m. O = [z,z']. Weset x=SUp{YE[z,e]:g(y)=y}

(Fig.34). Since m < m*, we have g(e) >e. Therefore, z::; x < e. For this point x, one can find a unique point x' E (e, z'] such that g(x') = x. It follows from the in

equality m < m* that g([x, x']) C [x, x'].

For z < x, we set J~:o = (z, x) and consider the cycle of intervals B~) = { J~:o'

J~:I' ... , J~:Pm-d, where J~:i = Ji(J~:o)' i = 1,2, ... ,Pm-1.

Since Pm+/pm=2, we have J;m+I,O=[ZI' zJ], where ZI E (x, x'), g(z{)= ZI'

g(ZI) = zj (Fig. 34a), and J;m+I.Pm = [ZI' zn, where zl' E (z{, x') and g (zl') = Zl'

In this case, the set R m defined above consists of countably many points which are pre

images ofthe points z, x, and ZI and belong to Rm.

a b

Fig.35

Thus, any trajectory of <1>: hits either the interior of an interval from the set A;m+!

or an interval from B~), or periodic points z, x, or z{ after finitely many steps. Note

that, for Pm+ I/Pm = 2, the dynamies of the map fl <1>* is much simpler than in the case m

where Pm+ I/Pm> 2, which is described below. In Fig. 34b, we displayaformal diagram, which may be regarded as an illustration of these conclusions. It is much simpler than the diagram depicted in Fig. 35b.

B. Pm+I/Pm>2. In Fig.35, wepresentanexampleofthemap g for Pm+/Pm = 3.

Unlike the first case, parallel with the cycles of intervals Ap* of period Pm + land m+1 B~) of period Pm' the map f possesses a cycle of intervals B;;) = { J;;,b, f( J;;,b), ... } ofperiod 2Pm' whichbelongsto <1>: butdoesnotbelongto <1>:+1' For Pm+/Pm=2, this is impossible.

The dynarnics of the map g in case B is schematically represented in Fig. 35b. As shown above, in this case, jm ::; 00 and km::; 00. In what follows, we construct examples of maps which illustrate some theoretical possibilities (according to the schematic diagram).

Section 1 Phase Diagrams of Unimodal Maps 121

Dynamics of the Map fl<l>~. for m* < 00. As in the case m < m*, we consider the

map g = fBn* I J* . We define an open set PI1I*'O

'Em* = {x E <1>:*: there exist a neighborhood U of x

and an integer N? 0 such that c Ci. Ji (U) for all i > N},

which may be empty, and the set 'R..m*= <1>:* \ 'Em*. As in the case m < m*, the co m

ponents of 'Em* can be decomposed into c1asses of equivalent components {B ~~ } {r;;i and {r ~~ } 7:"j (the first group of c1asses is formed by cycles of intervals, whereas the

second group consists of wandering intervals). Moreover, jm*::; 00 and km>::; 00. It

should be noted that one of the components of the set 'Ern> may contain the point c. Hence, for one of the c1asses of equivalent components, the image of a component is not necessarily a component (it may be apart of a component).

1

I

a b

Fig.36

Figures 36 and 37 display three possible cases for the map g. Note that, for the map

depicted in Fig 36 a, 'Em* is the set of internal points of the intervals from Ar, . There-m*

fore, in this case, we have jm* = 1, km* = 0, and R rn*= Uo<-< C)];m*,i' In Fig. 36b, _1_Pm*

we present a schematic representation of the possible dynamics of the map f on the set

<1>:* for m* < 00.

Case m* = 00. Consider the c10sed set <1>: = nrn~l <1>:. This set is nonempty and

f( <1>:) c <1>: because f( <1>:) c <1>: and CE <1>: for any m < 00. Denote 'E= =

int <1>:. As in the case m < 00, we define

'13~ = {x E : I there exist a neighborhood U of x

and an integer N ~ 0 such that c ~ Ji(U) for i > N}.

The set '13~ n Per(f) is empty because the periods Pm of the cycles of intervals APm

approach infinity as m increases. Hence, the components of '13~ are wandering inter

vals, i.e., the classes of equivalent components generate only the set {r 2)} ~:1 and

j==O.

a b

Fig.37

Alt - _____ _ {[(iJ}~ .. - i=/

Fig.38

By the definition ofthe set <1>:, its components are "almost periodic intervals", i.e.,

for any to-neighborhood UE of a component S ofthe set <1>:, one can indicate an inte

ger m ~ 1 such that fPm(s) C UE. This is why we write A.: instead of <1>: in Fig. 38

illustrating the dynamics of the map f on <1>: .

Seetion 1 Phase Diagrams of Unimodal Maps 123

Thus, the dynarnies of an arbitrary unimodal map ean be sehematieally represented in the form of a phase diagram, i.e., as an oriented graph whose vertiees are eycles of intervals (maximal by inclusion) or classes of equivalence for wandering intervals. The general form of the phase diagram of a unimodal map is displayed in Fig. 39. In what follows, we demonstrate that the phase diagram eontains essential (or almost all) information about the topologie al dynamies of a unimodal map.

_ {Bm}~, .," I ", -----< --_ {r.IiJ}~'

I ",

I It

Apml<"

{tI!J lim • .."." m/l j.T

- - - - -c ......... ....... {r.'ll}If",.

m* I-I

Fig.39

Properties of Phase Diagrams of Unimodal Maps:

(i) The eentral vertiees (A;m*' m::; m*) are linearly ordered.

(ii) If m < m* and Pm+ I/Pm = 2, then jm::; 1 and km = O.

(iii) If m < m* and Pm+ I/Pm> 2, then jm::;oo and km::; 00.

(iv) If m* < 00, then jm* ::; 00 and km*::; 00.

(v) If m* = 00, then j= = 0 and k=::; 00.

It is thus natural to seleet the most "simple" and "complieated" (from the viewpoint of the shape of their phase diagrams ) unimodal maps in the colleetion of maps of various

smoothness. Thus, quadratie maps x ~ ax2 + bx + c, a"* 0, prove to be simple in this

124 Topological Dynamics 01 Unimodal Maps Chapter 5

sense; for these maps, km = 0 for any m and im = 0 for m < m*; for m = m*, either

im = 0 or we arrive at the case displayed in Fig. 36a. Some examples of the most "com-

plicated" CO-maps (e.g., such that im =00 and km =00 for some m) are constructed below, where we also discuss typical problems encountered in this case.

2. Limit Behavior of Trajectories

Consider the dynamics of the map 1 on the sets that determine its phase diagram.

D . f th M 11' L B(i) - {JU) J(i) J U)} h . ynamlcs 0 e ap B~)' et m - m,O' m,l'"'' m.qi-l ' W ere qi IS

the period of a cycle of intervals. Obviously, qi = riPm for some Ij ~ 1. Let J~:o = (a, b).

If B~) doesnotcontain c, then g=lqil[a,bl isahomeomorphism. Itfollowsfrom

the maximality of B~) that either g(a) = a and g(b) = b or g(a) = band g(b) = a. In the first case, any trajectory of the map g approaches a fixed point. In the second

case, by considering g2, we conclude that any point is attracted either by a fixed point or by a 2-periodic trajectory of the map g.

If B,~) contains the point c, then it is not difficult to show that m = m* < CX) and

13m , = UO ' int Jp* i (i.e., this is the case depicted in Fig. 36a). Moreover, all tra-'5:1<Pm* m*'

jectories of the map g = Fm' 1 J' approach its fixed points. Pm*,O

Thus, the dynamics of the map IIBU) is very simple. In particular, the set of non-m

wandering points of I1 BU) is a sub set of Per (f). m

Dynamics ofthe Map 1 on r~). Any interval U from the set r~) is wandering, i.e., its different images are mutually disjoint. Consequently, the length of the interval

f (U) tends to zero as i increases. It follows from the definition of r~) that the ends

of the interval U belong to 2(m' Hence, the ffi-limit set ffiy of any point y E U E r~) does not depend on the choice of the point y and belongs to NW (f) n 2(m'

Dynamics ofthe Map Ion !l{m. We consider the case m < m*.

A. Pm + I/Pm = 2. As follows from the analysis of the corresponding case in the pre

vious section, the set 2(m consists of finitely many preimages of periodic points z and

x and infinitely many preimages of the periodic point z 1 (see Fig. 34a). Thus, for

Pm+l/Pm = 2, every point y E 2(m hits one ofthe periodic points z, x, or Zl within a

Seetion 2 Limit Behavior of Trajectories 125

finite period of time (i.e., after finitely many iterations). In this case, 2(m nN W(f) =

NW (JI 'l(,. ) C Per(J). Denote the cycle that contains z by n:, the cycle that contains

x by n;;:, and the cycle that contains z I by n;;:). It is clear that n: and n;;: may

coincide and n;;:) = n: + I' Hence,

where n:, n;;:, and n;;:) are cycles.

B. Pm+I/Pm>2. We representthe set ~ astheunionofsets 2(:,2(;;:, and ~), where 2(: and 2(;;: are, respectively, the finite sets formed by the preimages of peri

odic points z and x in 2(m (see Fig. 35b), and 2(;;:) denotes the set ~ \(2(: U 2(;;:). Denote 2(mnNWJ) by nm, ~nNW(f) by n:, 2(;;:nNW(f) by

n;;:, and 2(;;:) n NW (J) by n;;:). It is clear that the cycles n: and n;;: contain

points z and x, respectively. The investigation of the structure and properties of the set

n;;:) requires more detailed analysis.

Let g = fPm I J;mo,o' where j;m.o is an interval from the cycle of intervals A;m that

contains the point c. Let j~m.o be the closed interval with ends at g(c) and g2(c) (see,

e.g., Fig. 36a). It follows from the conditions m< m* and Pm+ I/Pm> 2 that CE int j~~.o d (j (O)) j(O) C 'd h I f' I A(O) { j(O) j(O) an g Pm. o = Pm,o' onSl er t e cyc e 0 mterva s Pm = Pm'O' Pm,I'"''

iO) } and the invariant set <1>(0) = U, iO) " where iO). = f(iO) ). Let Pm,Pm- 1 m O'>l<Pm Pm,l Pm.l Pm'o

us show that, for any point y E 2(;;:), its domain of influence Q (y, J) coincides with

the set <1>;;:).

Definition. The asymptotic domain of influence of a point y E I under the map

f: I ~ I is defined as the set

Q(y,J) = n U fi(U), j?o i?j U3y

where U denotes open neighborhoods ofthe point y.

Indeed, every point y E 2(;;:) hits the set <1>;;:) after finitely many steps and, conse

quently, Q (y,J) C <1>;;:). On the other hand, it follows from the definition of 2(m that,

for any neighborhood U of the point y E 2(m' there exists j = j (y) such that i (U)

contains the periodic point z I' which is an end of the interval jp* ° from Ap* m+l' m+1

(for the proof, see Lemma 5.3). Hence, it suffices to show that Q (z I' f) = ~). This equality is proved by the following two lemmas:

Lemma 5.1. Q(ZI,f) is a cycle ofintervalsfor which ZI is an internal point.

Proof. The point Z I is a periodic point whose period is at most Pm + I' Therefore,

its domain of influence Q(zl,f) consists of at most Pm+ I components permutable by

the map f, i.e., it is a cyc1e of intervals. If ZI belongs to the boundary of Q(ZI' f),

then, for any sufficiently small neighborhood U of Z I' either this neighborhood hits

~ll after finitely many steps or it is a cyc1e of intervals that does not contain c. In the first case, Z I must be a point of extremum of some iteration of the map f (recall that if

Co is a point of extremum of the map i, then f\co) = c for some k < j). Hence, this situation is impossible. The fact that the second case is also impossible follows from the proof of Lemma 5.2.

Lemma 5.2. A~~ is the minimal cycle of intervals (with respect to the ordering of

sets by inclusion) for which Z I is an internal point.

Proof. Let us prove that the intervals Jp' " of the cyc1e of intervals Ap* are m+l· m+l

mutually disjoint. Indeed, if this is not true, then, for some j, 0 <j < Pm + I' we have

J;m+l'O n J;m+l,j f:. 0. Hence, fj(J;m+l'O) C J;m+l,j and fj (J;m+l,j) C J;m+l'O'

Therefore, j=Pm+I/2 and the intervals Ji = J;m+l,i U J;m+l,i+j' i = 0,1, ... ,j - 1,

form a cyc1e of intervals of period j that contains the point c. Moreover, Pm <j =

Pm+I/2 <Pm+]' but this contradicts the assumptions made above.

Hence, z] is a periodic point with period Pm + l'

Let F = {F 0' F I , ••. , Fp_ 1} be a cyc1e of intervals for which z] is an internal point

and let ZI E Fo. Assume that c does not belong to any interval F i , 0 ~ i < p. If Fo has nonempty interseetion only with the interval Jp* 0 from Ap* , then P = Pm + 1

m+l. m+1

(because the period of ZI is Pm+ I)' Thus, the intervals Fo U J;m+l' O' F] U J;m+l,I"'"

p* U J* form a cyc1e of intervals of period Pm + I' This contradicts the Pm+l-] Pm+l'Pm+l-]

maximality of Ap* . m+l

Now assume that Fo n Jp* J' f:. 0 for some j > 0 (j < Pm+ 1)' In this case, one m+l'

can prove that j =Pm+l/2 and the images of the interval J;m+l,j UFo U J;m+l'O

form a cyc1e of intervals of period j. But this is impossible because Pm <j < Pm+ l'

Thus, for the cyc1e of intervals F, we have CE Fo. Therefore, P = Pm because

A;m+l is maximal. Hence, ipm(co) E Fo for all i ~ 0 and, consequently, J~~,o C Fo·

This completes the proof of Lemma 5.2.

Section 2 Limit Behavior ofTrajectories 127

It follows from Lemmas 5.1 and 5.2 that Q(ZI,f) = ~) and Q (y, f) = ~) for

all y E 2(~).

Corollary 5.1. Q (0) = IV (0) n (0) m ..l\..m m'

Proof. For y E 2(~) n ~), we have y E Q (y, f). Therefore, y E NW (f). On

the other hand, if y' E 2(~) and V is a sufficiently small neighborhood of y', then

i(V)C ~) forsome j:2.0. Hence, Q~) =NW(f) n 2{~) C ~). In view ofthe

inc1usion 2(~) n ~) C NW (f) established above, this yields the required equality.

Corollary5.2. Foranypoint YE 2{m' thereexists j =j(y) such that fj(Y)E

NW(f).

Proof. This statement is a consequence of the inc1usion

2(m C U ri(n: u n;;: U ~»). i~O

Lemma 5.3. Assume that y E 2(~), V is a sufficiently small neighborhood of y,

and S = Uo ' i(V). Thenthereexists j =j(V) suchthat i(S) = ~). 5,1<Pm

Proof. Since

2(~) C U ri(~» i~O

and the boundary of the set ~) is apart of a trajectory, one can indicate N such that

i(y)E int<l>~) for i>N. Hence, i(S) C ~) for i>N andsufficientlysmall V.

As above, let g = fPm I J* and let Z I be a periodic point lying on the boundary of Pm'O

lp' o. It follows from the definition ofthe set IV that ZI E fk(V) for some k:2. O. m+l' ..l\m

If f\ V) C lp' 0' then fk (y) E () lp' 0 because y E 2{m' Hence, y is a point of m+l' m+}'

extremum for l. In this case, i (y) = c for some i < k but this contradicts the con

dition y E 2(m' Therefore, l( V) contains either a neighborhood of the point zl (i.e.,

of the other end of the interval lp' 0) or a half neighborhood V- of the point ZI m+l·

which does not belong to lp* o. Note that, in the first case, V- is contained in m+l'

fk+Pm(V).

Thus, let V- be a half neighborhood of the point Z I lying in f\ V) and such that

V- n lp' 0 = {ZI}' It follows from the maximality of lp* 0 that V- C g (V-). m+l' m+1'

Let W = Ui~O gi(V} Then W is an interval and g(W) = W.

Let us show that W = J~~.o, where J~~.o is the closed interval with the ends g( c)

and g2(c). Indeed, the points g(Zt) and g2(Zt) belong to W (because Zt E W) and cannot lie on the same monotone branch of the map g because this contradicts the con

dition Zt E Per (I) (one must take into account the fact that g2(Zt):S; gi(Zt) < g(Zt)

for any i ~ 0; recall that c is regarded as the maximum point of the map g). Thus, c E

W. This implies the required equality W = J~~.o' Furthermore, CE int J~~.o and,

therefore, for some i ~ 0, we obtain CE gi(V-). Hence, gi+2(V-) = J~O) 0' i.e.,

i(V)= Jp(O)o andi(Uo<' iM) = <b~) for j=(i+2)Pm' Lemma5.3i;~roved. m' -l<Pm

Corollary 5.3. n~) n int <b~) C Per(f).

Proof. If a neighborhood V of the point YEn~) lies inside the set <b~), then,

by Lemma 5.3, we have V C i (V) for some j ~ O. Hence, the map i possesses a fixed point in the interval V.

---,.".,----

Denote the set n~) n Per(f) by C~). The following statement is a consequence

of Corollary 5.3 because the boundary of the set <b~) is a part of the trajectory of the point c.

Corollary 5.4. n~) \ C~) c { U i(x)}. 0<iS;2Pm

Corollary 5.5. For any point Y E !l{m' there exists j =j(y) such that i(y) C

Per(f).

The proof follows from Corollaries 5.2 and 5.4.

Lemma 5.4. C~) is the perfect part of the set n~), i.e., C~) is a perject no

where dense set (Cantor set).

Proof. If a point YEn~) is not isolated in n~), then it follows from Corollaries

5.1 and 5.3 that Y E C~).

Now let y be an isolated point of the set n~) that belongs to C~). Then y E

Per (f). If the period of y is p, then fP is a homeomorphism in a certain neighborhood

V ofthe point y. Therefore, y E int/'P(V) for all i ~ 1. Hence, y is not a boundary

point ofthe set <b~). By virtue ofLemma 5.3, the interval V contains infinitely many

preimages ofpoints ofthe set n~) and, by Corollary 5.1, the set V n n~) is infinite but this contradicts the assumption that the point y is isolated.

Section 2 Limit Behavior ofTrajeetories 129

Corollary 5.6. NW (fl g(O)) = C~). m

Lemma 5.5. 1f m + 1 < m*, then n~) \C~) = 0; if m + 1 = m*, then the

set n~) \ C~) is either empty or eoincides with the set {Uo . 2 tex)}, i.e., with <l~ Pm

the boundary of the set ~).

Proof. Assume that fee) Eint <1>:+1' Then the boundary of the set ~) does not

belong to n~) because n~) = 2(~) () ~). Hence, in this case, n~) = C~). If

fee) E 0<1>:+1' then g(Jp' 0) = Ip' 0' where g =fPm+IIJ' . Therefore, f has m+b m+b Pm+]'O

no cyc1es of intervals that contain e with periods greater than Pm + l' i.e., m * = m + 1.

This enables us to conc1ude that n~) = C~) whenever m + 1 < m*.

Let f(e) E 0 <1>:+1' For i = 1,2, ... ,2Pm' we consider half neighborhoods W of

the points i (e) that do not belong to ~) and half neighborhoods ~t of the points

i (e) lying in ~). We can assume that W = i (V), where V is a neighborhood of

the point e. Then the neighborhood ".-;- u i (e) U W of the point i (e) contains no

pointsoftheset n~) otherthan i(e) and the point i(e) isisolatedin n~). The statement of Lemma 5.5 now follows from Lemma 5.4.

By using Lemma 5.3 and the definition of the set C~), one can prove the following assertion:

Lemma 5.6. The set C~) is invariant and the map possesses the mixing property

on C~). Moreover, for any subset V of the set C~) open in C~), one ean find

j = j(V) such that icUo<' i(V)) = C~). _1<Pm

Proof. Since C~) is the c10sure of the set of periodic trajectories lying in n~), we have f(C~») = C~), i.e., C~) is a c10sed invariant set. Further, by Corollary 5.3,

the preimages ofpoints ofthe set C~) lying inside ~) also belong to C~). Hence,

by Lemma 5.3, the map f possesses the mixing property on C~). It follows from Lemma 5.6 that the set C~) is, in a certain sense, indecomposable.

The results established above can be formulated as folIows: For m < m* and

Pm + 1/ Pm> 2, the set 2(m () NW Cf) is nonempty and can be represented in the form

2(m () NW(f) = n: U n;;; U n~).

Moreover, for any point y E 2(m' there exists j such that i (y) E n: U n;;; U C~O),

where C~) = n;;;) n Per(f). These sets have the following properties:

Ci) n: and n;;; are cyc1es;

130 Topologieal Dynamies of Unimodal Maps Chapter 5

(ii) C~O) is a closed invariant set with the structure of the Cantor set and admits a

decomposition into Pm closed subsets cycIically permutable by the map f;

(iii) f pos ses ses the mixing property on C~);

(iv) if m + 1 < m* then n (0) = C(O). if m + 1 = m* then either n (0) = C(O) , m m ' , m m

or n;.?) \ C~O) = {lee), i = 1,2, ... , 2Pm}.

Dynamics of the Map f on the Set !J{11f" for m* < 00. As above, we consider the

map g = fPm'1 J' (for definiteness, we assurne that e is its maximum point). If Pm*'O

g(e) ~ e, then !l{m*fl NW(f) = n:* U n:* (see Fig. 37a), where n:* and n:* are cycles that contain points z and x, respectively. Furthermore,

fPm' (!l{m*) = n:. u n: .. If g (e) > e, then g2 (e) < e< g (e). (The case where e < g2 (e) contradicts the con

dition m = m* because, in this case, g has a cycIe of intervals of period two which

contains e.) Note that the sets !l{m*n NW(f) and !l{m.n NW(fPm'I~J may be

distinct:Indeed,if z:;t:x and g(e)=z' (i.e., g(e)E c)J;m',l)' thenthepoints l(e), i =

1,2, ... 'Pm" belang to the set NW(f) but not to the set NW(fPm'l ~J. This

follows from the argument used in the previous case. Moreover, in this case, the points

l(e), i = 1,2, ... ,2Pm*_!' belong to the set NW (f) \ Per(f).

Denote !l{m* n J;m"o by !l(m*,O and consider the set !l{m',O n NW(g). This set

b d · h c n' U n** U neO) h n* {} n** can e represente mt e lorm um*,o ~oI;m',O ~oI;m*,O' W ere Um*,O = Z , ~oI;m*,O = {x}, and n;.?2,0 = !l(m*,O n [g2 (e), g (e)]. The proof of this fact is similar to the proof

ofthe previous case. Considerthe set c~02,0 = n~2,0 n Per(g). It is cIearthat c~02,0 =

n;.?2,0 n Per(f).

Let n:. and n:. be the cycles ofthe map f that contain, respective1y, the points

z and x,

[g2(e),g(e)], <1>(0) m*

neO) m*

(/) n (0) .l\'m' m*' U l(C~02,0) = n~2 nPer(f).

O~i<Pm'

Then, for any point y E ~*, there exists j = j(y) such that i(y) E n:* u n:. u c~2. An analog ofthe assertion afLemma 5.3 holds for points of the set !l(m*P n (x, x')

(see Fig. 37). Therefore, the sets defined above has the following properties:

Seetion 2 Limit Behavior oJTrajectories 131

(i) c;;;2 is a closed invariant set of the map J, which can be decomposed into Prn*

closed sub sets cyclically permuted by the map J; this set has either the structure

of cycle of intervals (i.e., coincides with <1>;;;2> or the structure of Cantor set;

(ii) the map J possesses the mixing property on c~02;

(iii) the set Q;;;2 either coincides with c~o2 or

In order to prove these assertions, we use the following statement, which is similar to Lemma 5.3:

Lemma 5.7. Let V be a suJJiciently small neighborhood oJ a point y E

2{ *\ U· aJ* . and let S = Uo ' i(V). Then there exists j = j(V) m O'5:.l<Pm'" pm""l 5:t<Pm*

such that i (S) :::) ~02.

It follows from Lemma 5.7 that c;;;2 = ~o2 whenever 'Brn* () <1>;;;2 = 0 (the set

'Brn* is defined above in constructing the phase diagram of the map f). If 'Brn* () <1>;;;2 ::1=

0, then it follows from Lemma 5.7 that 0;;;2 is a nowhere dense set and c;;;2 is a perfeet nowhere dense set. By analogy with the case m < m*, one can prove the other

properties of the sets Q;;;2 and c;;;2. We only note that the equality m = m * implies

that if 0;;;2 \ c;;;2 ::1= 0, then a neighborhood of the point c is a wandering interval of

the map J. In this case, 0;;;2 \ c ~o2 = NW Cf) \ Per Cf).

The Case m* = 00. Here, <1>: = nrn>l <1>:, 'Boo = int <1>:, and 2{00 = <1>: \ 'Boo•

The investigation of the limit behavior of trajectories of the map J on the set <1>: is based on the use of the following lemma:

Lemma 5.8. For any point y E ~ and any its neighborhood U, there exist j;::O: 0

and m;::O: 1 such that

JPm( <1>:) c i ( U i(U»). 05,i<Pm

Proof. Let U be a neighborhood of the point y E '1\..00' Then U () a<l>: ::1= 0 for

some m;::O: 1. Therefore, JPm( U) () JLpm( U)::I= 0 and ni~OU j~i i (U) is a cycle of in

tervals. If c !i" i (U) for all j;::O: 0, then this cycle of intervals does not contain the point

132 Topological Dynamics oJ Unimodal Maps Chapter 5

c and, eonsequently, the ro-limit set of any point of the interval U is a eyde. However, for the point y E U, this is impossible beeause ro(y) C 2(= and 2(= () PerU) = 0.

Henee, c E i (U) for some j ;0: O.

On the other hand, i (U) () d <1>: =f. 0 for any i;O: 0 beeause J (d <1>:) C d <1>:.

Therefore, i+ 1(UO< i(U» eontains JPm(:). -1<Pm

Lemma 5.8 yields the following properties of trajeetories of the set <1>:.

Property 1. For any point y E 2(=, its domain oJ influence Q (y, J) coincides

with ~ = nm~l JPm( ~).

Proof. It follows from Lemma 5.8 that ~ C Q (y,j). Note that the set Q (y, j) is

invariant. On the other hand, it is not difficult to show that ~ is the maximal invari

ant sub set ofthe set <1>: beeause nm~l JPm(:) = ni~l i(nm~l <1>:).

Consider the set n~) = ~) () NW(f). Sinee eaeh component of the set '13= is

wandering, it follows from Property 1 that n~) = <1>: () NW (f).

Property 2. IJ Y E n~) and y =f. i(c), i = 1,2, ... , then y E Per(f).

Proof. Under the conditions of Property 2, the point y is an internal point of the set

JPm (<1>:) for any m;O: 1. By Lemma 5.8, for any sufficiently small interval U that

eontains the point y, we have U ci (U) for some j;O: O. Hence, the map i possesses

a fixed point in the interval U and y E Per (f).

Denote the set Q~) () PerU) by C~).

Proof. Assume that Q~) \ C~) =f. 0. If c ~ '13=, then c E 'R..sx, and we have c E

Q~) by Property land CE PerU) by Property 2. Hence, i(c) E Per(f) for any i ;0: 1. Therefore, we must eonsider the ease c E '13=.

If J (c) E '13=, then i (c ) E '13= for any i;O: 1 beeause the map J is monotone on

any eomponent of the set '13= that does not eontain the point c. Hence, J( c) ~ '13=. By

Property 1, J (c ) E Q~) and, therefore, i (c ) E n~) for any i;O: 1.

Let us show that i(c) ~ PerU) for i E {I, 2, 3, ... }. Let i;O: 1 and m > i. Then

i (e) is the end of the interval JPm(J;m,;}' The equality m* = 00 implies that the point

i (c) does not lie on the boundary of the set <1>:. Therefore, there exists a neighbor

hood U of the point i (e) such that JPm (U) C JlIn (<1>:). Let U+ be the part of the

neighborhood U that lies in JPm (J;m,;) and let [J be the remaining part of this neigh-

borhood. One can regard U+ as the image of the component of the set 'B"", which

contains the point c. Then I (u+) () er = 0 for any j;::: 1, i.e., U+ () Per(f) = 0. On

the other hand, U- () Per(f) = 0 because fPm(lr) C flln (4):), lr () f m (4):) = 0,

and the set j>m ( 4>:) is invariant. This completes the proof.

Property 4. F or any point y E 4>:, its OJ-limit set coincides with the set C~O).

Proof. Let J;m,i be an interval from the cyde of intervals A;m' There are Pm+lPm

intervals from Ap* . in Jp* i' The utmost left and right intervals in this collection are m+J m'

called one-sided in the sense that all other intervals from Ap* . lying in the interval m+)

J;m. i are located on the one side of the indicated intervals. All other intervals are called

two-sided. One can show that the intervals Jp* . k, k = 1,2, ... , 2Pm' are one-sided m+}'

intervals of the cyde of intervals Ap* . in the intervals of Ap* . m + ] m

Let Jp* . s be a one-sided interval in Jp* i' If, e.g., the interval 1* . contains m+J' m' Pm,l

no intervals of Ap* +). to the left of Jp* . s' then the trajectories of the points of the set m m+J'

4>: cannot have limit points in Jp* i to the left of Jp* . s but any trajectory of this m' m+J'

sort has limit points in J p* i to the right of Jp* . S' If the interval Jp* . s is two-m' m+J' m+.J'

sided in Jp* i' then the trajectories of all points of the set 4>: has limit points in m'

J;m. i both to the left and to the right of J;m+i'S'

Let [a, b] be a component of the set \I>: (a:::; b). Then there exists a sequence

{im};;;=1 suchthat imE {O, 1, ... ,Pm-d and [a,b]= nm",IJ;m.im.ltfollowsfromthe

indusion J;m+l,im+1 C J;m,im that im+1 = im + kmPm' km E {O, 1, ... , Pm+ ,I Pm - I}, and

{im} ;;;=1 is a nondecreasing sequence. Since PI = 1, we have i l = 0 and im + , = k, PI +

k2P2+ ... +kmPm form>l, where knE {0,1, ... ,Pn+IIPIl-1}, n=I,2, ... ,m. Hence, there exists a one-to-one correspondence between the family of components of the set

4>: and the family of infinite sequences of integer numbers of the form (k l , k2 , ... ),

where knE {O, 1, ... ,Pn+IIPn-1}.

Thus, let [a, b] = n I Jp* " . Assume that there exists m;::: 1 such that m~ nt'rn

Jp* . i . is a one-sided interval in Jp* i for any j;::: 1. In this case, im +). =2Pm for m+J·m+) rn'rn

any j;::: 1. This inequality implies the equality kn = 0 for sufficiently large n. If, for

any m, there exists j =j(m) for which Jp* . i . is a two-sided interval in Jp* i ' m+J'm+) rn'rn

then it is possible to show that either kn = 0 for any n;::: 1 or there are infinitely many

nonzero elements in the sequence (k" k2, ... ) that corresponds to the component [a, b].

Let [a, b] = n J/*, " be a component of the set <1>: and let (k l , k2 , ... ) be the m~l m' m

corresponding sequence. Assume that either all kn are equal to zero or the number of

nonzero elements in this sequence is infinite. We fix f. > 0 and choose m such that

Jp* i C (a - f., b + f.). By assumption, there exists j ~ for which the interval m'm

1* . is two-sided in JP*m ,,'m' Therefore, for any point y E <1>:, we have ffiJ(Y) n Pm+ j.lm+ j

(a - f., a) '* 0 and 0) J(y) n (b, b + f.) '* 0. Hence, the points a and b belong to the set

0) f(Y) because this set is closed.

If km '* 0 for some m ~ 1 and km+i =0 for all j ~ 1, then the component [a, b]

contains the point i m +J (e), where im+] = k]p] + k2P2 + ... + kmPm '* O. Thus, the inter

val Jp* ,i ,is one-sided in Jp* i because im +J' =im for any j ~ 1. Therefore, if m+J'm+} m'm

a,* b, then one end of the component [a, b] lies in ffi J(y) and the other end does not

belong to this set.

Following the proof of Property 2, we can show that i (e) i" O)f(y) for any i ~ 0

whenever e E 'B=. This completes the proof of Property 4.

Consider the dynamics of the components of the set <1>: under the map f in more details. As shown in the proof of Property 4, there exists a one-to-one correspondence

between the components of the set <1>: and the set of infinite sequences (k], k2 , ... )

with ki E {O, 1, ... , Pi+ dPi - I}. (Note that each of these sequences can be interpreted

as a digital representation of a number from the interval [0, 1] similar to its decimal representation.) In what follows, for the sake of brevity, the sequences (k], k2, k3, ... ) are written in the form O. k] k2 k3 ... and each ki is called the value of the ith digit.

We define the sum of two sequences (numbers) K = O.k] k2 ... and L = 0.1]12 ", as follows: The value of the ith digit in K is added to the value of the ith digit in L mod

uloPi+]/pi andtheoverflowunitisaddedtothe next digit. Thus,if Pi+]/pi=lO for

all i ~ 1, then 0.999 ... + 0.100 ... = 0.000 .... It is easy to check that the family of se

quences corresponding to components of the set <1>: equipped with this operation of addition is an Abelian group. The action of the map f corresponds to the operation of

adding the number F = 0.100 .... More precisely, if a component K of the set <1>: corres ponds to the number K = O. k] k2 ... , then the number K + F corresponds to the com

ponent of <1>: which contains f(K) and the number K - F corresponds to the preimage of the component K under the map f

Denote the family of components of the set <1>: by '1( and consider a map F: '1( ~

'1( defined as follows: For K, L E '1(, F(K) = L if f(K) C L. (As mentioned above, F(K) = K + F, where F =0.100 .... ) The distance p between elements K = O.k] k2k3 ...

and L = 0.1]1213 ", of the set '1( is defined by the formula

~ Ik. -1·1 p(K,L) = Li -'-i '-,

i=] (r;) where

Pi+]

Pi

By using the reasoning presented above, one can establish the following properties of

the set '1( and the map F:

(i) the set '1( has the cardinality of continuum;

(ii) the map F: '1( ~ '1( is a homeomorphism of '1( onto itself;

(iii) the <x-limit and ü)-limit sets of any trajectory of the dynamical system generated

by the map F on '1( coincide with '1(, i.e., '1( is the minimal set of this dynamical system;

(iv) the map F possesses the mixing property on '1(; more precisely, if UE is the

E-neighborhood of a point K E '1(, then one can indicate j = j(E) such that

Fj(UE ) = '1(.

Let us now summarize the results obtained in this section. For any unimodal map f, the set of its central motions admits the following repre

sentation:

m*

Per(f) = Po(f) U U C~o), m=l

where m* ~ 00 and

(a) C~o), m = 1,2, ... , m*, are nonempty c10sed invariant sets and Po(f) is an invariant subset of Per (f) (which may be empty);

(b) the sets Po(f) and C~), m = 1,2, ... , m*, are mutually disjoint except, pos

sibly, the sets c~o2 and C~OLI with m* < 00, which may have a common cy

c1e;

(c) the map flc(O) possesses the mixing property; in particular, this map is tran-m

sitive, i.e., one can indicate a point y E C~) such that ü)j(y) = C~);

(d) the set c~o2 contains ü)j(c);

(e) for m < m *, the set C~) is either a cyc1e or a Cantor set; for m* < 00, the set

c~2 is either a cyc1e, or a cyc1e of intervals, or a Cantor set; for m* = 00, the

set c~2 is a Cantor set, which is the minimal set of f

This decomposition is usually called the spectral decomposition of the set Per(f). Generally speaking, by using the phase diagram, one can construct similar decompositions for any invariant set of the map f Thus, for the investigation of the dynamics of a


map, it might be useful to have the relevant decomposition of the set of its nonwandering points; in the case of unimodal maps, this decomposition slightly differs from the de

composition of the set Per(f) , namely,

m*

NW(f) = Po(f) U U n~), m=l

where m* ~ 00, and the sets n~) coincide, respectively, with the sets C~) for all m

except, possibly, either n~l or n~tl (one of these sets may differ from the corres

ponding set in the decomposition of the set Per (f) by the presence of apart of the tra

jectory or of the entire trajectory of the point fee) that does not belong to Per(f)). The results established in this section imply the following important properties of the

dynamics of unimodal maps:

(i) the sets NW(f) and Per(f) may be different if and only if e e; NW(f) and

fee) E NW(f);

(ii) NW (f) \ Per (f) is the set of points from the set NW (f) \ Per (f) isolated in NW(f);

(iv) NW(fINW(f») = Per(f).

Note that it follows from Property 1 that if e E NW (f) or f(e) e; NW (f), then

NW(f) = Per(f). Another problem, which might be important for the investigation of the behavior of

trajectories of dynamical systems, is to describe the behavior of "typical" (generic) trajectories. In the most general case, a property of trajeetories should be called generie if it is observed for the trajeetories of the points of some set of the seeond eategory. For any unimodal map f: I ~ I, generic trajeetories possess one of the following three properties:

(a) after finitely many steps, a trajectory hits a eyde of intervals, where the map f possesses the mixing property;

(b) the trajectory is attracted by ())j(e);

(c) the trajeetory hits an open invariant set ofthe map f which does not eontain the point e.

To prove this assertion, we show that the set X of points of the interval I whose

Section 3 Maps with Negative Schwarzian 137

trajectories satisfy one of the conditions (a), (b), or (c) can be represented in the form of

at most countable intersection of open dense sub sets of the interval I. Indeed, it follows from the results of this section that

m* m*

I = U P('Bm,J) U U P(C::!),f) m=! m=l

(recall that P(A,J) = {x EI: (üf(x) CA} for A E Zl). For m < m*, we can write

i~O i~O

Note that Ui~O f-i(C~O» is a nowhere dense set because C~O) is a nowhere dense set.

One can easily show that

x = l\ ( U U r i ( C~O) ) ). 15:m<m* i~O

The assertion formulated above now becomes obvious.

3. Maps with Negative Schwarzian

The general form of phase diagrams of unimodal maps reflects dynamics possible for continuous maps. Thus, it is important to study the problem of realization of these possibilities for smooth maps. In particular, it is quite interesting to answer the following two questions: Wh at maps are characterized by the "most simple" phase diagrams (i.e., by phase diagrams without wandering intervals and cyc1es of intervals that do not contain

the point c)? Are there smooth maps characterized by the "most complicated" dynamics

(i.e., maps for which the estimates for the number of c1asses B~) and ['~) given in Section Z are attained)? These and other similar questions are discussed in the present section.

It is c1ear that, in typical situations, each c1ass of cyc1es of intervals B~) is associated with an attracting cyc1e. In the following lemma, this assertion is formulated rigorously:

Lemma 5.9. Assume that a map g: [a, b ] ~ [a, b] (a *- b) is continuous and

monotonicalty nondecreasing, g (a) = a, and g (b) = b. Then there exist a fixed point ZE [a,b] ofthemap g and E>O suchthateither z+Ea and g(x)?xforalt XE (Z-E<Z).

Proof. Assume that the assertion of the lemma does not hold for fixed points a and b. Then one can indicate points x, y E (a, b) such that x <y, g(x) > x, and g(y) < y.

In this case, it follows from the continuity of the map g that it possesses one more fixed point in the interval (a, b). Hence, if the assertion of the lemma is violated for two fixed points of the map g, then the interval between these points contains a fixed point.

Thus, if the lemma is not true, then the set Fix (g) = {x E [a, b]1 g (x) = x} is dense in [a, b] and, consequently, g (x) = x for all XE [a, b] but this contradicts the assumption of the lemma. Lemma 5.9 is proved.

Corollary 5.7. lf [c, d] is a component of Fix (g), then g (x) = x for alt XE

[c,d]. If (c,d) isacomponentof [a,b]\Fix(g), then g(c)=c, g(d)=d, and either g(x»xforall XE (c,d) or g(x)<xforall XE (c,d).

Note that the case of an orientation-reversing map g (i.e., the map g is nonincreas

ing) can easily be reduced to the case described above by passing to the map g2.

Thus, the question about the number of classes B~) in the phase diagram is closely related to the question about the number of attracting or serniattracting cycles of a map.

It is weIl known that unimodal maps defined by quadratic polynomials may have at most one attracting or semiattracting cycle. This property of quadratic maps was established as early as at the beginning ofthe century by Julia [1] and Fatou [1] when investigating rational endomorphisms of the Riemann sphere. It is important to clarify which property of quadratic maps is responsible for the restrictions imposed on the number of sinks. Julia and Fatou showed that the number of sinks of the maps under consideration is bounded by the number of critical points of these maps. However, by using Lemma 5.9, one can easily construct even a monotone map of the interval (of any smoothness) with any (finite or countable) number of attracting cycles.

Another important property of quadratic maps is their convexity. This property also cannot playa decisive role in this case because convexity is not invariant under iterations of maps. This observation is clarified by the following examples:

lExWDIDJPllle S), ll. Considerthe orientation-reversing homeomorphism generated by

thefunction f(x) = ~, XE [-t, 2]. We have f([ -t, 2]) = [-t, 2] and, for XE [-t, 2],

j'(x)=- ~ ~- t and j"(x) = 4-::::: t. On the other hand, f2(x) = x and, conse-x x

quently, (J2)(x) = ° forany XE [-t,2].

lExWDIDJPllle s),2, Consider the map gE E C~([ -t, 2]) generated by the equality

jO, 1/2 ~ x ~ 1,

gE(X) = €e 11l - x sin _1t_, 1< x ~ 2. x-I

It is not difficult to show that, for any r::::: 0, 11 gE 11 er ~ ° as € ~ ° and the set of

zeros of the function ge (x) coincides with the set

z = [~, IJ U {i~I}~ . 2 1 ,=1

Moreover, the function ge(x) changes its sign on passing through each isolated point of

the set Z (g~ c : 1) :;o!: 0 for all i:? 1).

Wechoose c>O suchthat IIge 1l C2< t andconsiderthemaP!J(x)=f(x)+ge(x),

where f(x)= ~. Then

For the map f?, we arrive at the equality

Fix U?) = {I} U {_. 1_' , i ~ 1}= . 1 + 1 1 i=1

which can be established by direct computation. Thus, if x E [ !, 1) \ {i ! I};: I' then

{ '+I}OO f l (x) = f(x) E (1,2]\ T i=1

and

because ge(f(x» :;o!: O. For any i:? 1, we have g~C: I) -:f. 0; therefore,

As follows from Corollary 5.7, the map f? has countably many attracting fixed points

and countably many repelling fixed points. In particular, fJ has countably many attract-

ing cyc1es of period two (these cyc1es are formed by the pairs of points {i! I' i: I} with

odd i:? 1).

lExM:Dl.plle 5.3. Let

rrn (i )(i + 1 ) gn,e(x)=c(x-l) -.--x -,--x, i=1 1 + 1 1

Forany n21 and r21, 11 g11, E llcr-7 ° as f-7 0. Wefix n21 andchoose f>O

such that 11 g11,Ellc2 < k. Consider the map hex) = fex) + g11,E(X), where fex) = ±' XE [ t, 2]. Acting as in Example 5.2, we conclude that

Fixuh = {I} U {~i~I}~ I + 1 I ;=1

and the pairs of points ; i 1 and ; ; 1 form 2-periodic attracting cycles of the map 12 for odd i and repelling cycles far even i.

Note that the monotone map 12 constructed in Example 5.3 is analytic. Hence, analyticity, as weB as convexity, cannot guarantee the uniqueness of attracting cycle for unimodal maps.

On the other hand, unimodal maps from the family fa,ß: x -7 axe-ßx, x 2 0, a > 0,

ß > 0, are neither convex nar concave. Nevertheless, they have at most one sink just as quadratic maps (Jakobson [2]).

Maps from this family and quadratic maps are characterized by the foBowing co mmon property:

Their Schwarzian

f'" 3 [f"]2 Sf = f' - 2 l'

(which is also called the differential Schwarz invariant or Schwarzian derivative) is negative in the entire domain of its definition. A remarkable property of the Schwarzian is the invariance of its sign under iterations of the map: Since

S(fo g) = Sf(g)(g') + Sg

(this equality can be verified by direct ca1culation), we have Sr< ° (> 0) whenever Sf < ° (> 0). Below, we show that just the negativity of the Schwarzian and the fact that the corresponding maps possess a unique critical point are responsible for the existence of at most one sink.

Before studying dynamical systems, we consider some properties of the maps whose Schwarzian preserves its sign.

Parallel with quadratic functions, there are many other primary functions with sign

preserving Schwarzian. Thus, it is negative for x3, e X, sinx, and tan- 1 x and positive

for the corresponding inverse functions VX, In x, sin-1 x, and tanx (at aB points where the functions and their Schwarzians are weIl defined). The indicated property of inverse

functions is explained by the formula Sr l (x) = -Sf(x)![f'(x)] 2.

Section 3 Maps with Negative Sehwarzian 141

It is not difficult to show that Sf(x) = 0 on an interval 1 if and only if fex) is a linear-fractional function on this interval.

Let fE C3 (I, 1). Assurne that the Schwarzian of the map f is weIl defined and preserves sign on an interval 1 (i.e., it is either always negative, or always positive, or al

ways equal to zero). In this case, the map f has the following properties:

Property 1. If f'(x)Sf(x) < 0, then thefunetion f'(x) has no loeal minima on

an interval I; if f' (x) Sf(x) > 0, then it has no loeal maxima on this interval; if f' (x) S f (x) = 0, then the funetion f' (x) is monotone.

Proof. Assurne that f'(x) possesses a local minimum at a point a. Then f"(a) = 0

and the condition of minimum implies the inequality f'(a) Sf(a) = j"'(a) ?:: O. The other assertions are proved similarly.

Property 2. If Sf(x) < 0, then min If'(x)1 = min If'(x) I. If Sf(x) > 0, then XEl xEal

max If'(x)1 = max If'(x)l· XEl xE al

Property 3. The funetion fex) has at most one point of infleetion in 1 (i.e., at

most one point where f" (x) = 0).

These properties immediately follow from Property 1. An exclusive place occupied by maps with negative Schwarzian in the collection of

maps with sign-preserving Schwarzian is explained by the following assertion:

Proposition 5.1. Let fE Cl (I, I) be a unimodal map and let K (f) = { XE 1 I f'(x) = O}. Suppose that fE C3 (1\K(f)) and the Schwarzian of the map f pre

serves its sign on the set I\K(f). Then Sf(x) < 0 for XE I\K(f).

Proof. It follows from the definition of Schwarzian that if g(x + d) = af(x) + b

for some a E lR \ {Ol, b, d E lR, then Sg(x + d) = Sf(x). Hence, without loss of generality, we can assurne that 0 E 1 and e = 0 is the maximum point of the map f: 1 ~ I.

Let 1 = [y, y']. Then f(O) > f(y'). We choose a constant b such that the map g (x) = f(x) + b satisfies the inequalities g (0) > 0 and g (y') < O. Then there exists a unique point z E (0, y') for which g(z) = O.

Consider the function g2 on the interval [0, z]. It is easy to check that g2 increases

on (0, z) and satisfies the equalities (g2)'(0) = (g2)'(z) = O. If Sg(x)?:: 0, then, by

Property 2 of the maps with sign-preserving Schwarzian, g2 (x) ?:: 0 for all x E [0, z]

but this contradicts the inequality g2(0) = g(g(O)) < g(O) = g2(Z).

Remark. The condition of unimodality of f in Proposition 5.1 is not essential: It follows from the proof that it suffices to impose the condition that f is not a con-

142 Topological Dynamics oJ Unimodal Maps Chapter 5

stant. One can also omit the inc1usion J(I) C L According to the proof of Proposition 5.1, unimodal maps with positive and zero

Schwarzian cannot be differentiable at the point c. Moreover, for these maps, both one

sided derivatives are not equal to zero at the point c. In what folIows, unimodal maps satisfying the conditions of Proposition 5.1 are re

ferred to as S U-maps or S-unimodal maps.

To establish restrictions that should be imposed on the number of sinks for S U-maps, we consider some properties of periodic trajectories of maps with negative Schwarzian.

Lemma 5.10. Let JE c3 (/, 1), let SJ(x) < 0 Jor XE I\K(f), where K(f) =

{x EIl J' (x) = O}, and let B = {ßo, ß I' ... , ßn- d be a cycle oJ the map f. A s -

sumethat IIl(B)I~ 1, where Il(B)=J'(ßo)f(ßI)· ... -f'(ßn-l) is the multipli

er oJ the cycle B. Then B is either an attracting cycle or a semiattracting cycle oJ the map f

Proof. It suffices to consider the case IIl(B) 1= 1. If Il(B) = 1, then, for the map

g = r, we have g' (ßo) = 11 (B) = 1. If, in this case, g" (ßo) * 0, then the cyc1e B is

semiattracting. If g"(ßO) = 0, then it follows from the condition SJ(x) < 0 that

S g (ßo) = g'" (ßo) < 0 and, consequently, B is an attracting cyc1e.

If Il(B) = -1, then we consider the map g2 = J2n. For this map, (g2),(ßO) =

(g'(ßO))2 = 1 and (g2)"(ßO) = g"(ßO)(g'(ßO) 2 + g"(ßO)g'(ßo) = O. The condition

SJ(x) < 0 implies that S g2 (ßo) = (g2)'" (ßo) < 0 and, consequently, in this case, the

cyc1e B is also attracting.

Lemma 5.11. Assume that JE cl (/, 1), JE c3 (I\K(f)), and SJ(x) < 0 Jor x E

I\K(f). Let B = {ßo, ß I' ... , ßn- d be an n-periodic attracting or semiattracting

cycle oJ the map J and let Po (B) be its domain oJ immediate attraction. IJ n > 2,

then po(B)n K(f) * 0 andif n~2, then po(B)n {K(f)U Cl!} * 0.

Proof. As fOllows from the results established in Chapter 1, the set Po(B) consists

of disjoint intervals Jo, J1, ... , Jn_ l , which form an n-periodic cyc1e of intervals. If n >

2, then the indicated collection of intervals contains an interval Ji such that Ji n ClI = 0. Let J 0 be an interval of this sort and let a and b be its ends. Then the following three

cases are possible for the map g = Jn..

(i) g(a) = a and g(b) = b;

(ii) g(a) = b and g(b) = a;

(iii) g(a) = g(b).

Let us show that, in all cases, the interval 10 contains the critical point of the map g.

In case (iii), this is obvious. Case (ii) is reduced to case (i) if we consider the map g2.

Let g(a) = a and g(b) = b. Assume that 10 n c(g) = 0 and the cycle B is at

tracting. Then ßOE (a,b), theinequality g(x»x holdsfor XE (a,ßo), andtheinequality g(x) < X holds for XE (ßo, b). Hence, by the law of mean, there exist points

ZI E (a, ßo) and Z2 E (ßo, b) such that g' (ZI) = g' (Z2) = 1. By Property 2 of maps with sign-preserving Schwarzian, we have g' (x) ~ 1 for all XE [ZI' Z2], which is im

possible in view of the fact that g(Z2) - g(zl) < Z2 - ZI.

If B is a semiattracting cycle, then either ßo = a or ßo = b; moreover, we have

g'(ßo) = 1. If ßo = a, then g'(z) = 1 for some point Z E (a, b). Hence, we again conclude that g'(x) ~ 1 for all XE [ßo, Z], which is impossible because g(x) < x for

XE (ßo, b). The case ßo = b can be investigated sirnilarly.

Thus, for n > 2, we have 10 n K(g) 7: 0. Let cI E 10 n K (g). It is easy to show that

n-I

K(g) = U f- I (K(f)). ;=0

Hence, fk(cI) E K(f) for some k <n, i.e., f\10) n K(f) 7: 0 and Po(B) n K(f) 7: 0.

Similar reasoning is applicable in the case where n ~ 2. However, in this case, it is possible that the ends of intervals of the domain of immediate attraction are not fixed

points of the map g (and an end of the interval I is attracted by the cycle B).

Corollary 5.8. Let fE cl (1,1) and let fE c3 (1\K(f)). Assume that Sf(x) < 0

for x E 1\ K (f). Then the number of attracting and semiattracting cycles of the map f does not exceed the number of components of the set K (f) plus two.

The assertion of Corollary 5.8 immediately follows from the statement and proof of Lemma 5.11.

lEJtUDljpl[e 5.~. Consider a map f: I ~ I defined by the formula (see Fig. 40):

fex) = j-J2 sin 1t (x - .!..) +.!.. x E [0 .!..] 4 4 4' , 2 '

-J2 sin 1t (x _ ~) + 2 + -J2 x E (.!.. 1] 4 4 8' 2'·

We have fE Cl (1, 1) and fE C3 (1\ { ! }). The Schwarzian of the map f is negative

everywhere except the point ! (where it is not defined) and f is a monotone function

without critical points. Nevertheless, the map f has three attracting fixed points: 0, !, and 1.

144 Topological Dynamics of Vnimodal Maps Chapter 5

y ~r I~------~------~ 1~------'-------~

o x 0 a C T x

Fig.40 Fig.41

Similarly, one can construct a homeomorphism f: I ~ I with negative Schwarzian

which belongs to the dass C3 everywhere except countably many points (where fE

Cl) and has countably many attracting cydes.

where

(see Fig. 41).

XE [0, ~],

XE(~,l],

x (1 )3 (1 )3 g(x) = 140 f "2 - x "2 + x dx o

The map f belongs to C3 (I, l) and has three attracting points a, b, and c. It is not difficult to show that the Schwarzian Sf is negative everywhere except the points 0, t, and 1, where the first three derivatives vanish. Thus, the map f: [a, c] ~ [a, c] has

exactly one critical point x = t and three attracting fixed points.

For S V-maps, the assertion of Corollary 5.8 implies the following theorem:

Theorem 5.1. Let f: I~ I be an SV-map. Assume that j'(x) =/:. 0 for x=/:. c.

Then f has at most two attracting or semiattracting cycles. Moreover, if there are two

cycles of this sort, then one of these cycles is a fixed point attracting the trajectory of at least one ofthe ends ofthe interval 1.

Proof. Let 1= [a, b], let c be the maximum point, and let n be the period of an at

tracting or semiattracting cycle B of the map f As follows from Lemma 5.11, we must

consider only the cases n = 2 and n = 1. If n = 2, then Po(B) = Jo U J1. If, in this

case, Ji n aI = 0 for some i E {O, I}, then it follows from the proof of Lemma 5.11

that CE Po(B). Now assume that a E Jo and bE J1• In view ofthe unimodality of the

map f, in this case, one can also prove that CE Po(B). Hence, under the conditions of the theorem, CE Po(B) for any n ~ 2.

a b c

Fig.42

For n = 1, let x* denote a fixed point of the map f such that a \i" P(x*). In this case, by using Property 2 of maps with sign-preserving Schwarzian and the unimodality of the map f, we can also prove that CE P(x*). This completes the proof of the theorem.

Corollary 5.9. lf, under the conditions ofTheorem 5.1, 1'(x) \i" [0, 1] for XE a I, then f has at most one attracting or semiattracting cycle.

This statement follows from the proof of Theorem 5.1. Indeed, if a E Po (x* ) for some attracting fixed point x*, then Property 2 implies that 1'( a ) < 1 (provided that c is the maximum point). Simple analysis demonstrates (see Fig.42) that if 1'(a) < 1, then either the structure of the map f is quite simple or the investigation of this map can be reduced to the investigation of a map that satisfies the conditions of the corollary.

Theorem 5.2. Let f: I ~ I be an S U-map. Then the set Fix (r) is finite for

any n ~ 1. In particular, the set Per (f) is at most countable.

146 Topological Dynamics of Vnimodal Maps Chapter 5

Proof. Under the conditions of the theorem, the map fn has finitely many intervals of monotonicity. If the set Fix (f") is infinite, then one can find an interval of monoton

icity of the map r that contains infinitely many fixed points. Hence, this interval con

tains infinitely many points at which the derivative of the map fn is equal to one. By

virtue of Property 2, there exists an interval on which fn coincides with the identity map but this contradicts, e.g., Lemma 5.10.

Remark. By using Theorem 5.2, one can collect a lot of information about the structure of the set Per (f) for S V-maps. Thus, it follows from Theorem 5.2 and results established in Sections 5.1 and 5.2 that, for S V-maps, the set Per (f) is closed if and only

if m* < 00 and Pm+ /Pm = 2 for all m < m* in the phase diagram of the map f. In this

case, the periods of periodic orbits of the map f are uniforrnly bounded. The following assertion is a consequence of Theorem 5.1 and theorems on coexis

tence of cycles of various periods (and types) (see Seetion 3.2).

Theorem 5.3. Assume that an S V-map f possesses a simple cycle of period n and that this cycle is a sink. Then f has no cycles of periods m such that n <l m. Moreover, the map f has no cycles whose types are greater than the type of the indicated attracting (or semiattracting) cycle.

The following statement provides a simple sufficient condition for the negativity of Schwarzian in the case of polynornial maps.

Proposition 5.2. If fex) is a polynomial whose degree is greater than one and alt roots ofthe equation f'(x) = 0 are real, then Sf(x) < 0 for alt x such that f'(X):j:. o.

Proof. Let n + 1 be the degree of the polynornial f(x), n ~ 1. Then, by the condition of Proposition 5.2, we can write

n

f'(x) = ao TI (x - ai), ;=1

where ao:j:. 0 and ai E lR. Therefore,

Sf(x) = 2 I I_i (i 1 J2 < 0 i=1 j=2 (x - ai)(x - aj) 2 i=1 (x - ai)

because

In particular, the Schwarzian of a polynomial is negative whenever all its zeros are real. The well-known fact that the Schwarzians of quadratic maps are negative also follows from Proposition 5.2.

Note that polynomial maps have the following property, which is not based on the characteristics of the Schwarzian:

Let f(x) = anXn + an_1 xn- I + ... + ao, where n:?: 2 and ai E IR. Then the map f:

IR ~ IR has at most n - 1 attracting or semiattracting trajectories. More exactly, if

F(z) = anzn + an_1 ~-I + ... + ao, where z E ([ I, then the number of sinks of the real

map f: IR ~ IR does not exceed the number of different complex roots of the equation

F'(z) = O.

Indeed, let z = x + iy. Then F(z) = q>(x, y) + i'Jf(x, y), where q>: IR 2 ~ IR and

'Jf: IR 2 ~ IR are such that <p(x, 0) = fex) and 'Jf(x, 0) = 0 for x E IR. It follows from

the Cauchy-Riemann conditions for complex functions that ~~ = ~ and ~~ = - ~~ . Hence, for z = x + i· 0, we have ~~ = - ~~ = 0 and dF(z) = f'(x)dz. Therefore,

every attracting cyclc of f: IR ~ IR is an attracting cycle of the map F: ([ 1 ~ ([ 1 and

every neutral cycle of the map f: IR ~ IR is a neutral rational cycle of the map F: ([ 1 ~

([ I. As shown by Julia [1] and Fatou [1], the number of attracting and neutral rational

cycles of the polynomial map F: ([ I ~ ([ 1 does not exceed the number of its critical

points, i.e., the number of different roots ofthe equation F'(z) = O. Note that this fact is not a consequence of Corollary 5.8. Thus, Singer [1] construct

ed an example of a unimodal map given by a polynomial of the fourth degree with two sinks (see also the re mark at the end of this seetion).

Dur interest to thc class of maps with sign-preserving Schwarzian is not restricted to maps with negative Schwarzian. Thus, some problems are connected with the study of iterations of maps that consist of finitely many pieces of linear-fractional functions (see

Alicv et al. [1]), i.e., of maps of the form g (x) = ax + ~. Their Schwarzian is equal to yx +u

zero, and one can formulate the following analog of Theorem 5.1 for these maps:

TheoremS.4. If f: I~ I is a unimodal map, fE C3 (l\{c}), and Sf(x) = 0

for XE 1\ {c}, then the set of all cycles of the map f that are not repelling is either empty, or consists of one attracting or semiattracting cycle, or is a cycle of closed intervals B = {Jo, J1, ... , Jn- 1} of period n :?: 1 such that the point c is one of the ends

ofthe interval Jo and fn(x) = x for any point XE Jo.

Theorem 5.4 can be proved just as Theorem 5.1 by using Properties 1-3 of maps with sign-preserving Schwarzian.

For any unimodal map f with positive Schwarzian, the set Fix er) is also finite for

any n:?: 1. At the same time, unimodal maps with positive Schwarzian may have more than one attracting cycle. (Properties 1-3 do not impose any direct prohibition even on the existence of countably many attracting cycles of these maps.)· Thus, let

1 ~ 1

A - O~x<-2' 2'

A~, ~~X~l,

For any XE [0, 1] \ { 112}, we have SgA,(x) > O. One can easily show that, for A<

JJ, the map gA, possesses an aUracting fixed point (other than the fixed point x = 0).

For l;ff < A < .fi, this map possesses two cycles of period two (aUracting and repel-

. f ~ ( 1 +.J5 2) lmg). Hence, or I\, E 2-J2'.f3' the map gA, has both an attracting fixed point and

an attracting cycle of period two.

Remark. The importance of the function

S x = f'''(x) _ ~ [f"(X)]2 J() f'(x) 2 f'(x)

was first noticed by Herman Schwarz [1] in the second half of the 19th century in connection with the investigation of conformal maps. Allwright [1] and Singer [1] almost simultaneously applied the notion of Schwarzian to the study of one-dimensional dynamical systems. Thus, Singer [1] gave the first proof of Theorem 5.1 and constructed the following example of a unimodal convex map with two sinks:

fex) = 7.86x - 23.31~ + 28.75x3 - 13.30x4.

For this map, ß = 0.7263986 ... is a sink of period one and ßl = 0.3217591 ... , ß2 = 0.9309168 . .. is a sink of period two. It is clear that the Schwarzian of this map cannot preserve its sign (this is a consequence of Proposition 5.1).

It is worth noting that, in Chapter 8, the expression for the Schwarzian appears in a natural way as a characteristic of the period doubling bifurcation for periodic trajectories of smooth one-dimensional maps.

Note that the constancy of sign of the Schwarzian is a sufficient condition for the validity ofProperties 1-3 used in the study ofthe dynamics of maps. There are more general conditions, which, in particular, do not require the existence of the third derivative of a map but their verification is more complicated. Some of these conditions are considered below.

Matsumoto [1] studied C2-maps such that any their iteration has at most one inflection point in each interval of monotonicity of this iteration (cf. Property 3 presented above).

For a map fE cI (/,1), one can use the properties of concavity, convexity, or

linearity of the function g(x) = If(x)I- 1I2 instead of the negativity, positivity, or equality to zero of the Schwarzian, respectively.

Seetion 3 Maps with Negative Sehwarzian 149

Indeed,if fE C3(I), then g"(x)= h(x)Sf(x) (thisequalityeanbeeheekedbydireet eomputation).

If the existenee of derivatives of a map f is not required, then one ean use the so

ealled hypergraphie property, whieh is defined as folIows: Let fE CJ(I) be a monotone

funetion defined on the interval I and let xl < x2 < x3 < x4 be points of this interval. Consider a funetion

A funetion fex) is ealled hypergraphie on I ifthe quantity

has the same sign for any set of four points xl < x2 < x3 < x4 from the interval /. If fE

C3, then the negativity, positivity, and equality to zero ofthe quantity 'l(jCXI, x2, x3, x4)

on the interval I are, respeetively, equivalent to the negativity, positivity, and equality to zero of the Sehwarzian Sf(x) of the funetion f on the interval l. This fact can be proved by using the relation

This observation enables us to reformulate Properties 1-3 and Theorem 5.1 for a broader dass of unimodal maps.

In the next seetion, we shall prove that unimodal maps from the dass C2 with nondegenerate eritical point eannot have wandering intervals. This result and Theorem 5.1 imply the following assertion:

Theorem 5.5. Let f be an SU-map, let f'(x) *- 0 for x*- e, and let f"(e)*- O. Then the phase diagram of the map f eonsists only of central nodes and, possibly, of

eycles ofintervaIs B[I) of period one and B~l of period Pm' (in this ease, m * < 00) eorresponding to the domains of immediate attraetion of attraeting or semiattraeting eycles of the map.

Corollary 5.10. If a map f: I ~ I satisfies the eonditions of Theorem 5.5, then

NW (f) = Per (f).

This assertion follows from Theorem 5.5 and the results established in Section 5.2.

150 Topologieal Dynamies 01 Unimodal Maps Chapter 5

Note that the corresponding statements can also be formulated for the spectral de

composition of the set NW (f) of maps of this sort.

4. Maps with Nondegenerate CriticaI Point

Let I: I ~ I be a map from the c1ass Cl (I, I) and let K (f) = {x EI: f' (x) = O} be the set of its critical points. A point e E K(f) is called nondegenerate if there exists a

neighborhood U of the point e in the interval I such that I E C2 (U) and f" (e) =1= O. Similarly, a point e E K (f) is called nonflat if there exist a neighborhood U of the

point e in the interval land r> 1 such that I E Cr (U) and

d r I (e) =1= O. dx r

In this seetion, we describe the dynamics of maps from the class C2 (I, I) with nonflat critical points.

To forrnulate our principal results, we introduce the following notation:

For a unimodal map I: I ~ I, let 'Ba(f) denote the set of points from the classes of 1 f · 1 BCi) . cye es 0 mterva m' l.e.,

m* jm

'Ba(f) = U U m=l i=l

The domain of attraetion of 'Ba(f) is denoted by 'B(f). It follows from the results

ofSeetion 5.2 that 'B(f) = Ui~O.ri('Ba(f»). Aninterval U is ealled a wandering in-

terval of the map f if U<t: 'B(f) and fi (U) () P (U) = 0 for i =1= j. The set of points that belong to wandering intervals is denoted by ['(f). It is clear that the intervals from

the classes ['~) belong to the set ['(f). It should be noted that significant results in this field were obtained after the appear

anee of the Russian version of this book. Thus, the results obtained by Martens, de Melo, and van Strien and by Blokh and Lyubich made it possible to formulate the following two theorems, which establish the fact that the dynamics of smooth maps is, in a certain sense, similar to the dynamies of maps with negative Schwarzian derivative:

Theorem 5.6. Let f be a C= -map with nonflat eritieal points. Then [' (f) = 0, i.e., the map f has no wandering intervals.

Seetion 4 Maps with Nondegenerate Critical Point 151

Theorem 5.7. Let f be a C~ -map with nonflat critical points. Then the set 'Bo(f)

consists of finitely many intervals, i.e., the periods of attracting or semiattracting periodic points of the map f are uniformly bounded.

For phase diagrams of unimodal maps, these theorems imply the following assertion:

Corollary 5.11. The phase diagrams of C~ -unimodal maps with nonflat critical points satisfy the relations

m* m*

2,jm < 00 and 2, km = O. m=1 m=1

For the original ideas of the proof of Theorems 5.6 and 5.7, we refer the reader to the works by de Melo and van Strien [1] and van Strien [2, 3]. A detailed presentation of the dynamics of smooth maps of an interval can be found in the book by de Melo and van Strien [2].

The following two theorems demonstrate that the conditions of nonflatness in Theorems 5.6 and 5.7 cannot be omitted.

Theorem 5.8. There exists a unimodal C~ -map of type 2~ with three critical

points (two critical points that are not extrema are flat) such that NW (f) \ Per f * 0.

Prooj. We fix E E (0, 1) and choose a sequence ßI > ß2 > ... > ßi > ... > 0 satisfying the condition

For example, let ßi = E2-i- l .

~

2, ßi < E. i=1

Assume that an interval I I = II = [xo, Yo] CI is such that mes II = E. We define

12 = 1120 U I?I C I{, where I?o and I?l are closed disjoint intervals such that one end of the first interval coincides with the point Xo and an end of the second interval coin

eides with the point Yo. The lengths of the intervals I?o and I?I are, respectively, (E

ßI)(Ol+l)-l and (E-ßl)OI(OI+l)-I, °1 >1. Denote UI =/I\/2.

The sets Im with m> 2 are defined by induction. Assume that we have already con-

structed the set Im_1 = U(/~-l, where U = UI U2 ... U2m-2 is a sequence of zeros

and ones such that, for any i = 1,2,4, ... , 2m - 3, we have either UI ... Ui = Ui+ I'" U2i

or UI",Ui= (Xi+I'" (X2i' where (Xi=l-ui' Inthiscase,theset Um- 2 = UaU~-2, where U~-2 C 1~-2 are open intervals, is also defined. Open intervals U:;:-I C I~-I are defined so that their lengths are equal to Ym-I . ßm-I, where


( 2 J- I Ym-I = mes 1;:-1 t - 'I, ßi '

,=1

and the intervals from the set 1;:-1 \ U;:-I satisfy the conditions

(i) Iß' U I{y = 1;:-1 \ U;:-I, where ß = aa, ß' = a a, and the interval I{y is

locatedtotheright(left)of U;:-I if al···a2m -3 = a2m -3+1 ... a2m -2 (al'"

a 2m-3 = a 2m-3+1 ... a 2m-2);

(ii) mes I{y = (t- 'I,I ßi)Ym-IOm-I(Om-1 + 1)-1, .=1

mes Iß' = (t -'I,I ßi)Ym-1 (Om_1 + 1 )-t, Om_1 > 1. .=1

Also let om' m ~ 1, be such that

00 rr 0m(om + 1)-1 ::f. O. m=1

For example, let om = 2m. It follows from the construction that

contains countably many intervals (denoted by Ji ) and Lx, \ Ui Ji is a Cantor set.

A map feh) = I1 satisfying the conditions of the theorem is determined as folIows:

First, we take two sequences xi' Yi i ~ 1, ofpoints from I such that

2i+1 x2i+2 = x2i+ I + ß2i+2 rr o/Oj + 1 )-1,

j=1

Y2i+1 = Y2i - (t -2i' o/Oj + 1)-I)(02i+1 + 1)-1, }=I

Section 4 Maps with Nondegenerate Critical Point

2i hi+2 = Y2i+ 1 - ß2i+ 1 TI 0j(Oj + 1 )-1,

j=1

153

The sequences xb Yi i;:::: 1, are forrned by the ends of the intervals U:;: such that a l ...

aj = Uj+I'" U2j forallj= 1,2,4, ... ,2m - 2.

At the points xb Yi i;:::: 0, the map f is defined as follows:

In the intervals (x2i' x2i+ I) and (Y2i+ I' Y2i)' we set fex) to be equal to

2i+1 f(x2i) + 02i+2 TI Ojl(x-X2i)' XE (X2i' X2i+I);

j=1

2i

f(Y2i) + 02i+1 TI Oj\Y2i- X)' XE (Y2i+I'Y2;). j=1

We extend the map f: I] ~ I1 by continuity to the intervals (x2i+ I' x2i+z) and (Y2i+2'

Y2i + I) as monotone functions and to the interval (lim Xi' lim Yi) as a segment of the

straight line Y = Ya. It is not difficult to show that the map thus constructed is a map of

type 2= with "flat" extremum (the interval where the map f attains its maxinum value). For the problem under consideration, we define the map f on the intervals [0, xa)

and (Ya,l] as L-smooth monotone functions. In the interval (limxi' lim Yi~ the i --7 00 i --7 00


"flat" extremum is replaced by a unimodal c-function whose derivatives at the points

lim xi and lim Yi are equal to zero. Thus, we defined a unimodal map f: I ~ I. This i-.:;oo 1-':;00

map is oftype 2~ ifthe trajectory ofthe interval J = [Yo,f(c)] lies in I\I~.

Note that the trajectory of J belongs to the set U:=I Um' Moreover, it "cuts off' the

left and right ends of each interval U;: from this set. In our construction of a c-map

f, we do not change its values on the intervals [x2i' x2i+ d and !Y2i+l' Y2d and assurne

that each interval from the trajectory of J constitutes a third part of the interval U;: such that JeU;:.

Denote the derivative ofthe function fl[x . y_. 1 by 0i' Let 07 denote the "aver-2Z"'LI+ 1

aged derivative" of f on the interval from the trajectory of J whose end coincides with

the point x2i+ I (this interval is denoted by (x2i+ I' x2i+I))' let 01+2 denote the "aver

aged derivative" of f on the interval from the trajectory of J whose end coincides with

the point x2i+2 (this interval is denoted by (xZi+2' x2i+2))' and let 0; denote the "av

eraged derivative" of f on the interval (x2i + I' x2i + 2) (the "averaged derivative" is de

fined as the ratio of the length of the image of an interval under the map f to the length of the original interval). One can easily show that

i-2 i-I

0: = I1 0-:-1 I J ' 0; = I1 0/,

j=1 j=1

and, hence, 0; > 07 > 0;+2 > 01+2'

We extend the rectilinear segment

2i+3

ß i+1 ~ ~n ~-:-I(~. 1)-1 Ui+2 = ß. uJ Ua 2 + ,

I j=1

I1 ~-I Y = f(X2i+2) + 02i+4 Uj (X-X2i+V'

j=1

to the interval (xZi + 2' X2i + V so that the length of the projection of its extended part onto

the abscissa is equal to tl X2i+2 - xZi+21. The rectilinear segment

2i+1

Y =f(X2;)+02i+2 I1 0- 1 (X-X2i)' XE [X2i'X2i+d,

j=1

is extended to the interval (x2i+ I' xZi+l) so that the length of the projection of its ex

tended part onto the abscissa is t 1 xZi + 1 - x2i + I I. Finally, the rectilinear segment

Seetion 4 Maps with Nondegenerate Critical Point 155

is extended to the intervals (x2i + I, x2i + I) and (x2i + 2, X2i + ~ so that the length of the

projections of its extended parts onto the abscissa are, respectively, tl X2i+1 - X2i+ 11

and tl x2i+2 - X2i+21· Further, we connect the values of f at the points

and

and

and

by rectilinear segments.

Let ao = Öi and

By using the function

where

[ ry 1 x JI g(x)dx <p(x) = f 1 - ~ dy,

o fog(x)dx g(x) = exp { - [xCI-x)]} -I,

we connect the points

by C'" -smooth curves. In the same way, we connect the points

and { x2i+2 + x2i+2 (X2i+2 + X2i+2)} 2 ' f 2 '

and

Sinee

sup Il(x)1 ~ Ck<\(X2i+2-X2i+I)I-k sup I <p(k)(X) I, k> 1, xe(x2i+]' X2i+2) xe[O, I]

where Ck is a number that depends only on k, we have

for any k? 1.

By repeating the same procedure for the intervals (Y2i+2' Y2i+ I)' we obtain a map

fE C2 (I, 1) of type T with wandering interval. Now assume that J is the maximal wandering interval of the map f In this case, it

is easy to check that r (c ) E NW (f) and r (c ) ~ Per f for any n? 1. Hence,

NW(f) \ Perf

Theorem 5.9. There exists a unimodal C= -map with single flat critical point,

countably many sinks, and wandering interval.

Proof. Here, we present the proof of a simpler assertion (which can be regarded as a modification of a result established by Sharkovsky and Ivanov [1]), namely, we construct

an example of a unimodal L-map with countably many critical points, countably many sinks, and wandering interval.

First, we construct a map with attracting cycles of arbitrarily large periods.

1. We choose an arbitrary number 'A> 1 and a sequence Xo = 'A/(1 + 'A2 ) > xI > x2

> ... > xn > xn+ I > ... ~ 1/2 and set

fex) = { Ax, XE [0, Xo],

'A(l- x), XE [1 - xo, 1].

Section 4 Maps with Nondegenerate Critical Point 157

Let NI < N2 < ... be a sequence of natural numbers specified below. We define the

quantities fn = 1 - xn/ A Nn and hn = fn+ I - fn and the functions

and

Xn+l

in = f gn(x)dx, n = 1,2, ....

On the interval [xo, 112], we set

fex) = {fn + ~ ( gn(x)dx,

0,

XE [Xn , Xn+r1, n = 1, 2, ... ,

X=1.

It remains to extend the definition of f( x) to the remaining part of the interval to ob

tain a unimodal C"'-map of the interval into itself. Thus, on [1/2, 1 - xo], the map f(x)

can be defined as an arbitrary monotonically decreasing function from the class C'" sat

isfying the conditions f(1I2) = 1, fm)(l/2) = 0 for m ~ 1, f(l - xo) = Axo, 1'(1-

xo) = - A, and fm)(l- xo) = 0 for m> 1. The function fex) thus defined is continuous in the entire interval I = [0, 1] and has continuous derivatives of all orders everywhere except, possibly, the point x = 1/2. Each point xn' n ~ 0, is a periodic attract

ing point of period Nn + 1 for the map f: I ~ I, i.e.,

Let us show that, for a properly chosen sequence {Nn, n = 1,2, .,. }, the derivatives

fm l( x), m > 0, exist for x = 112 and are equal to zero. For this purpose, it suffices to

provethat lim maxlfm)(x)1 = 0 for m>O; here, In = [xll'xn+I]. n~oo XE In

Consider the interval (x/1/ xn+I)' By successive differentiation, we obtain rex) =

(hn!Jn)gn(x) and fm)(x) = (hn!Jn)(Qn.m(x)/ p"Lm (x»), where Qn,m is a polynomial

withcontinuouscoefficientsthatdependonlyon m, x", and xn+I, Pn(x) = (x-xn)(x

xn-I), and Lk + 1 ~ 2Lko k = 2,3,4, .... Since xn ~ 1/2, we have

max I Qn,m(x)1 ~ Cl (m), XE In

158 Topological Dynamics 0/ Unimodal Maps Chapter 5

where Cl (m) is a constant that depends only on m.

Denote an = Xn_ 1 - XIt Then

Xn+l

Jn = f exp{-I/(x-xn)2(X-Xn+I)2}dx

1/2 { 1 2 1 2 } = an f exp - 1/ ( x + "2 ) (x -"2 ) a~ dx.

-1/2

By using the Hölder inequality, we conc1ude that

where

K = 1/2 { 1 2 1 2} f exp - 1/ ( x + "2 ) (x -"2) dx.

-1/2

Therefore,

max lJ<m)(x)1 ::; Cl (m)(hn/JJ max 1 gn(x)/rm(x) I. XE In XE In

Since lim [exp { - + } / f] = 0, k > 0, we can write n.-?=

lim max Ign(x)/PnLm(x)1 = o. n-t oo XE In

Further,

·f ~ Nn a- 4 > 1· ·f 1 I'v an K n _ , l.e., 1

(5.1)

Hence, if inequality (5.1) is true, then

lim max lJ<m)(x)1 = 0 n-+ oo xE In

Section 4 Maps with Nondegenerate Critical Point 159

for any sequence xn ' n ~ 1. Thus, by setting "A = l/K and ~n+1 -xn) = l/n2, we

conclude that Nn ~ n8 + 2ln n /ln "A for sufficiently large n. This means that, for the in

dicated sequence of xn ' it suffices to set Nn = n8 + n.

2. To construct a unimodal C~ -map with an wandering interval, we choose an

arbitrary number "A> 1 and sequences in and Zn' n = 1,2, ... , such that ~2 > YI > 1+/1.

N' ZI>Y2>Z2> ... >Yn>Zn>Yn+I>Zn+I> ... --t1l2. Weset an = Zn+I/"A n and ßn=

Yn+ 1/ "A N~, where N~, n ~ 1, is a sequence of natural numbers.

Note that, in the construction realized above, one can set N2k - 1 = N2k, k ~ 1, in ad

dition to inequality (5.1). Further, if we set x2n-1 = Y", x2n = zn' hn-I = an' hn = ßn'

hn = fn+ 1 - fn' and Nn = N[\n+ 1)/2]' n ~ 1, then, for any sequence x", there exists a se

quence N~ such that the function fex) constructed as in the previous case belongs to the

class C"'[O, 1]. In this case, the interval (YI' ZI) (as weIl as any other interval (Yn' zn)'

n > 1) is a maximal wandering interval.

6. mETRIC aSPECTS OF DvnamlCS

The phase space of dynamical systems under consideration, i.e., the interval I, is endowed with Lebesgue measure. It is thus useful to establish some properties of dynamical systems that are typical with respect to this measure, i.e., properties exhibited by trajectories covering sets of full measure.

1. Measure of the Set of Lyapunov Stable Trajectories

By using the phase diagram constructed and studied in Sections 5.1 and 5.2, we have already described the lirniting behavior of trajectories of unimodal maps. Let us now describe the set of points of an interval that generate Lyapunov stable trajectories (recall that the trajectory of a point x E I of the map f: I ~ I is called Lyapunov stable if, for any E > 0, there exists ö> 0 such that the inc1usion fi (y) E (fi (x) - E, fi (t) + E) holds

forany YE In (x-ö,x+Ö) andali i 2:: 0).

Let f: I ~ I be a unimodal map and let .t: (f) be the set of the points of the interval I whose trajectories are Lyapunov stable. If XE r(f), i.e., if the point x belongs to a wandering interval U, then x E .t: (f) because Ifi (U) I ~ 0 as i ~ 00. Hence, r(f) C

.t: (f).

Assume that XE 'E(f), i.e., there exist a cyc1e of intervals B~) = {Jo, J1, ... , J n- d which does not contain the point of extremum of f and a number k < 00 such that fk(x) E Jo. If fk(x) ~ Per (f), then the point fk(x) belongs to the domain of immedi

ate attraction of some point of the set Jo n Fix (f'ln). In this case, XE .t: (f). Now let

fk(x) E Per (f). Then fk(x) E Jo n Fix (f'ln). We set ß = fk(x) and assume that there

exists E > 0 such that either f'ln(y) < Y for all y E (ß - E, ß) or f2n(y) > Y for all Y E

(ß, ß + E). In this case, ß ~ .t: (I) and, hence, x ~ .t: (f). One can easily show that x ~ .t: (f) in all other cases. The set of all periodic points of 'E(f) whose trajectories are not Lyapunov stable is denoted by Ao = Ao(f). The set Ao is invariant and consists of at most countably many cyc1es because, for each of these cyc1es, one can indicate its half neighborhood that does not contain periodic points of the map.

161

162 Metric Aspects 01 Dynamics Chapter 6

The results established in Section 5.2 imply that

1= 'B(f) U r(/) U U U ricd~» m<::'m* i;;"O

for m* < 00 and

1= 'B(f) U r(f) U PCC~O),f) U U U l-iCC~O» m<::'oo i;;"O

for m* = 00. (Recall that the elements of the spectral decomposition of the set Per(f)

are denoted by C~), and P C C~O),J) denotes the domain of attraction of the set C~O).) Therefore,

m<oo i;;"O mSm*

i;;"O

where A oo = 0 if m* < 00 and Aoo is a subset of the set P (C~O) ,f) if m* = 00 (it can

be shown that A oo is empty ifthe set <1>: is nowhere dense and nonempty if <1>: con

tains intervals). Note that trajectories from the set Ui;;"O l-iCC;:(» are not Lyapunov

stable for Pm + d Pm> 2 because, in this case, C;:() is a Cantor set and the map I1 c (0) m

possesses the rnixing property.

The representation of the set 1\ :CU) constructed above demonstrates that the answer to the question as to whether Lyapunov stability is a generic property of trajectories of a given unimodal map I essentially depends on the Lebesgue measure of elements of

the spectral decomposition of the set Per Cf). The following theorem is a simple consequence of this observation:

Theorem 6.1. Let I: I ~I be a unimodal map. Assume that I has no cycles whose periods are not apower 01 two and has no wandering intervals. Then the set 01 points olthe interval I whose trajectories are not Lyapunov stable is a set 01 Lebesgue measure zero.

Proof. Under the conditions of the theorem, in the phase diagram of the map I, we

have Pm+llPm = 2 for any m < m* (here, we use the same notation as in Seetion 5.2).

Hence, for any m< m*, the set C~O) is a cycle or a pair of cycles, i.e., mes C;:() = O.

Consider the set c~o2 for m* < 00. Since the map I has no cycles whose periods

are not apower of two, the set c~2 can be neither a cycle of intervals nor a Cantor set.

Therefore, c~2 is aperiodie trajeetory and mes c~2 = O. Thus, mes (1\ :CCf) = 0

whenever m* < 00.

Section 1 Measure of the Set of Lyapunov Stable Trajectories 163

If m* = 00, then the fact that wandering intervals are absent implies that the set

<1>: = U <l>m" is nowhere dense (see Section 5.2). Hence, the Lebesgue measure of m~l

each component of the set ~ tends to zero as m increases. On the other hand, if x E

P( C~O),f), then, for any m < 00, there exists a neighborhood U = U(m) of the point

x such that fk( U) c ~ for some k < 00. Therefore, in this case, we have x E ;C (f), and the proof of Theorem 6.1 is completed.

If the map is not "simple", i.e., if it has cyc1es whose period is not apower of two, then both the situation where Lyapunov stability is a typical property of trajectories of a given map and the situation where this is not true are possible. Simple examples of maps of both kinds are presented below.

For maps with negative Schwarzian, the following assertion is true:

Theorem 6.2. Let fE C3 (I, /) be a unimodal map such that its critical point c is

notflatand Sf(x) < 0 for XE I\{c}. Thentheinequality mes;C(f)<mesI holds

if and only if m* < 00 and c~2 is a cycle ofintervals.

The proof of Theorem 6.2 immediately follows from the properties of the spectral de

composition of the set NW (f) for unimodal maps with negative Schwarzian, Theorem

5.5, and Theorem 6.3 formulated below. Note that if mes;C (f) < mes land both fixed

points of the map f are repelling, then mes;C (f) = O.

lB}!'WllDlplle l6.11. Consider the map g (x) = 1 - 21 x - t I, x E [0, 1], encountered

somewhat earlier. The map g is unimodal and consists of two linear pieces. Moreover,

it is expanding, i.e., 1 g' (x) 1 = 2 for XE [0, 1] \ { t }. It is easy to check that g pos

ses ses the mixing property on the interval [0, 1]. Therefore, Per(g) = [0, 1] and

;r; (g) = 0, i.e., the map g has no Lyapunov stable trajectories.

The point x* = i is a repelling fixed point of the map g. We fix arbitrary E < li and replace the right branch of the function g (x) by a piecewise linear function g (x)

such that

g(x* - 2E) = x* + 2E, g(x* + 2E) = x* - 2E,

g(x* - E) = x* +~, and g(x* + E) = x* - ~ (see Fig. 43).

For XE [0, tJ, we set g(x) = g(x). The function g(x) obtained as a result is uni

modal, the point x* is an attracting fixed point of the map g, and the interval (x* - 2E,

x* + 2E) is the domain of immediate attraction of the point x*. We have

164 Metric Aspects of Dynamics Chapter 6

where ciO) = 1\ Ui~O g-i((x* - 210, x* + 2t) is a Cantor set. By direct calculation,

one can show that the Lebesgue measure of the set Ui~O g-i((x* - 210, x* + 1:) is

equal to the measure of the entire interval [0, 1], i.e., mes ciO) = 0 (independently of

the choice of 10). Hence, for the map g, we have mes ~ (g) = mes ([0, I]) because

~ (g) = Ui~O g-i((x* - 210, x* + 2t)).

Fig.43 Fig.44

Thus, from the metric point of view, the limiting behavior of a "typical" trajectory of

a unimodal map f: I ~ I is determined by the structure of a certain sub set .521. (f) of the

set Per(f) and the dynamics of the map fl51.(f)" It is natural to choose the smallest pos-

sible .521. (f). Thus, for the map g constructed in Example 6.1, we must take .521. (g) = {x* }. The exact definitions are presented below.

Recall some definitions from Chapter 1. The probabiJistic limit set M (f) of a map

f: I ~ I (according to Milnor) is defined as the smallest c10sed set that contains the ffilimit sets of trajectories of almost all points of the interval I (with respect to the Le

besgue measure). It is c1ear that, for any unimodal map j: the set M(f) is an invariant

subset of Per (f). On the other hand, if, for any measurable set A C I such that f (A) C

A, we define Il(A) = mes {x EI: ffijCX) CA}, then M(f) is the smallest c10sed in

variant subset of Per(f) such that 11 (M(f) = mes!.

For any set M C I, let

n-l

p(x, M) = lim inf.!. L XMUk(x), n--7= n k=O

Section 2 Conditionsfor the Existence of Absolutely Continuous Invariant Measures 165

where x EI and XM is the indicator ofthe set M. We say that M is the center of at

traction (of almost all trajectories) of the map f if, for any neighborhood U of the set M, the equality p(x, U) = 1 holds for almost all points x E I with respect to the Lebesgue measure. The minimal center of attraction of almost all trajectories .9l(f) is defined as the center of attraction that contains no other centers of attraction.

It is not difficult to show that .9l (f) c M (f) and these sets are distinct even in the case of unimodal maps. Thus, for the map

{X + 2m xm+1 XE [0,1/2],

x ~ g(x) = ' 2(1- x), XE [1/2, 1],

where m?1 (seeFig.44),wehave M(g) = Per(g) = [0,1] and .9l(g) = O. In the neighborhood of the nonhyperbolic fixed point 0, the motion of trajectories is significantly decelerated and the time 1: (E) of expansion of the E-neighborhood of the point 0

to its 2E-neighborhood infinitely increases as E ~ 0, i.e., 't(E)- E-mjm as E ~ o. One can also construct an analytic map which pos ses ses the indicated property.

As shown above, metric properties of a given unimodal map f depend on the metric

properties ofthe elements of the spectral decomposition of the set Per(f). Numerous important results were obtained in this direction for smooth maps.

As usual, for a map fE C' (1,1), a c10sed set A such that f(A) C A is called hy

perbolic if there exist A, > 1 and C > 0 such that either I (fY (y) I ? CA n or I (f")' (y) I :s; A, -n j C for any point y E A and all n? O.

Theorem 6.3. Assume that fE C2(1,I), A is a cIosed set, f(A) CA, and A ()

K(f) = 0, where K (f) = {x EI: f'(x) = O}. Then there exists N < 00 such that all periodic orbits in A whose periods are greater than N are hyperbolic and repelling. Moreover, if A does not contain any nonhyperbolic periodic orbits, then A is a hyperbolic set, and if, in addition, A does not contain attracting periodic orbits, then mesA = O.

This theorem was first proved by Mane [1]. Another proof was suggested by van Strien [3].

2. Conditions for the Existence of Absolutely Continuous Invariant Measures

We study the asymptotic behavior of trajectories. This type ofbehavior can be efficient

ly described, e.g., for maps preserving a measure 11 (such that 11 (j-l (A)) = 11 (A) for

any measurable set A). It is weIl known that the support of a measure of this sort must

belong to the set Per (f) .

In order to exdude trivial cases, we require that !.l (Per (f» = O. The best possibility is to guarantee the absolute continuity of a measure !.l with respect to the Lebesgue measure, i.e., to require that the condition mesA = 0 imply the equality !.l(A) = O.

There are some general results conceming the existence of absolutely continuous invariant measures for nonsingular maps of an interval. (RecaIl that a map f is caIled

non singular ifthe equality mesA = 0 yields the equality mesf-l (A) = 0 for any measurable set A.)

Theorem 6.4 (Foguel [1]). Let f: I ~ I be a nonsingular map. Then the follow

ing assertions are equivalent:

(i) There exists an invariant measure of the map f absolutely continuous with respect to the Lebesgue measure;

(ii) there exists E< 1 such that the condition mes (A) < E implies the inequality

lim sup mes (rn(A» :s; ! mes I for all n ~ 0; n---7=

(iii) there exists E< 1 such that the condition mes (A) < E implies the inequality

lim sup (.!.. Imes(rk(A»):S; .!..mesI. n---7= n k=O 2

The absolute continuity of a given measure with respect to the Lebesgue measure can be established by using the Radon-Nikodym theorem:

A probability measure 11 is absolutely continuous with respect to the Lebesgue

measure if and only if there exists an L I-function p (x) such that

Il(A) = f p(x)dx A

for any measurable set A.

By using this representation of absolutely continuous measures, one can prove the existence of invariant measures absolutely continuous with respect to the Lebesgue mea-

sure for expanding maps of an interval from the dass C 2 .

Theorem 6.5 (Lasotaand Yorke [1]). Let f: [-1, 1] ~ [-1,1] be such that

0, 1, ... , n - 1;

(ii) 11'1 > 1 on [-1,1]\ {Cl, ... , cn-d.

Then f possesses an invariant measure absolutely continuous with respect to the Lebesgue measure.

The proof of Theorem 6.5 is based on the use of the Frobenius-Perron operator. This operator realizes a transformation in the set of densities of measures corresponding to the transformations of the Lebesgue measure under iterations of the map f By applying the

Frobenius-Perron operator to the original density Po(x), we obtain a density Pn(x) given by a function of bounded variation. The conditions of smoothness imposed on the

map f enable us to conclude that the functions P n(x) converge to a limit function P (x)

which is a function of bounded variation and, hence, an Ll-function. Note that, in the

conditions of Theorem 6.5, 0-smoothness can be replaced by Cl +E-smoothness. By using Theorem 6.4, one can show that a smooth map similar to the map displayed

in Fig. 44 has no finite invariant measure absolutely continuous with respect to the Lebesgue measure. This example clarifies the importance of the requirement of hyperbolicity of periodic trajectories in the conditions of theorems establishing the existence of absolutely continuous measures (see Theorem 6.7 below).

Consider a simple unimodal map f: I ~ I, i.e., a unimodal map with topological en

tropy equal to zero. In this case, in the phase diagram of the map f, we have Pm+ d Pm = 2

for any m < m*. If m* < 00, then NW (f) = Per (f) and, moreover, the periods of all points of the set Per (f) are uniformly bounded. Therefore, it remains to consider the

case m* = 00 where PerU) = Per (f) U C~o) (C~o) is a Cantor set). The following assertion is true (see also Collet and Eckmann [1] and Misiurewicz [2]):

Theorem 6.6. Let f: I ~ I be a unimodal map in the phase diagram of which

m* = 00 and Pm+l/Pm =2 for alt m < m*. Then there exists a unique invariant prob

ability measure !l on Per(f) equal to zero on any subset of the set Per(f). The

following equality holds for any point y E P( Cr;!), f) and any continuous function g(x) defined on I:

The proof of Theorem 6.6 is split into two parts: first, we construct the measure !l and then study its properties. The construction of the measure !l depends on the form of the phase diagram of the map f It follows from the results established in Seetion 5.2

that, under the conditions of the theorem, dO) c n m* , where = m<:l

Pm-I

~ = U J~) i=O

. (0) (I) (Pm-I) . d . and the mtervals J m ' J m , ... , J m form a cyc1e of peno Pm (m the case under

'd' - 2m) Th f h' I J(O) J(l) J(2 m-I). conSl eratlOn, Pm - . e measure 0 eac mterva m' m" .. , m IS as-

sumed to be equal to 2-m. For a more detailed proof, see Misiurewicz [2]. If 1 is a convex unimodal map with a single nondegenerate critical point, then

mes C~O) = O. This fact and some other possibilities connected with the Lebesgue mea

sure of sets of type C~O) are discussed at the end of this section. If the topological entropy of the map 1: I ~ I is positive, then we have the following

sufficient condition for the existence of an invariant measure absolutely continuous with respect to the Lebesgue measure:

Theorem 6.7. Let 1 E c2 (l, l) and let all critical points 01 the map 1 be nonflat. Assume that 1 has no attracting and nonhyperbolic periodic trajectories and

K(f) n Ui~1 /(K(f» = 0.

Then 1 possesses an invariant measure absolutely continuous with respect to the Lebesgue measure.

Note that if a unimodal map 1 satisfies the conditions of Theorem 6.7, then, in its

phase diagram, m* < 00 and c:!;2 is a cyc1e of intervals (the invariant measure is concentrated just on this cyc1e of intervals). Guckenheimer [3] conjectured that, at least for unimodal maps with negative Schwarzian, an invariant measure absolutely continuous

with respect to the Lebesgue measure exists if and only if m* < 00 and c:!;2 is a cyc1e of intervals. The following example suggested by Johnson demonstrates that this is not true. It is c1ear that, for the map constructed in this example, the condition

cannot be satisfied.

Theorem 6.8 (Johnson [1]). For thelamily olmaps h,: x ~ Ax(l - x), one can

indicate a value 01 the parameter A = such that the set c~02 (fl-..o) in the phase dia

gram 0111-..0 is a cycle 01 intervals and any finite invariant measure' 0111-..0 is singular.

In [1], by using the lemmas formulated below, Johnson proved that, for the map 11-..0:

I ~ I, one can indicate a sequence of sets {G mJ i = I such that A~ ( Gm) C I \ Gm i and

mes (1\ Gm) 4 0 as i 4 00. Therefore, any finite invariant measure of the indicated

map is not absolutely continuous with respect to the Lebesgue measure. Let ß E Per(f,) and let ß/7= ß be such that f(ß/) =f(ß). The cJosed interval

(ß, ß/) with ends at ß and ß I is denoted by h. If f;:(J.;.) C 1-;... for some n ~ 1,

then we say that J.;. is a periodic interval (of period n if n is the least possible number

with the indicated property). If f;:(J.;.) = J.;., then the interval J.;. is called strictly periodic.

Let [A l1' Ab] be the maximal interval such that the map fA possesses a periodic

interval 1-;... of period no for any A E [A l1' Ab].

Lemma 6.1 (Guckenheimer [2]). For any (j > 0, there exists Ö> 0 such that, for

any A E (AlP Ab + ö), one can indicate a set PA and mE N for wh ich mes (1\ PA)

< (j and ftPA C Uo .< f~(h). <!-no

Lemma 6.2. For any E > 0 and (j > 0, there exist A' E (AlP Ab + Ö) and a

strictly periodic interval J.;., such that mes (orb J.;.,) < (j.

As A~, we take the limit ofthe sequence {Ai} formed by the values of the param

eterequalto Ab forperiodicintervals h; ofperiods ni, where ni4°O as i 4 00. In

this case, by Lemma 6.2, the intervals h can be chosen so that (Uo . fj(Jn)) 4 0 I <J'5,ni

as i 4 00 and the set Gi is chosen so that 1\ Gi = (l\P,..) U Uo .< ß(f.;.,J, where <J_n,

PA; are prescribed by Lemma 6.1.

For maps with negative Schwarzian, we have the following theorem, which charac

terizes the probabilistic limit set A (f):

Theorem 6.9 (Blokh and Lyubich [1]). Let fE C3 (I, I) be a unimodal map and

let S f (x) < 0 for x E 1\ { c }. Then, for almost alt points of the interval I with respect to the Lebesgue measure, only one of the foltowing three possibilities is realized:

(i) co/x) is an attracting or semiattracting cycle;

(ii) co f(x) is a cycle of intervals (and coincides with the set c;:(h

The proof of Theorem 6.9 follows from the estimates established in the previous chapter and the proof of Theorem 6.7 (see van Strien [3]).

3. Measure of Repellers and Attractors

The results of this section can be regarded as a supplement to the results established in Section 6.2. The following assertion demonstrates that the condition of continuity of the second derivative in Theorems 6.3 and 6.7 is fundamental:

Theorem 6.10. There exists a unimodal map from the dass Cl wh ich possesses a

repeller K* in the form of an invariant hyperbolic Cantor set of positive Lebesgue measure; moreover, the Lebesgue measure is invariant on K*.

Prooj. We fix an arbitrary number E E (0, 1) and a sequence of numbers ßo> ß I

> ß2 > ... such that

~

(i) L ß i = ß < E and (ii) lim ßi+l = 1. H~ ßi i=O

Thus, we can take

i = 1,2, ....

First, we construct a set K* C I homeomorphic to the standard Cantor set K and such that mes K* = 1 - ß.

In the ternary notation, the Cantor set K takes the form {O. i 1 i 2 ... , where i s = ° or

2, s = 1, 2, ... }. Denote by Uij "' ik the intervals (0. i I ... i k022 ... , O. i I'" i k200 ... )

"removed" at the (k + l)th step, k;::: 1 (for each k;::: 1, there are 2k intervals of this sort).

Let h beahomeomorphismfrom I into I. Denote h;ji2 ... = h(0.i l i 2 ... ), u=

(h022 ... , h 200 .. J, and Uij ... ik =h(Uij ... ik ). Assumethat h(l) = land

(i) the interval U is equidistant from ° and 1 and mes U = ßo;

(ii) the intervals Ui j ... ie k;::: 1, are equidistant from the points h ij ... ikOO ... and

h i j .•. ik 22 ... , respectively, and mes U i j .•• ik = ~Z .

The set K* = h(K) is homeomorphic to the Cantor set. Indeed,

K* = (1\ U) \ u U' .. mes K* = 1 - ß > 1 - E. I} .. . ls'" lk'

i =0'2 I S;s'S;k,'k?:l

Seetion 3 Measure of Repellers and Attractors 171

The map f: R ~R is constructed as follows:

(1) fex) == 2.x for x::;; 0;

(2) for 0 < x::;; I-fo, the function fex) is monotone continuous and such that

(a) f(h i ]i2i3 .. J == hi2i3i4 ... (hence, f(V i ]i2 ... ik )= V i2 ... ik for k > 1, f(Vo) ==

V);

(b) on the intervals Vi] ... ie the function f (x) is defined as an arbitrary func

tion from the dass cl satisfying the conditions

(b') j'(x) ~ 2;

(b") limX~(w. . j'(x) == 2 (this condition can be satisfied because, for ,] .. 'k

k> 1, we have

mesV·· ß ___ 1-,,-2_"--,,·lk~ == 2 ~ > 2 mesu.·· ßk 1]12 .. •lk

mes V == 2 ßo > 2); mes Vo ßI

(b"') SUPXEU" . j'(x) ~ 2 as k ~ 00 (this condition can also be sat-'1'2··· 1k

isfied because ßk-I /ßk ~ 1 as k ~ 00);

(3) for

the function fex) is defined as an arbitrary function from the dass Cl satisfying the conditions

(a) fex) > 1;

(b) lim j'(x) == 2; ] - Po

x~-2-

(c) 1'(1) == 0;

172 Metric Aspects of Dynamics

1 (4) fex) = f(1 - x) for x 2: -.

2

It follows from the construction of the function f that

{xEI:fi(x)EI,i=I,2, ... } = (/\V)\ u is =0,2

!~s~k, k=I,2,3 ....

Chapter 6

and mes K* > 1 - E. It remains to show that f is a function from the class Cl. For this purpose, it suffices to check that !'(x) exists for XE K* and is equal to 2 for X::; 1/2 and to - 2 for x> 1I2. Indeed, if this is true, then f' (x) is continuous in K* (by virtue of (2b") and (2b"'» and, hence, in the entire interval I.

We introduce the following notation: <X! = 1 + ß 0' <X i = 1 - ß 0 - ... - ß i-2 + ß i-I'

i = 2, 3, .... Then <Xi ~ 1 - ß as i ~ 00 and, hence, <Xi+l / <Xi ~ 1. By the construc

tion of the set K*, we have

Therefore, if x' = hiii2 ... E K* and i; = is for s = 1,2,3, ... , m, then

fex') - f(hi1iz .. )

x' - h· . '1'2 ...

( ., ') (" . )<Xm +l + Im+l - lm+! <Xm + Im+2 - lm+2 -2- ... 2 <X

( " ') (" ') m+2 lm+l - lm+l <Xm+l + lm+2 - lm+2 -2- + ...

The smaller the difference I x' - h i 1 iz ... I, the closer this ratio to two. At the same time,

if X'E V ii ... i", i.e., hii ... i,,022 ... <x'< hii ... iPOO ... and,fordefiniteness, x'> h i1iz ... ,

then

f(h·, ., 022 ) - f(h· ) fex') - f(hi1 iz ... ) f(h i1, ... ik' 200 .. ) - f(hi1 iz .. ,) '1 .. ·'k·" '1'Z'" < ____ -'--"-_ < _--2-:..:.c..:..!c.::..:.::..:.:.::. __ -'-!..:.L..:"-

hii ... i" 022 ... - 11;1 i2 ...

In this case, the smaller the difference I x' - h·· I the smaller the quantity '}'2 ... '

Il x'-h.· I-llf(x') -f(h·· )1- 21· '1'2'" '1'2'"

Thus, !,(hiliz ") = 2 whenever h iliz ... < 1/2.

It follows from the construction of the map f that K * is a hyperbolic invariant set of positive Lebesgue measure.

Section 3 Measure of Repellers and Attractors 173

Let A i l i2 ..• ik denote the maximal c10sed interval such that

k-1

U u· .. 11 ···ls ···ll

1=1

Then

2mes (A·· . n K*) = mes(A·· . n K*) 1112 ···lk 1213··· lk

and, hence, the Lebesgue measure defined on K* is invariant under the map f By using Theorem 6.3, one can estimate the measures ofthe sets c<~) with m < m*

in the spectral decomposition of the set Per f. The set c~2 contains the critical point and, therefore, cannot satisfy the conditions of the theorem. The following theorem de-

termines the measure of the set c~o2 with m* = 00 for maps with negative Schwarzian.

Theorem 6.11 (Guckenheimer [3]). Let f be an S-unimodal map with the follow

ing properties:

(i) it possesses a unique nonflat critical point (the point of extremum);

(ii) f is a map oftype 200•

Then the Lebesgue measure of the quasiattractor of the map f is equal to zero.

Proof. We prove this assertion for maps symmetrie with respect to their point of extremum. For the sake of convenience, we assume that the point of extremum of the map fis located at the origin and f(O) > O. Denote the points fj(O) by Cj' It follows from

the conditions of the theorem that, for any n > 0, there are 2n- 1 mutually disjoint inter

vals Jn,j = [Cj' cZ'-I+j]. Each ofthese intervals contains a single (repelling) point of

period 2n- 1. Denote the left fixed repelling point of f by P -1' the periodic point of pe

riod 2n c10sest to the point of extremum by p n' and the points P (P n) by p n,j' For any

n, wehave Jn+1,jUJn+1,2n-l+jCJn and K= nn>O(Uj~oJn,) isaquasiattractor

(see Barkovsky and Levin [1] and Misiurewicz [2]). Let qn denote the first point to the

rightof Cl suchthatf2n-1(qn) = -Pn-1' Notethatpnbn.I,/k) isahomeomorphism.

Let I JI denote the length of the interval J, To prove the theorem, it suffices to

show that there exists a O. Indeed, in this case,

174 Metric Aspects oj Dynamics

and, henee, mes K = O.

2n - 1

L.. IJn) < anlJI,11 j=1

The proof is split into several steps.

Step l. I D j2" (qn) I > 1.

Chapter 6

We proeeed by induetion. Sinee j is symmetrie, we have q 0 = - p _ 1 and, therefore,

11'(qo) I = 11'(p-dl > 1. Supposethat I Dj2n(qn) I > 1. Then

Note that qn+1 lies in the interval (Pn+I,I, qn+l) and j2n+1 is monotone in this inter

val. Sinee an iterations of j have negative Sehwarzians, I D j2"+1 (qn+ I) I is greater than

the minimum of I D f2n + 1 (qn) I and I D f2n +1 (p n+ 1 I) I. Sinee an periodie orbits are re

pelling,wehave I Dj2n+I (Pn+I,I) I > 1 and,henee, I Dj2n+1 (qn+l) I > 1.

Note that this inequality remains true for all nunder weaker eonditions than the

symmetry of f Thus, it suffiees to require that 11'( qo) I > 1.

Step Il. P n/ c2n < 0.71 for an suffieiently large n.

Sinee the point of extremum is nonflat, the funetion j on the interval [0, P n-I] ean

be approximated (as n ~ 00) with any desired degree of aeeuraey by a funetion of the

form a - bx 2. This enables us to eonclude that

Moreover,

beeause IDj2n (x)1 > 1 forany XE (Pn,I'C1) C (Pn,I,ql)'

Consequently,

and, henee,

Seetion 3 Measure of Repellers and Attractors 175

1 -J2 + e < 0.71

for large n if we set e = 0.002.

Step III. 112 1 > .!.. c2n 3

Since f is symmetrie with respect to the origin, the map f2 n is symmetrie in the in

terval [Pn-l' -Pn-l]· Sinee Sf< 0, the map f2n is expanding on [Pli' -Pn-d. The

inc1usion C2n+IE (O,-p,J impliestheinequality IC2n -Pnl < IPn-C2n+11 < 21Pnl·

Henee, I C2n I = IPnl + I C2n - Pnl < 31Pnl and I Pn/C2n I > 1/3.

Lemma 6.3. Let h be a C3-dijfeomorphism on [0, 1] such that S h < 0, h(O) = 0, and h(l) = 1. Then the inequalities

I h"(x) I 2 (h'(x))2 < ~ and I h'(x) I < exp {~}

h'(y) Ö

holdfor any x and y from the interval h-1 (Ö, 1- Ö).

A similar assertion was proved by van Strien [3] and we refer the reader to this paper for the proof.

The following statement is an immediate consequenee ofLemma 6.3:

Step IV. There exists e > 0 such that I Pn / Pn-ll > e for all n.

Step V. There exists a eonstant ß > 0 such that

If Pn-l < 0, then the points are ordered as follows: Pn-l < -Pn < C2n+1 < 0 < C2n+2 <

Pn < C2n < -Pn-l· By using Step II, we obtain Jc2n+1 / Pn-11 e > 0 for any n, the results of Steps

II and III imply that the quantities 1c2n+1 /c2n I are also separated from zero for any n. This proves the existence of the eonstant ß.

All preliminary steps of the proof of Theorem 6.11 are now eompleted, and we can make the following eonc1usions:

(ii) I C2n+2 - C2n+1 I/I C2n +1 - C2n I are separated from zero (see Step IV);

(iii) I C2n +2 - P n 1/ I C2n +2 - C2n +1 I are separated from zero (see Step V).

Thus, a constant y defined as the minimum of the ratio of the length of the "removed" interval to the length of the original interval exists and is positive. Hence,

2n 2n - 1

L IJn+l,jl < a L I Jn), j=1 j=1

where a = 1-y.

Theorem 6.12. There exists a unimodal e ~ -map with flat extrem um, which pos

sesses a quasiattractor of positive Lebesgue measure.

Prooj. The corresponding example was suggested by Misiurewicz [4]. However, the map constructed in that example is characterized by a property that seems to be nontypical of smooth unimodal maps with nonflat extremum, namely, the multiplicator

lim sup Idr(X)1 n---7~ xeK dx

of the quasiattractor K of this map is unbounded. An example presented below is free of this shortage. At the same time, the smooth

ness ofthe map at the point of extremum is not higher than er, r ~ 0 (see Kolyada [1]). It is worth noting that, in this example, one can also show that any invariant measure is singular.

We fix E E (0,1) and EO E (0, E) and take a sequence of numbers ~) > ~2 > ... > ~ i > ... > 0 such that

We construct a Cantor set

~

L ~ i = ~ < E - Eo· i=1

such that mes I ~ > 1 - E as follows:


Let m = 1. We choose an arbitrary interval [xo, Yo] eint 1 of length 1 - Co and set I . 2 U 2 I 2 1 I = I I = [xo, Yo]. For m = 2, we defme a set 12= 110 I II C 11 , where 110 and

Ifl are closed disjoint intervals, Xo E i)Jfo, Yo E d/fl' mes Ifo = (1 - Co - ßI)Ö(Ö +

1)-1, and mes Ifl = (1- Co - ßI)Ö(Ö + 1)-1, where ö> 1. Let VI = I I \/2. For m > 2,

the set Im is constructed recursively. Assurne that we have already constructed the set

I m - 1 = Ua 1;:-1, where (X = (XI (X2 ... (X2m-2 is a sequence of 2m - 2 zeros and ones

such that, for any i E {I, 2, 4, ... , 2m- 3 }, either (X I ... (Xi = (Xi + I ... (X2i or (X I ... (Xi =

ai+1 ... a2i' where ai = 1 - (Xi. Then the set Vm _2 = U a V;:-2, where V;:-2 C

1;:-2 is an open interval, is also well defined.

Let us now construct the set Vm - 1 = U a V;:-l. We choose an open interval V;:-l

C /;:-1 such that mes V;:-I = Ym-l ßm-l' where

( m-2 J- I

Ym-l = mes/;:-l 1-co- Lßi ' l=1

and the intervals from the set /;:-1 \ V;:-I have the following properties:

(i) I;U/; = I;:-I\V;:-l, where ß'=(Xa, ß=(X(X, andtheinterval I; is

located to the right (left) of V;:-I if (X I ... CXzm -3 = a2m-3 + I ... a2m-2 ((XI ...

(X2 m - 3 = (X2 m- 3 + I ... (X2 m - 2);

(ii) mes I;

mes I;

Thus, we have constructed the set I"" = U:=l U a I;:. Since

( m-I)( Ö )m-I sup mes (I"" n I;:) = 1- Co - L ßi -- , a i=1 Ö + 1

the set I"" does not contain intervals. Hence, by construction, it is a Cantor set of posi

tive Lebesgue measure: mes I"" = 1 - Co - ß > 1 - c.

We now construct a map f: 1 ---71 of type 2"" whose quasiattractar coincides with

I"". Far this purpose, we choose two sequences {xi } i': I and {y;}i':I' where Xi and y i

are the ends of the intervals U;: such that a l '" a j = <Xj+1 ... <X2j for all j = 1, 2,

4, .... Then xO<xi<",<x2i<'" and YO>YI>"'>Y2i>'" aresuchthat

- ß 1:2i+I(1 1:)-(2i+l) x2i+2 - x2i+1 + 2i+2U + U ,

ß 1:2i( 1 1:)-2i Y2i+2 = Y2i+1 -- 2i+IU +U ,

( 2i+l) 2i -(2i+l)

Y2i+ 1 = Y2i - 1 - co - L ß j 0 (1 + 0) , J=I

where i = 1,2, ... and Xo and Yo are the points used in the construction of the set l~.

At the points xi' Yi, i = 0,1,2, ... , we define the values ofthe map 1 as folIows:

2i

I(yu) = Yo- (l-co- Lßj )(O+1)-2i, J=I

In the intervals [xu, xu+ d and [Y2i+ I' Y2J, the map 1 is defined as folIows: I(xu) +

0-2i(x-X2i), x E[X2i,X2i+I], and I(Y2i)+OI-2i(Y2i -X),XE [Y2i+1,Y2J Letus

extend the definition of the map to the remaining intervals (xU+I' x2i+0 and (Y2i+2,

Y2i+I)' Denotetheinterval (x2i+I,X2i) by K i andtheinterval (YU+2,Y2i+l) by Kr·

Section 3 Measure o! Repellers and Attractors 179

To paste the relevant parts of the map in these intervals, we determine the coordinates of

the points ofintersection ofthe straight lines Y =!(x2i) + 0-2i(x -x2i) and y=!(x2i+V + s:-2i-2 I U (x -x2i+V' name y,

and of the straight lines Y = !(Y2i) + 0 1-2i (Y2i - x) and Y = !(Y2i+V + 0-2i-1 (Y2i+2 -

x2), i.e.,

Let us now compute the quantities

and

Wehave

for large i (in particular, for 0 = 2, we have ßi = 1- E02 for any i = 1,2, ... ), (I + I)

[ 02i-1 (( 2i) )] sign 2i+2 1- Eo - L, ßj 0 - ß2i+I(03 + 0-1) + ß2i+2 > 0 (0+1) j=1

.. I . (. . I .. s: - h ß - 1- EO f . - 2 3 ) F lor arge I m parheu ar, lor u - 2, we ave i - -. -2 or any I - , ,.... or (I + I)

i = 1, we can write

180 Metrie Aspeets 0/ Dynamies Chapter 6

y = Y2 + ~y, and ~y ::; ~. Hence, only in the last case, it is possible to define / on

the interval [Y3' Yo] as a convex function. Consider a segment [a,b] andstraightlines y=a(x-a)+A and y=ß(x-b)+B

such that ! =: > a > ß > O. Then the coordinate x of the point of intersection of these

straightlinesbelongstothesegment (b,c], c>b. Wedenote !=: by y and 2y-a

by y' and construct a straight line Y = y' (x - a 0) + b 0 such that the coordinate x of the point of intersection of this line with the straight line y = ß (x - b) + B satisfies the

condition 1 < Qo - X < k = const. In view of the fact that Qo - x = yY - aß, this is pos-x-a X-Q -

sible only in the case where y > k~ ~Iß. Since a = 8 2 ß, for the construction of the re-

quired example, one must check the inequality y > kf_~ ß ß. Indeed, in the intervals

(x2i-l,x2i)' i= 1,2,3, ... , wehave

y= ß = 8-2i

ki>2 - ß and, hence, y> k"=lß for large i and all k> 1. (Note that, for 8 = 2, k = 6 and

8 = ~, k = 2, this inequality holds for any i = 1,2, .... ) Similar reasoning is applicable

to the intervals (Y2i' Y2i-l)' i = 2, 3, ....

Denote the point of intersection of the lines Y = ß (x - b) + Band Y = y' (x - a 0) + b o by {al,b 1}. Inthesegment (a,ao), wepastetheselinesbythefunction

= b o + (2y-a)(x- a o) + 2(y-a)(ao-a)<pCo-_Q~),

in the segment (ao, 2a 1 - b], for this purpose, we use the function

and in the segment (2a 1 - b, b), these lines are pasted by the function

Since

sup I gyr)(x)1 = max{(2y-a-ß)[2(b-al)P-r sup 1<p(r)(x)l, xe[a,b] xe[O,I]

2(y-a)(ao-a)l-r sup 1<p(r)(x)l}' j=0,1,2, r=2,3, ... , xe[O,I]

we condude that g j E C~ ([ a, b]).

By applying this construction to the segments (x2i-l' x2i) and (YZi+2' Y2i+ I)' i = 1,

2, one can easily show that, for any r;::: 0, there exists 8 such that

!im sup I g~r)(x)1 = 0 and , I

l~~ xe(x2i_l,x2i) !im sup I gY) (x) I = O. i~~ xe[Y2i+2' Y2i+l)

The map constructed as a result belongs to the dass er, r;::: O. By connecting the point

o with Xo and the point Yo with 1 by monotone e~-functions (under the relevant sewing conditions for the derivatives at the points Xo and Yo), we obtain the required map.

Let us now discuss in brief the example suggested by Misiurewicz [4]. We use the notation introduced in the proof of Theorem 6.12 with certain modifications. Thus, the

sequences xO>xI>x2>'" and YO<YI<Y2<'" aredefinedas

1 1 =--~

(2i + 3)2' = ------.,...

(2i + 2)(2i + 3)2 '

Y2i - YZi+ 1 1

= (2i+2)2'

Y2i+ 1 - Y2i+2 (2i + 1)(2i + 2)2 '

i;::: O.

It is not difficult to show that

r r def ,1m Xi = ,1m Yi = C. l~OO l~oo

Let us construct the map f For this purpose, we set

1 1 fex) = 1, f(xZi) = 1 - (2i+2)(2i+2)!' f(xZi+I) = 1 - (2i+2)(2i+3)!'

f(yZi) = 1 - (2i + 1)(2i + I)!' f(YZi+I) = 1 - (2i + 1)(2i + 2)!

Then

182 Metrie Aspeets 01 Dynamies Chapter 6

and

lim I(xi) = !im I(Yi) = I(e) = 1. l~OO i-7OO

After this, the map 1 is constructed as in the previous example: First, in the intervals

[X2i' x2i+ rJ and [Y2i+ l' Y2J, it is defined as a linearfunction and then, in the remaining

intervals, the relevant linear segments are C= -smoothly pasted. The map 1 E C=(I, l) obtained as a result of this procedure possesses a quasiattractor of positive Lebesgue measure equal to

7. LOCAL STABILITY OF InVARIAnT SETS. STRUCTURAL STABILITY

OF unImOD8L m8P5

1. Stability of Simple Invariant Sets

1.1. Stability of Periodic Trajectories. Let f: I ~ I be a continuous map and let B =

{ ßo, ß I' ... , ßn-I} be its cycle of period n ~ 1. One can distinguish between two types

of stability of the cycle B, namely, between stability under perturbations of the initial data and stability under perturbations of the map. First, we consider the first type of stability.

Recall some definitions. A cycle B is called asymptotically stable or attracting if there exists a neighborhood U of this cycle such that

n fi(U) = B. i~O

A cycle B is called repelling if there exists a neighborhood U of this cycle such

that, for each point XE U\B, one can indicate i ~ 0 for which fi(x) ~ U.

A cycle B is called semiattracting if there exists a neighborhood U of B such that,

for any point ß j E B, one can indicate its half neighborhood Uj such that if x E

nO~j<n uj, then / (x) ~ U for some i ~ 0 and the other half neighborhoods Ui' of

the points ßj satisfy the equality

n i( U Ui') = B i ~o 0 ~j< n

As indicated in Chapter 1, these definitions do not exhaust all possibilities in the behavior of trajectories.

Theorem 7.1. An n-periodic (n ~ 1) cycle B = {ßo, ß I' ... , ßn-d of a con

tinuous map f: I ~ I is attracting if and only if, for any point x from some neigh-

183

184 Local Stability of Invariant Sets. Structural Stability of Vnimodal Maps Chapter 7

borhood V 0 of the point ß 0' the inequality f2n (x) > x holds for x < ßo and the

inequality f 2n (x) < x holds whenever x> ßo. A cycle B is repelling if and only if, for any point x of some neighborhood V 0

ofthe point ßo' f 2n (x) "- [x, ßol for x < ßo andf2n(x) "- [ßo, xl whenever x> ßo. A cycle B is semiattracting if and only if, for any point x of some neighborhood

Vo of the point ßo' either f 2n(x) E (x, ßol for x< ßo and f 2n (x) > x for x> ßo or, vice versa, f 2n(x) < x for x< ßo and f 2n(x) E [ßo, x) for x> ßo.

Theorem 7.1 can be proved by the direct investigation of the behavior of trajectories

of the map f2n in a neighborhood of its fixed point ßo under the conditions of the theorem.

Consider the case where the map f is smooth in more details. Let fE er (I, 1), r;::: 1,

and let B = {ßo, ßl'"'' ßn-d be a cycle of the map f of period n. For k E {I, 2} and i E {I, 2, ... , r}, we define the quantities

The quantity Il(B) = IlF) (B) is called the multiplier of the cycle B. The theorem below

establishes the relationship between the values of Il~) (B) and the type of stability of the cycle B.

Theorem 7.2. Let fE er (I, I), r;::: 1, and let B = {ßo, ß l' ... , ßn-d be a cy

cle of f with period n;::: 1. lf IIl(B) I > 1, then B is repelling.

Suppose that Il (B) = 1 and there is s > 1 (s::::; r) such that Ills)(B):f::. 0 but

ll\i\B) = 0 for 1 O.

Suppose that Il(B) = -1 and there exists s> 1 (s::::; r) such that ll~s)(B):f::. 0

but 1l~)(B) = 0 for 1 O.

Theorem 7.2 is proved by the direct verification of validity of the conditions of Theorem 7.1 under the conditions of Theorem 7.2. Here, we restriet ourselves to the proof

of the following statement: If Il(B) = -1, then ll~s)(B) = 0 for even s. Indeed, if s = 2, then

If s even and 1l~)(B) = 0 for 1 < i < s, then

Seetion 1 Stability of Simple Invariant Sets 185

Hence, in this case, s must be odd and the cycle B cannot be semiattracting. Parallel with the concept of asymptotic stability, one can also use the concept of Lya

punov stability.

Definition. A cyc1e B = {ßo, ßI' ... , ßn-I} of period n ~ 1 of a map fE

C°(/, I) is called Lyapunov stable if, for any neighborhood V of B, there exists a neighborhood V of B, V ~ V, such that fi(V) C V for all i> O.

It is clear that any attracting cycle is Lyapunov stable and any repelling or semiattracting cycle is not Lyapunov stable. It follows from Theorem 7.1 that if a cycle B of period n is Lyapunov stable but not attracting, then the points of this cycle are not isolated in the set of periodic points of period n or 2n. Hence, if this cycle B is a cycle of

amap fE Cr(I,I), r~l, theneither fl(B) = 1 and Illi)(B) = 0 for l<i~r or

fl(B) = -1 and fl~)(B) = 0 for 1< i ~ r. Thus, it follows from Theorem 7.2 that if a map f: I ~ I is analytic and f (x) =1= x at

least at one point x E I, then any cycle of this map iseither attracting, or repelling, or

semiattracting. Note that this is not true even for maps from the class C= because, for these maps, the set of periodic points of the same fixed period can be infinite.

Consider the problem of stability of periodic trajectories under perturbations of the

map f

Definition. We say that a cyc1e B of period n ~ 1 of a continuous map

f: I ~ I survives under CÜ -perturbations of the map f if, for any neighborhood V of the cyc1e B there exists a neighborhood 'U of the map f in

CÜ(/, I) such that any map j E 'U possesses a cycle of period n lying in the neighborhood V.

Theorem 7.3. A cycle B = {ßo, ß I' ... , ßn-I} of period n ~ 1 of a continuous

map f survives under CÜ-perturbations of the map f if and only if, for any neighbor

hood V o of the point ßo' there exist xl' x2 E Vo such that (r(x l ) - XI) (r(x2)

x2) < O.

Proof Suppose that (r(x l ) - XI )(r(x2) - x2) ~ 0 for some neighborhood Vo of the point ßo and all points x I' x2 E V 0' Without loss of generality, we can assurne

that rCx) ~ x for XE VO' If n = 1, then, for any E> 0, the map j = f - E has no

fixed points in Vo' If n> 1, then we choose a neighborhood V n-I of the point ßn-l

such that f(V,,_I) C Vo' Let V~_I be a neighborhood ofthe point ßn-I which lies in

186 Local Stability 01 Invariant Sets. Structural Stability 01 Vnimodal Maps Chapter 7

Vn_1 together with its dosure and let <p(x) be a eontinuous funetion taking values from

the interval [0, 1], equal to zero outside V n--l' and equal to one inside V~-l' Then, for

all suffieiently small E> 0, the map J = 1 - E<p has no periodie points of period n in

a eertain neighborhood Vü of the point ßo beeause P (x) ~ x - E for x E Vü. The other statements of Theorem 7.3 are obvious.

eorollary 7.1. If a cycle is attracting or repelling, then it survives under cf! -per

turbations 01 the map f

Note that the proof of Theorem 7.3 implies the following assertion: If a eyde does

not survive under cf! -perturbations of the map, then it does not survive under er-pertur

bations of the map. Indeed, the funetion <p(x) used in the proof of Theorem 7.3 ean be

taken eve,n from the dass c. If a eyde B does not survive under er-perturbations of the map 1, r ~ 1, then, by

virtue of Theorem 7.2 and Corollary 7.1, we ean write J.l (B) = 1 and either there exists

an even number s ~ r such that IlY)(B) ;:f. ° but J.llil(B) = ° for 0< i < s or J.lli\B)

= ° for 1 < i ~ r. It is worth noting that the survival of eydes under perturbations of a map is not eon

neeted with the preservation of the strueture of a dynamieal system in the neighborhood of a eyde (i.e., with the behavior of trajeetories): The behavior of trajeetories of a perturbed map in the neighborhood of a eyde may signifieantly differ from the behavior of the original map in the neighborhood of the original eyde even if this eyde survives (for example, a eyde may change the type of stability). For this reason, we introduee the following definition:

Definition. A map 1 E er (I, l) is called er -structurally stab1e in the neighborhood of its cycle B if there exist a neighborhood V of the cycle

B and a neighborhood 11 of the map 1 in er (I, I) such that, for any J E 'l1,

one can indicate a homeomorphism h = h(J) of the interval I onto itself

forwhich Johlu=hollu.

The homeomorphism h translates trajeetories (or parts of trajeetories) of the map 1 lying in V into trajeetories (or their parts) of the map J and preserves the mutual arrangement of the points of these trajeetories. This remark immediately implies the following assertion:

Theorem 7.4. A map 1 E er (1,1), r ~ 1, is er-structurally stable in a neighbor

hood 01 its cycle B if and only if I J.l (B) I ;:f. 1 and J.l (B) ;:f. 0.

Note that the eoneept of eO-struetural stability is meaningless beeause there are no

Section 1 Stability of Simple Invariant Sets 187

cf> -structurally stable maps: Indeed, for any point Xo E I, we can modify the map f to

guarantee that j (y) = const for all points in a certain neighborhood of xo. If f *- const

in this neighborhood, then the dynamics of trajectories of the map fundergoes significant changes near the indicated point. In all other cases, one can also easily construct the

required cf> -perturbation of the map f

1.2. Stability of Cycles of Intervals. By analogy with the stability of periodic trajecto

ries, we now consider the problem of stability of cycIes of intervals. Let A = {1o, / 1, ••• , In-I} be a cycIe of intervals of period n of the map fE C°(l, l). Without loss

of generality, we can assume that the intervals 1; are cIosed. In order not to introduce

new notation, we denote the set Uo . /. also by A if this does not lead to misunder-$,<n ,

standing. Finally, any open set that contains the set A is called a neighborhood of the

cycIe of intervals A. By analogy with the general definitions of attractor, repeller, and quasiattractor, we

introduce the corresponding definitions for cycIes of intervals in order to characterize the behavior of tnuectories in the neighborhood of a cycIe of intervals.

Definition. We say that a cycle of intervals A = {1o, 11, ••• , In-I} of a

map fE COU, J) is an attractor if one can indicate a neighborhood U of A

such that ni ~ ° f;(U) ~ A.

A cycle of intervals A is called arepeIler if there exists a neighbor

hood U of A such that, far any XE U\A, one can find i = i (x) for which the point fi (x) does not belong to the set U .

A cycle of intervals A is called a quasiattractor if, for any its neigh

borhood U, there exists a neighborhood U' of A such that fi(U') CU for all i;;::: O.

For cycles of intervals, one can formulate an analog of Theorem 7.1.

Let A = { 11, 12, .•. , In-I} be a cycIe of intervals of a map fE CO and!et I(A) be

the component ofthe set U . /. which contains L(} Note that I(A) = 1o whenever O$,<n ,

the intervals I; are mutually disjoint; otherwise, n is even and I(A) = 10 U In/ 2 ·

Theorem 7.5. For a map fE C°(l, I), let A = {Io, Ip ... ,In-d be a cycle of

intervals ofperiod n and let I(A) = [a, b]. The cycle ofintervals A is an attrac

tor if and only if there exists a neighborhood 1.1 of the interval I (A) such that

f 2n (x)if. [x,b] if XE 1.1 and x<a and f 2n(x)if. [a,x] if XE 1.1 and x>b.

The proof Theorem 7.5 is similar to the proof of Theorem 7.1.

188 Local Stability 01 Invariant Sets. Structural Stability 01 Unimodal Maps Chapter 7

Corollary 7.2. Let 10 = [ao, bol. 11 ao <r(ao) < bo and an< r(bo) < bo, then

the cycle 01 intervals A = {la, 11, ... , In-I} is an attractor.

It is obvious that if a cyc1e of intervals A is an attractor, then it satisfies aB conditions in the definition of quasiattractor. If a cyc1e of intervals A is a quasiattractor but

not an attractor, then Theorem 7.5 implies that at least one end ofthe interval 10) is not isolated in the set of periodic points of period n or 2n and, consequently, either a E

Per (f) or b E Per (f). Let us now consider the problem of preservation of cyc1es of intervals under pertur

bations of a map.

Definition. We say that a cycle of intervals A = {1o, 11, ••• , In-I} 0 f

period n of a map I E eO(I, l) is preserved under cD-perturbations of this

map if, for any 10 > 0, one can indicate a neighborhood 'li = 'li(e) of the

map I in e°(l, I) such that any map J E 'li pos ses ses a cycle of intervals

A of period n and the Hausdorff distance between the sets A and Ais less than e.

We say that a cyc1e of intervals A does not vanish under cD-perturbations of the map f if, for any neighborhood U of A, there exists a neigh

borhood 11 of the map f in cD (I, l) such that any map JE 'li has a cy

c1e of intervals A of period n and U is a neighborhood of this cycle.

As foBows directly from this definition, a cyc1e of intervals preserved under perturbations of the map does not vanish in the indicated sense. It is also easy to show that at-

tractors are preserved under cD -perturbations. We say that a cyc1e of intervals A of period n of a map I is maximal if the map I

has no cyc1e of intervals A of period n such that A c A and A *" A . In what follows, we restrict ourselves to the c1arification of conditions under which maximal cycles of intervals of unimodal maps are preserved or do not vanish.

Let A = {1o, 11, ••• , In-I} be a maximal cycle of intervals of period n for a map

I E cD (I, 1). Suppose that the map rl 10 is monotone. Consider the interval I(A) = [a, b 1 introduced above. Obviously, rU(A)) c I(A). Let s be the least positive integer of

the form n, 2n, 3n, ... for wh ich r I 10 is nondecreasing. It is clear that s is equal ei-

ther to n or to 2n. It follows from the maximality of A that F (a) = a and F (b) = b.

Moreover, the invariant interval [a, b 1 of F must be arepeIler, i.e., the inequalities

F(x) < x for XE (a - 10, a) and rex) > x for XE (b, b + 10) must hold for some sufficiently small 10 > o.

Theorem 7.6. Assume that a cycle 01 intervals A 01 a unimodal map I E eO(I, I)

does not contain the point 01 extremum. Then A does not vanish under cD -perturba-

Section 1 Stability of Simple Invariant Sets 189

tions ofthe map f if and only ifthe interval I(A) = [a, b] contains points xl and

X2 suchthat Xl <X2, F(Xl) > Xl' and F(X2) < X2'

The eycle of intervals A is preserved under CJ -perturbations of the map f i f

and only if, for any 10 > 0, one ean indieate points xl E (a, a + 10) and X 2 E (b - 10,

b) such that r(xl) > xl and f S (X2) < X2'

The proof of this theorem is similar the proof of Theorem 7.3.

Now assurne that a cycle of intervals A = {1o, 11, ..• , In-l} of a unimodal map f contains its point of extremum e and is maximal. Let 10 = [ao, bol. Then it follows

from the results of Chapter 5 that the map r/ [ao' tu]' where n is the period of A, is uni-

modal and either r(ao) = ao and r(bo) = ao or r(ao) = bo and r(bo) = bo (with obvious exceptions n = 1 and n = 2).

For the interval I(A) = [a, b] defined above, there exists a unique number s:2: 1

such that F (I(A)) C I(A) and F /I(A) is unimodal. It is clear that s = n if the intervals

ofthe cycle A are mutually disjoint and s = nl2 whenever 10 n In/2 *" 0.

Let <I' (e), f2s (e) be the interval with ends at I' (e) and f2s (e). Denote this inter

val by [al' b l ]. Then [al' bl ] C I(A) and the following theorem is true:

Theorem 7.7. For a unimodal map fE C°(l,I), let A be a eycle of intervals of

period n that contains the point c. If there are points xl E (a, al) and X2 E [bi' b]

such that I' (xl) > Xl and I' (X2) < X2' then the eycle of intervals A does not van

ish under CO -perturbations of the map f Moreover, any unimodal map 1 suffiei

ently close to f in C 0(1, I) has a eycle of intervals of period n that eontains the

point of extremum of the map J. The eycle of intervals A is preserved under cD-perturbations of the map f if and

only if, for any 10 > 0, there are points Xl E (a, a + 10) and X2 E (b - 10, b) such that

Xl, X2 ~ [al' btl, F(Xl) > Xl' and r(X2) < X2'

Prooj. Without loss of generality, we can assurne that the point e of the map F/ I(A)

is a point of maximum. First, we consider the case s = nl 2. If there are no points Xl and X2 indicated in

the conditions ofTheorem 7.7, then r(e) = band, as in the proof of Theorem 7.3, one

can construct a small continuous perturbation of the map f such that the resulting per

turbed map J has no cycles of intervals of period n that contain the point e.

If s = n and there are no suitable points Xl and X2' then either r (e) = b or r (X)

$ X for XE (a - 10, PS (e)] with some 10 > O. It is c1ear that, in both cases, the cycle of

intervals A disappears under small CO-perturbations ofthe map f

Corollary 7.3. A eycle of intervals A of period n of a unimodal map f survives

190 Local Stability 01 Invariant Sets. Structural Stability 01 Unimodal Maps Chapter 7

under eO -perturbations 01 the map if and only if, Ior any e > 0, the map I possesses

a cycle 01 intervals ii 01 period n such that

(a) the Hausdorff distance between the sets A and ii does not exceed e;

(b) there exists a neighborhood U oithe cycle oIintervals ii which lies in A;

(c) the cycle 01 intervals ii is an attractor.

This corollary is a consequence ofthe assertions and proof ofTheorem 7.7. As in the case of periodic trajectories, it follows from the proof of Theorem 7.7 that

if a cyc1e of intervals A of a unimodal map I E er (I, 1), r ~ 1, is not preserved under

cD -perturbations of the map f, then it is not preserved under er-perturbations of the map

I even if the perturbed map J remains in the c1ass of unimodal maps. This observation is used in what follows.

Generally speaking, the problem of structural stability of the map I in the neighborhood of a cyc1e of intervals A under perturbations of the map I is not simpler than the

problem of structural stability of the map I in the entire interval I. Therefore, we consider this problem in Seetion 3.

2. Stability of the Phase Diagram

2.1. Classification of Cycles of Intervals and Their Coexistence. In Chapter 3, we used the c1assification of cyc1es in terms of permutations to study the coexistence of periodie trajectories of continuous maps. Similar c1assification can be applied to the investigation of cyc1es of intervals.

Let A = {1o, I], ... , In-]} be a cyc1e of intervals of period n of a map I E e O (I, 1).

This cyc1e of intervals is associated with a permutation

1t(A) = (1 2 ... n) t] t2 ... tn

as follows:

(a) the intervals ~, i = 0, 1, ... , n - 1, are renumbered in the order of their location

in the real line; as a result, we obtain an ordered collection of intervals Ä =

{I], 12 ,,,,, In};

Section 2 Stability of the Phase Diagram 191

(b) we set t; = j if f( 1;) c 1j , i = 1, 2, ... ,n; the permutation 1t (A) obtained as a

result is called the type of the cycle of intervals A.

If apermutation

(1 2 ... n)

1t = t1 t2 ••• tn

is the type of a cycle of intervals of a continuous map, then the set { 1, 2, ... , n} is the

minimal set of the map 1t of this set onto itself, i.e., it contains no proper invariant subsets. Permutations of this sort are called cyclic permutations. They were studied in Chapter 3. For any cyclic permutation 1t, one can easily construct a continuous map f:

I ~ I which possesses a cycle of intervals A whose type n( A) coincides with a given permutation 1t.

In Chapter 3, for continuous maps, we established several theorems on the coexistence of periodic trajectories of various periods and types. The following statement demonstrates that, for cycles of intervals, the situation is somewhat different because, unlike periodic orbits, cycles of intervals consist of nondegenerate intervals.

Proposition 7.1. For any cyclic permutation

1t = ( 1 2 ... n) t1 t2 ••• tn '

there exists a continuous map f: IR ~ IR which has a cycle of intervals of type 1t but has no other cycles of intervals.

Proof. Consider apermutation

(1 2 ... n)

1t = t1 t2 .•. tn

For i = 1, 2, ... ,n, we define 1; = [4i - 2, 4i]. The map f: IR ~ IR is first defined at

points with integer coordinates j E {I, 2, ... , 4 (n - 1) + 5} as folIows: If j = 4 i - 2 or

j = 4i, then f(j) = 4 t; - 2; if j = 4 i-I, then f(j) = 4t;; at all other points, we set

f(j) = O. Then we extend f to the components of IR \ {I, 2, ... ,4n + I} by linearity.

As a result, we obtain the required piecewise linear map f: IR ~ IR. This map is expanding because its derivative is greater than two at all points of its domain of definition. Hence, the trajectory of an arbitrary interval U either eventually hits one of the intervals

1;, i E {I, 2, ... , n} or covers the point of extremum of the map f which does not be

long to these intervals. In the second case, 0 E I(u) for some k. Since the intervals

192 Local Stability of Invariant Sets. Structural Stability of Unimodal Maps Chapter 7

11, lz, ... , In form a eycle of intervals and jm (0) ~ -00 as m ~ 00, this completes the

proof of Proposition 7.1. Nevertheless, under certain additional restrictions, the fact that a continuous map has

cycles of intervals of a given type implies that it also has cycles of intervals of some other types. (The exact formulations are presented below.)

Let

1t(I) = (1 2 ... n) t l t2 •.• tn

be a cyc1ic permutation. We say that a cyclic permutation

(2) (

1 2 ... k) 1t

sI Sz ... sk

divides the permutation 1t(l) if there exists m ~ 1 such that n = m· k and, for any jE

{1,2, ... ,k}, themap 1t(I) mapstheset {mj-m+l, mj-m+2, ... ,mj} ontotheset

{msj - m + 1, mSj - m + 2, ... , mSj }.

It is c1ear from the definition that any permutation divides itself and that the permutation

divides any other permutation. A nontrivial example is given by the permutations

( 1 2 3 4 5 6) 1t6 = 4 6 5 3 2 1 and 1tz = (~ ~}

It follows from the definition that if apermutation 1t(3) divides apermutation 1t(Z) and

the permutation 1t(2) divides the permutation 1t(l), then 1t(3) divides 1t(l).

Proposition 7.2. Let fE eO(I, I). Assume that the map f has cycles of intervals

A = {10' 11' ... , In-d and A = {Ja. 11 •...• 1k- l } of periods n and k, respec

tively, such that

U l i C U 1i · O~i<n O~i<k

Then the permutation 1t (A) divides the permutation 1t (A).

Proof. Proposition 7.2 is a consequence of the definition of cycles of intervals.

Proposition 7.3. Let A = {1o, / 1, ••• , In-I} be a cycle of intervals of period n of

a map fE eO(/, I). Assume that the map f is monotone in any component of the set

1/ UO";i<n li' Then, for any permutation 7t' which divides the permutation 7t (A),

h ,/" l' {I' I' '} , t ere exists a cycle OJ interva s A = 0' 1, ... ,lk-1 of the map f such that 7t (A)

= 7t' and

U li C U 1i· O";i<n O";i<k

Proof. Let the conditions of Proposition 7.3 be satisfied and let k be the length of

the permutation 7t'. Then n = k· m for some m ~ 1. We enumerate the intervals of the

cycle A in the order of their location in the real line. As a result, we obtain an ordered

collection of intervals A = {11' 12"" , In}. Let

(1 2

7t' -tl t2

Then, under the conditions of the proposition, for j = 1, 2, ... , k, the intervals of A with

indices j m - m + 1, j m - m + 2, ... , j mare mapped into the intervals with indices

tjm- m + 1, tjm- m + 2, ... , tjm, respectively. For j = 0, 1, ... , k - 1, let 1j be the

smallest interval that contains the intervals of the set A with indices j m + 1, j m +

2, '" ,j m + m. Since f is monotone in components of the set 1/ Uo ' I", we have ";,<n

f( 1j ) c 1tj for j E {O, 1, ... , k - I} and the intervals 1j form a cycle of intervals A'

of period k such that 7t (A') = 7t' and

U li C U 1i· O";i<n O";i<k

Note that Proposition 7.3 gives information about the nonlocal behavior of maps, which is used in what follows.

The following statement establishes conditions for the coexistence of periods of cycles of intervals and periods of periodic trajectories of continuous maps.

Proposition 7.4. Let A = {/O, I" ... , In-I} be an n-periodic cycle of intervals

of a map fE e°(l, I). Then the map f possesses a periodic trajectory of period s, where s = n if the intervals of the cycle Aare mutually disjoint and s = n / 2 if this is not true.

ProoJ. Under the conditions of the proposition, we have fnUo) c 10. Hence, the

map fn possesses a fixed point ßo in the interval 10 . If the intervals of the cycle A

are mutually disjoint, then ßo is an n-periodic point of the map J. Otherwise, it is not

difficuIt to show that n is even and ßo is an intern al point of the interval 10 U In/2 = I(A). Thus, the period of the trajectory of the point ßo under the map f is not less than n12. Clearly, in this case, the period of ßo is equal either to n or to nl 2. It follows from the resuIts of Chapter 3 that, in both cases, the map f possesses a periodic trajectory of

period n12.

2.2. Conditions for the Preservation of Central Vertices. As shown in Chapter 5, the central vertices of the phase diagram of a unimodal map (i.e., vertices corresponding to the cycles of intervals that contain the point of extremum of a given map) are linearly ordered and their number is at most countable. In Chapter 5, central vertices of the phase

diagram were denoted by A;m' m :'0: m*. They were identified with maximal cycles of

intervals of period Pm covering the point of extremum. In this section, we formulate

conditions under which central vertices do not disappear under CÜ-perturbations of the map. These conditions, together with results established in Section 5.2, enable us to make some conclusions about the structural stability of uni modal maps, i.e., about the nonlocal behavior of dynamical systems.

In this section, we denote central vertices of the phase diagram of a unimodal map f by A;m (f) and their number by m*(f) (recall that m*(f) :'0: 00). The following asser

tion establishes the relationship between the behavior of trajectories for unimodal maps whose phase diagrams are characterized by central vertices of the same types.

Proposition 7.5. If the equality 1t (A;m (f)) = 1t(A;n (f)) holds for unimodal

maps fand g for some m:'O: m*(f) and n:'O: m*(g), then m = n and

forany k:'O:m.

ProoJ. Proposition 7.5 imrnediately follows from Propositions 7.3 and 7.4 and from the construction of phase diagrams in Chapter 5.

Proposition 7.6. Let f be a unimodal map. Then, for any m< m*(f), there exists

E = E(m) > 0 such that m *(g) ~ m for any unimodal map g with Ilf-glico < E

d * * an 1t(Apn (g)) = 1t(Apn (f)) faral! n:'O: m.

ProoJ. If we assume that a cycle of intervals A;m (f) vanishes under CJ -perturba

tions of the map f, then it follows from the proof of Theorem 7.7 that m = m* (f) but

this is impossible by the condition of the proposition. Hence, the conditions of the first statement ofTheorem 7.7 are satisfied. By virtue ofTheorem 7.7, one can indicate E > 0

such that any unimodal map g with Ilf - g 11 Cl < E has a cyde of intervals A of period

Pm which contains the extremum point of the map g. It is clear that, in this case, 1t (A) =

1t (A;m (f)). The required assertion now follows from Proposition 7.5.

Proposition 7.7. Let f be a unimodal map.lfthe point e is not periodie and lies

in the domain of attraetion of an attraeting eycle, then m*(g) = m*(f) and

for any unimodal map g with sufficiently small Ilf - g 11 co.

Proof. It follows from the results established in Chapter 5 that m*(f) < 00 under the conditions of Proposition 7.7. By Theorem 7.7, any unimodal map g sufficiently

dose to the map f in the metric of the space CO (I, I) has a cyde of intervals of period Pm*(1l that contains the point of extremum ofthe map g. Hence, m*(g) ;::: m*(f).

Assurne that the trajectory of the point of extremum e of the map f is attracted by the trajectory of a periodic point ß. Denote the period of the point ß by k. According

to Theorem 7.1, there is a neighborhood U of the point ß such that fk( U) C U for

XE U, f2k(x) > x if x< ß, and f2k(x) < x if x> ß. Let U be the largest neighborhood of the point ß with the indicated property. Then the domain of attraction of the

trajectory of the point ß coincides with the set Ui;::o f-i (U). The trajectory of the in

terval U forms a cyde of intervals, which is denoted by B. Hence, under the conditions of the proposition, there exists j;::: 0 such that f j (e) E

U. By Theorem 7.6, the cyde of intervals B is preserved under sufficiently small CÜperturbations of the map f In this case, if the perturbed map g is unimodal, then the first j iterations of its point of extremum C are slightly different from the first j iterations of the point e of the map f Hence, the point gj (c) also belongs to a cyde of intervals which does not contain the point of extremum of the map g. This means that, under the conditions ofthe proposition, we have m*(g) = m*(f) and

Let us now make several remarks. Let g be a unimodal map sufficiently dose to a

uni modal map f in CO(I, 1). If e E Per (f), then, by virtue of Proposition 7.6, we have

m*(g) ;::: m*(f) - l. One can easily construct an example of g such that m *(g) > m*(f); moreover, for any k;::: 1, one can find a map g such that m*(g) ;::: m*(f) + k. On the other hand, it is not difficult to show that, for smooth unimodal maps fand g

sufficiently dose in Cl (1,1), we have m*(f) ::; m*(g) ::; m*(f) + 1.

If w(c) is not a cyc1e and there is a neighborhood U of the point c such that fi (!J) ()

fj (U) = 0 for all i:#= j, then itfollows from the proof of Theorem 7.7 that the cyc1e of

intervals A;m' (f) (f) (with m*(f) < 00) does not vanish under Cl-perturbations of the

map f Consequently, m*(g) ~ m*(f). It is not c1ear whether the equality m*(g) = m*(f) is true under these conditions for smooth unimodal maps f and g sufficiently c10se in er (1,1), r ~ 1. The same question remains open for m*(f) = 00.

3. Structural Stability and Q-stability of Maps

In this section, we study the problem of stability of the dynamical structure of dynarnical systems. In order to compare the dynamics of various systems, we use the concept of topological equivalence introduced in Chapter 1.

We recall the corresponding definition. Maps f: I ~ land g: I ~ I are called to

pologically conjugate if there exists a homeomorphism h: I ~ I such that g 0 h = hof

in I.

It follows from this definition that if maps f and gare topologically conjugate, then the homeomorphism h transforrns trajectories of the map f into trajectories of the map g. This means that topologically conjugate maps generate topologically equivalent dynamical systems.

By using this relation of equivalence of maps, one can introduce all necessary char-

acteristics of the stability of the structure of trajectories in er (I, 1), r ~ O.

Let A: er (f, I) H 21 be a map which associates every point f E er (I, l) with a

c10sed set A (f) E 21 such that f (A Cf)) C A (f). We say that a map fE er (I, I) is er _

structurally A-stable if there exists a neighborhood V(f) of the map f in er (f, I) such

that, for any gE Vif), the maps flA(f) and g IA(g) are topologically conjugate.

We consider the cases where a role of the set A Cf) is played either by the entire in

terval I (this corresponds to er-structural stability) or the set of nonwandering points

(this corresponds to the so called er -structural Q-stability). Note that, parallel with structural A-stability, it might be interesting to study A-stab

ility regarded as the stability of the set A (f), i.e., to test the map A: f ~ A (f) for con

tinuity or upper semicontinuity at the point fE er (I, 1).

In what folIows, we assurne that the spaces er (I, 1), r ~ 0, are equipped with metric

p/f,g) = L. maxi Dif(x) - df(x) I, 05i5r xel

where

Section 3 Struetural Stability and Q-Stability of Maps 197

if dxi .

In what folIows, main attention is paid to the problems of C--structural stability and cistructural Q-stability for the following reason: It is clear that the class C°(l, l) contains

no maps that are cD -structurally stable: Indeed, by small cD -perturbations of the map in a neighborhood of a fixed point, one can always change at least the qualitative behavior

of trajectories in this neighborhood. On the other hand, if a map from the class Cl (I, l)

pos ses ses a critical point, then there are maps close to this map in Cl (I, I) which possess an interval of critical points. Therefore, these maps are not topologically equivalent

to the original map. A similar situation is also possible for maps from the class C2 with degenerate critical points. At the same time, it may happen that either NW (f) contains

no critical points of fE cI (I, I) or all critical points of the map f are periodic, i.e.,

isolated in NW (f). In this case, it seems reasonable to study the problem of cistructural Q-stability of the map f

Suppose that fE C2 (I, I) is a unimodal map. The map f cannot be C2 -structurally stable if f" (e) = 0 or if it possesses a nonhyperbolic periodic trajectory. If there exists a

point x E I such that e E O)j(x), then the map f mayaIso be C2 -structurally unstable.

Structural stability is also impossible in the case where fj (e) E Per (f) for some j ~ O.

At the same time, if the indicated possibilities are excluded, then the map f is C2 -structurally stable. Moreover, the results established in the previous chapter imply the following assertion:

Theorem 7.8. Assume that a unimodal map fE C2(I, l) satisfies the eonditions

(a) f'(x)=F-O for XE I\{e} and j"(e)=F-O;

(b) the set Per (f) does not eontain nonhyperbolic orbits;

(e) efi NW(f) andfi(e)fi Per(f)forall i~1.

Then f is C2 -strueturally stable.

Proof. Since the critical point is unique and e fi NW (f), one can indicate i ~ 0 and a periodic interval L such that fi (e) E L. Since f i (e) fi Per (f), the point f i (e) belongs to the domain of immediate attraction of a certain attracting cycle (by the condition, the map f does not have any nonhyperbolic periodic orbits). By Theorem 6.3, the set NW (f) is hyperbolic, i.e., there exist C> 0 and 'A > 1 such that, for any

point XE NW(f), either I Dnf(x) I ~ C'An or IDnf(x)1 :s; LI 'A-n forall n ~ o. By Theorem 5.6, f has finite1y many attracting cycles. Let Bo(f) denote the union

198 Loeal Stability of Invariant Sets. Struetural Stability of Unimodal Maps Chapter 7

of the domains of immediate attraction of all attracting cycles of the map f. Then Bo(f)

consists of finitely many open intervals. We choose an integer number n such that

e E Bn(f) = U ri(Bo(f)) OSiSn

and the inequality I D F'''(x) I ~ /1 < 1 holds for any single-valued branch of the map

r n for all x which do not belong to the set Bn(f). Hence, for any map j sufficiently

close to the map f in cl (I, I), the set Bn(j) and the constant Ci are close to Bn(f)

and /1, respectively. This implies that the maps f I NW(f) and j INW( j) are

topologically equivalent and the maps fand j are topologically conjugate.

Corollary 7.4. Let fE C3(I, I) be a unimodal map, let Sf(x) < 0 for XE I\{e},

andlet IDf(x)l> 1 for XE (JI. If e~ Per(f) andthereexistsapoint ß E Per(f)

suehthat 1J'(ß)f'(f(ß)).·.f'(fIl-I(ß))I < 1, where n istheperiodofthepoint ß,

then f is C2 -strueturally stable.

Note that a theorem similar to Theorem 7.8 is true for an arbitrary map from the class

d(/, I) (Jakobson [1]).

Theorem 7.9. The set of C2 -strueturally stable maps is dense in the spaee Cl (I, I)

with metrie PI'

Forthe complete proof ofTheorem 7.9, see (Jakobson [1]). Here, we prove this theorem only for unimodal maps.

First, we show that the collection of maps which have attracting cycles is dense in

C l (/, I). Assurne that e E Per(f), where fE cl (/, I), and that e is the point of maxi

mumofthemapf. Thereisaneighborhood U ofthepoint e suchthat I Df(x) I <E/3 for XE U. By the assumption, the neighborhood U contains a periodic point ß of the

map f. Without loss of generality, we can assurne that f(ßl) < f(ß) for any ßI E

orb (ß), ßl"* ß· Thus, one can find points x, x' E U such that x< e < x' and (x, x') n orb (ß) = {ß}. Under these conditions, the function f can be replaced in the interval

(x, x') by a function j such that PI (f, j) < E, j(x) = fex), j(x') = fex'), j(ß) =

f(ß), and the point ß is a unique extremum point of the map j. Note that the maps from the class C2 are dense in the space cl(/, I) with metric PI'

Hence, we can assume that fE C2(/, I).

If e ~ Per (f), then either there is i ~ 0 for which fi (e) belongs to a periodic homterval or e E U, where U is a wandering interval. In the first case, the required assertion is obvious. In the second case, one can assurne that U is the maximal wandering

Section 3 Structural Stability and o.-Stability of Maps 199

interval that contains the point c. It follows from the results of the previous chapter that

there exists a critical point cl which lies in the interval Per(f), which contradicts the assumption that the extremum point c is unique.

The set of 0-maps with nondegenerate critical points is dense in the space Cl (I, I). Therefore, by using the reasoning presented above, we have actually proved that the set

of C2-unimodal maps whose single critical point lies in the domain of attraction of an at

tracting cyc1e is dense in the space of cl-unimodal maps. The fact that maps without nonhyperbolic periodic orbits are typical is established by

using the Sard theorem.

Thus, C2 -structurally stable maps form a dense sub set of the space Cl (I, l) with met

ric PI' The answer to the question as to whether C2-structurally stable maps are dense in

the space C2 (I, l) with metric P2 remains unclear even in the case of unimodal maps. Note that the argument presented above yields the following assertion for structural

o.-stability:

Theorem 7.10. Assume that a unimodal map fE Cl (I, I) satisfies the conditions

(a) j'(x) *- 0 for XE I\{c};

(b) the set Per (f) does not contain nonhyperbolic orbits;

(c) for any i;:: 0, either f i (c) II NW(f) 0 r f i (c) is a periodic point isolated in Per (f).

Then f is cl-structurally o.-stable.

Consider the problem of o.-stability. It is c1ear that 0. -stable maps cannot have wandering intervals vanishing under perturbations of a map; moreover, the elements of the spectral decomposition of this map should not undergo significant changes under these perturbations.

Theorem 7.11. Let fE C 3(I, I) be a unimodal map, let Sf(x) < 0 for XE I\{c}, and let f"(c) *- O. Assume that f has no semiattracting periodic orbits and

a( C~2) n Per(f) = 0. Then the map 0.: f ~ NW(f) in the space C 3(1, l) with

metric P3 is continuous at the point f

ProoJ. Note that the conditions of Theorem 7.11 are, in fact, necessary and suffici

ent conditions for the C3 -0. -stability of maps with negative Schwarzian. Both cases

where f is not C3-o.-stable are displayed in Fig. 37 (where the graph ofthe map fPm' is

depicted in a neighborhood of the cyc1e of intervals A;m* which contains the point c).

In these cases, the trajectories of almost all points of the interval I (with respect to the

200 Loeal Stability of Invariant Sets. Struetural StabiZity of Unimodal Maps Chapter 7

Lebesgue measure) are attracted by an invariant set which is not an attractor: In the first case, this is a serniattracting periodic orbit formed by an attractor "coupled" with a repel-

ler. In the second case, this is the set e~oJ which is, in this case, a repeller. Note that, in both cases, m* < 00.

If m* < 00 and the conditions ofTheorem 7.11 are satisfied, then it follows from the

results obtained in the previous section that the sets CP: and cp~) undergo small

changes under small e3 -perturbations of the map f for all m ~ m*. This means that, under the conditions of the theorem, the set NW (J) cannot become much larger.

On the other hand, Per(J) = NW(f) for maps with negative Schwarzian. Therefore, under the conditions of the theorem, for any E > 0, one can choose a finite Enet formed by hyperbolic periodic orbits in the set NW (f). Under sufficiently small perturbations of the map f, the periodic orbits of the indicated net do not vanish and the

corresponding periodic orbits ofthe perturbed map J form a 2E-net in the set NW(J). If m* = 00, then the set

CP: = n CP: m~l

contains no intervals. Therefore, the required assertion follows from the argument used in the case m* < 00.

Note that, for maps satisfying the conditions of theorems presented in this section, we

have Per(f) = NW(f). The following theorem demonstrates that this is a typical property of smooth maps:

Theorem 7.12 (Young [I]). Let r ~ 0 and fE er (I, I). Then, for any E > 0, 0 n e

eanfindamap gE er (I, I) suehthat pr(g,J) < E and NW(g) = Per(f).

ProoJ. We consider only the case of unimodal maps. If NW (g) '* Per(f), then e(l; NW(J) and f(e)E NW(J) (see Section 5.2); moreover, there exists a neighborhood U of the point e such that fi (U) () U = 0 for any i ~ 1. The map g is defined as follows: For XE 1\ U, we set g (x) = fex). For XE U, the map g is

defined so that e remains its unique point of extremum, g(U) C f(U), and g(e) (I;

o(J(U». (Note that fee) E o(J(U» because fee) E NW(J». Clearly, the quantity Pr (g, f) can be made as small as desired and, by construction, we have g( e) (I; NW (J)

and, therefore,

NW(g) Per(g) Per(f).

8. OnE-PRRRmETER FRmILIE5 OF UnImODRL mRP5

1. Bifurcations of Simple Invariant Sets

If a dynamical system describes areal process or phenomenon, then, as a rule, its properties depend on parameters. Any variation of the parameters inevitably results in a certain perturbation of the trajectories of a dynamical system under consideration. It is worth noting that small changes in the parameters may lead to significant changes in the structure of dynamical systems, i.e., to bifurcations or qualitative changes in the behavior of trajectories. In many cases, it is quite useful to know the values of the parameters for which "small errors" are admissible and the qualitative behavior of trajectories is not affected as well as the values of the parameters for which these "small errors" significantly distort the original dynamical picture.

Here, we consider the simplest case of one-parameter families of maps. As becomes clear from our subsequent presentation, these families are characterized by all types of bifurcations typical of one-dimensional maps.

Let ft.. be a family of maps from the class er (I, 1), r ~ 0, and let A be a parameter

that takes values from an interval A. We say that a value AO E A of the parameter A is regular if there is E > 0 such that the maps JA and J'A..J are topologically conjugate

for any A E (AO - E, AO + E). Denote the set of regular values of the parameter by AR'

The set AB = A \ AR is called the set of bifurcation values of the parameter.

Bifurcations of cycles are the simplest type of bifurcations. Their investigation can be reduced to the study of the local behavior of maps in the neighborhood of points that form a cycle.

For one-parameter families of smooth maps, there are several typical bifurcations of periodic trajectories. One of these has already been encountered in Chapter 1, where we

studied the family A: x ~ AX (1 - x). Indeed, as the value of the parameter A increases

from 0 to A* "" 3.57, one observes the successive appearance of attracting cycles of pe

riods 1, 2, 22, 23, .... These bifurcations of cycles can be described as follows: If An is the bifurcation value of the parameter corresponding to the appearance of a cycle B of

period 2n, then the cycle B is attracting for An< A< An+ 1 and its multiplier varies from

201

202 One-Parameter Families oJ Unimodal Maps Chapter 8

+ 1 (for A = An) to - 1 (for A = An+ 1)' For A> An+ [, we have !l (B) < -1. Therefore,

the cycle B becomes repelling. The period of the attracting cycle B' that appears for

A > An+ 1 is twice as large as the period of B). This cycle is attracting for An+ I < A :S;

An+2 and

!im !leB') = 1. "-t "n+l

As A increases, this process is repeated again and again.

For A > A * '" 3.57, the map x ~ Ax(l - x) has cycles of periods that are not powers of two. For A = 4, this map has cycles of all periods. It is clear that the period doubling bifurcation cannot be responsible for the appearance of all these cycles. Thus, it cannot result in the appearance of cycles with odd periods. In general, the bifurcation that generates cycles of odd periods (including fixed points) can be described as follows: For

A < AO' the map J" has an interval J which does not contain fixed points of the map Jr.:

(i.e., Jr.: (x) :f- x for XE J). For A = AO' the curve y = Jr.: (x) touches the line y = x

at a point Xo E J, i.e., we observe the appearance of a fixed point Xo of the map Jr.:

(its multiplier is equal to + 1). For A> AO' this fixed point decomposes into two fixed points one of which is attracting and the other one is repelling.

It is worth noting that these two types of bifurcations are substantially different. In fact, period doubling bifurcations are local and qualitative changes in the behavior of trajectories are observed only in a small neighborhood of the cycle (mild bifurcation). Bifurcations of the second type (bifurcations of creation of cycles) arrest the motion of

points from the domain {x < xo} to the domain {x > xo} near the point x = Xo as soon

as the indicated lines touch each other and lead to global (i.e., not only in the neighbor

hood xo) qualitative changes in the behavior of a system (rigid bifurcation). Following Guckenheimer [1], we now formulate the conditions which lead to bifur

cations of cycles, in the form of two theorems.

Theorem 8.1. Let h: I ~ I be a Jamily oJ C2 -maps with smooth dependence on

the parameter A E (AI' A2)' let ßo be a Jixed point oJ the map h o' AO E (A1' A2)'

and let Jio (ßo) = 1. lf

1) Ji~ (ßo) > 0 and

d 2) dA h(ßo),,="-o < 0,

then there exist c > 0 and 0 > 0 such that

(a) Jor A E (AO - 0, AO)' the map J" has no Jixed points in the interval (ßo - c, ßo + c);

Section 1 Bifurcations of Simple Invariant Sets 203

(b) for A E (Ao, Ao + 0), the map h has two fixed points in the interval (ßo - 10,

ßo + E); one ofthese points is attracting and the other one is repelling.

The statement of the theorem remains valid if both 1) and 2) are replaced by the inverse inequalities. If only one of these inequalities is replaced by the inverse inequality,

then fixed points appear as A decreases. In other words, fixed points appear or disap-

pear as A increases in accordance with the sign ofthe product f{'(x) :A h.cx) for A =

Ao and x= ßo.

Proof. Considerthefunction h(x,A) =hJx)-x. Wehave

dh dA "* 0 and

dh = 0 dx

at the point (ßo, Ao). By the implicit function theorem, there exists a smooth function

A = <p(x) such that Ao = <p(ßo) and h (x, <p(x» = 0 in a certain neighborhood of the

point ßo. By differentiating the last identity two times, we obtain

Since

for x = ßo' the curve A = <p (x) lies on the one side of the tangent at the point ßo. The

last statement of the theorem follows from the fact that ! (~;) "* 0 at the point

(ßo' Ao)·

Theorem 8.2. Let h: I ~ I be a family of C3 -maps with smooth dependence on

the parameter A E (A" A2), let ß o be a fixed point of the map h o' Ao E (A" A2),

and let Ro (ßo) = -1. If

d3 ff(X) I) < 0 and

dx3

d 2) dA (J{(x») < 0,

for A = Ao and x = ßo, then there are 10 > 0 and 0 > 0 such that

204 One-Parameter Families of Unimodal Maps Chapter 8

(a) for A E (Ao - 0, Ao), the map h has exactly one fixed point in the interval

(ßo - E, ßo + E) and this fixed point is attracting;

(h) for A E (Ao, Ao + E), there are three fixed points of the map h in the inter

val (ßo - E, ßo + E); moreover, the middle point is a repelling fixed point of

the map hand the other two points form an attracting cycle of period two.

If inequality 2) has the opposite sign, then the assertions of the theorem remain true

but the eycle of period two appears as A deereases. If we change the sign in inequality 1), then it is necessary to replace the word "attracting" by "repelling", and vice versa.

Proof. Since Ro (ßo) = -1, we can write

for x = ßo. Consider the function hex, A) = Ilex) - x. For x = ßo and A = Ao, we have

h = 0,

and

oh = 0 ox '

where, as above, SA denotes the Sehwarzian of fA:

By the implicit function theorem, there exists a function x = <p(A) such that ßo = <p(AO) and hJ <p(A)) = <p(A) for all A close to AO. Consider the function

hex, A) hex, A) x - <p(A)"

For x = ßo and A = AO' we have

h = 0, oh ox = 0, and oh oA "#0.

By applying the reasoning used in the proof ofTheorem 8.1 to hex, A), we arrive at the required assertions.

Section 2 Properties oIthe Sets oI Bifurcation Values. Monotonicity Theorems 205

Note that the Schwarzian appears in the proof of Theorem 8.2 as a natural characteristic of the period doubling bifurcation.

The formulations of Theorems 8.1 and 8.2 presented above are adjusted to the case of fixed points. To cover the case of bifurcations of cyc1es of period n, one must replace

there I'/.. by Ir:· It is worth noting that condition 1) of Theorem 8.2 is always satisfied for quadratic

maps (as weIl as for general maps with negative Schwarzian). Hence, these maps are characterized by a single type of bifurcations, namely, by period doubling bifurcations in the course of which an attracting cyc1e of period n becomes repelling and generates an attracting cyc1e of period 2n.

2. Properties of the Set of Bifurcation Values. Monotonicity Theorems

We find it reasonable to anticipate the investigation of arbitrary smooth one-parameter families of one-dimensional maps by the analysis of the behavior of some very simple families of maps, e.g., of the farnily of quadratic maps or more general families of maps with negative Schwarzian.

Let I'/..(x) = Aj(x), where ft..: [0, 1] ~ [0, 1] is an S-unimodal map such that I(O) =

I (1) = 0 and A E A = (0, I/I (c)). In addition, we require that the inequality f" (x) < 0 must hold for all XE (0, 1). The importance of this assumption is c1arified in what follows. Note that the farnily of quadratic maps often encountered earlier satisfies the indicated conditions.

For the family I'/.., let ARbe the set of regular values of the parameter A (defined in

the previous section). It follows from the definition ofthe set AR thatit is open in A.

As shown in Chapter 6, for maps from the family I'/.., the probabilistic limit set 51. (J'/..)

(i.e., the smallest set which contains the (J)-limit sets of almost all points with respect to the Lebesgue measure) is either an attracting (or semiattracting) cyc1e or a cyc1e of intervals in which the map I'/.. possesses the mixing property or coincides with the set (J) Jl. (c)

(in this case, (J)Jl. (c) is a Cantor set and CE (J)Jl. (c); see Theorem 6.9); moreover, in all

cases, we have (J) iA (c ) c 51. (J'/..). Thus, the range of the parameter A can be split into

the following mutually disjoint subsets:

Ao = {A E AI 5I.(f'/..) is a cyc1e},

Al = {A E AI 5I.(J'/..) is a cyc1e of intervals },

A2 {A E AI 5I.(J'/..) is a Cantor set}.

Let us now formulate several hypotheses concerning the problem of alternation of

regular and stochastic behavior in the family ft ...

Proposition 8.1. AR is a subset of Ao and Al U A2 is a subset of AB·

Note that the proof of Proposition 8.1 is closely connected with the investigation of the problem of structural stability of maps with negative Schwarzian.

Let Adx be the set of values of the parameter I. E A for which ft.. possesses an in

variant measure absolutely continuous with respect to the Lebesgue measure.

Proposition 8.2. mes Adx > o.

For a special case, this assertion was formulated and proved by Yakobson.

Theorem 8.3. Let f be a C3-map which is sufficiently close in C3 (I) to the map

x ~ x(l-x). Then, for the family x ~ Af(x), the Lebesgue measure of the set A dx

is positive; moreover, the point I. = 4 is a density point ofthis set.

The proof of this theorem can be found in Jakobson [4].

Note that it follows from Proposition 8.2 and the inclusion Adx C Al U A2 that

mes (Al U A2 ) > O. It is thus interesting to find the measures of the sets Al and A2 (it

is known that both these sets are uncountable) and to check the validity of the inclusion

Adx C Al.

Let fand g be S-unimodal maps. We say that the map fis not simpler than g if f is semiconjugate to g, i.e., there exists a monotone continuous map h: I ~ I such that

go h = hof (see Seetion 2.4). In this case, h maps the trajectory of a point x of the map f into the trajectory of the point h (x) of the map g. Therefore, if f is not simpler than g, then the kneading invariants satisfy the inequality VI ::; v g (recall that the points

of extremum are assumed to be the points of maximum).

Proposition 8.3. The dynamics of the map h.. becomes more complicated as I. increases, i.e., if Al ;:::: 1.2 ' then fA! is not simpler than ftv;..

This proposition is completely proved only for the families of quadratic maps. In this case, it is a consequence ofthe following theorem (see Milnor [1] and Jonker [2]):

For a quadratic map f: x ~ A x2 + Bx + C, A *- 0, we define its "discriminant" by

the formula I),.f = B 2 - 4AC - 2B.

Theorem 8.4 (monotonicity theorem). Let fand g be quadratic maps. If I'1f<

I),g, then, for any n ;:::: 1, the number of fixed points and the number of extrem um

points of the map fn do not exceed the number of fixed points and the number of extremum points of the map gn, respectively.

Seetion 3 Sequence oJ Period Doubling Bifurcations 207

The proof of this theorem is based on the following statements (see Milnor [1]):

I. J and g are linearly conjugate (i.e., J = h- 1 0 g 0 h, where h is a linear func

tion) if and only if ~J = ~g.

11. J possesses an invariant interval if and only if ~ JE [ -1, 8].

111. If the extremum points of J and g are periodic and form equivalent cycles, then

J and g are linearly conjugate.

The main problem encountered in proving Theorem 8.4 is connected with the proof of the third statement. Although this statement seems to be obvious, the proof suggested by Milnor [1] requires the transition to the complex plane.

In conclusion, we present the formulation of another monotonicity theorem (Matsumoto [1]) for general families of smooth unimodal maps.

Theorem 8.5. Let hex) = Aj(x), JE C2(l, 1), J(O) =J(I) = 0, and f'(x)<O

Jor alt x Eint I. IJ h has a cycle oJ odd period k, then, Jor any Il > A, the map JIl

also possesses a cycle oJ period k. Moreover, if J is an S-unimodal map, then this

assertion holds Jor any k"* i, i = 1, 2, ....

3. Sequence of Period Doubling Bifurcations

Consider a farnily of continuous unimodal maps J" = Aj(x), where J: [0, 1] ~ [0, 1],

J(O) =J(I) = O,J(c) = 1 (c isthepointofextremumofthefunctionf), and AE [0,1]. Denote

A[n] = inf {A E [0,1]1 J" has a cycle of period n}.

Then, for any A< A[n], the map h has no cycles ofperiod n. Therefore, A[n] may

be called the value of the parameter for which a cycle of period n appears in the family

h. By the theorem on coexistence of cycles, the following statement is true:

Theorem 8.6. Let hE CJ(l, I). Then the inequality A [nd :-s; A[n2] holds Jor

any n1 and n2 such that n1 <l n2.

In this section, we consider families of maps of the indicated type without any further comments and explicitly mention only additional restrictions imposed on the maps.

If a map f belongs to the class cl, then cycles of different periods appear in the family A for different values ofthe parameter, i.e., A[n] "* A[m] if n"* m (Milnor

and Thurston [1]; see also Chapter 2). In particular, this is true for the families of maps with negative Schwarzian.

Theorem 8.7. Let A be afamily ofmaps with negative Schwarzian. Then

1.[1] < 1.[2] < 1.[4] < ... < 1.[5·2] < 1.[3·2] < ... < 1.[5] < 1.[3].

Theorem 8.7 implies that infinite sequences of period doubling bifurcations may appear in families of maps with negative Schwarzian.

At the same time, Theorem 8.8 demonstrates that infinite sequences of period doubling bifurcations are impossible for one-parameter families of unimodal maps whose Schwarzian is equal to zero, i.e., for maps composed of two pieces of linear-fractional functions.

Theorem 8.8. Let fA. be a one-parameter family of unimodal maps whose Schwarz

ian is equal to zero such that, for some 1.0 E A, the map f~ has cycles whose peri

ods are not equal to powers of two. Then there exists an integer number n ~ 0 such

that A[2n+j ] = A[2n] foral! j>O.

Proof. Suppose that the assertion of the theorem is not true. Then one can indicate

a sequence ni, i = 0,1, ... , such that A[2'b] < 1.[2":t] < ... < 1.[211;] < ... and there

exists A~ = lim I. [2 11i ] :::; 1.0. Hence, the central branch of the phase diagram of the

map f'A,.: I ~ I consists of infinitely many vertices formed by the cycles of intervals

A;m' m ~ 1, Pm = 2m. In view of the fact that unimodal maps whose Schwarzian is equal

to zero have no wandering intervals, any neighborhood U E (c ) = (c - e, c + e), e > 0, of

the point c contains infinitely many intervals from A;m.

Letf'A,.(x) beasymmetricfunction,i.e.,if f'A,.(xo) =f'A,.(xI)' then 1 Df'A,.(xo) 1 =

1 D f'A,. (Xl) I. Hence, either 1 D h,,.,(x) 1 > 1 for any XE 1\ {c} (the map f"", is expand

ing) or inf I DJ{2 (x) I > 1 because XE! =

where CPJ. is the nearest right fl~ -preimage of the point c.

If f'A,.(x) is an asymmetrie function, then there exists a neighborhood UE(c) such

that either

Seetion 3 Sequence of Period Doubling Bifurcations 209

for any XE (c - E, c) and x' E (c, C + E) such that f(xo) = x'. For the neighborhood

UE(c), thereexists m<oo suchthat A;m = [xm,Ym]C UE(c) and,henee,either

forall XE [xm,c) and X'E (c,Ym]' Theeasewhere !DJ{':(x)! = ! DJ{': (X')! for

f{m[A' has already been eonsidered above (f{m[A' is symmetrie). 00 Pm 00 Pm

In the other eases, the map f{:lA;m may take the shape displayed in Fig. 45.

Fig.45

Without loss of generality, we ean assume that xm is a fixed point of the map f{~':

I ~ I. Then all possibilities depieted in Fig. 45 ean be eharaeterized by the expression

whiehmaytakethevalues (-,-,+), (-,+,+), (+,-,+), (+,+,+), (-,-,-), (-,+,-), (+, -, -), and (+, +, -).

One ean easily show that the map f{m[A' is expanding if u{:' A;m} is equal to 00 Pm ~

(+,-,-) orto (+,+,-). In the remaining eases, we have the following pieture:

1. If Ulm,Ap* } = (-,-,+) or (+,-,+), then U/m+!,Ap* } = (+,-,-) or (+, f'l.,oo m /\'00 m+l

+, -) and, henee, the map f{:+! [A' is expanding. Pm+l

2. If {J/m, Ap' } = (-, +, +) or (+, +, +), then {hPm +1 , Ap* } = (-, -, -). In this /\'00 m /\'00 m+l

ease, we have u{m+l, u~Pm+l)} = (-, -, +), where u?m+l) c fA Ap' is a neighbor-00 p p 00 m+l

hood of the right preimage of the point c (this point is denoted by cp ). The map

J{m+! 11: A* can be taken as the original map. By the assumption, the middle part of 00 A"", Pm+l

the phase diagram of this map consists of infinitely many vertices formed by cycles of

intervals of periods Pm+i' i = 2, 3, .... We denote these intervals by ApO .' Then, for m+,

any i:?: 2 and some jE 1, Pm+i+I - 1, we have hj A p* = ApO . Thus, if ApO ~ /\'00 m+k m+k m+2

U~Pm+!), then u{m+2, ApO } = (+, -, +) or (+, +, -) and, hence, f!:m+2IAü is an p 00 m+2 00 Pm+2

expanding map. Otherwise, u{m+2, u~Pm+!)} = (+, +, -) and u{m+3, U:Pm +!)} = (-, ~ p ~ p

-, +) and we return to the initial state. Therefore, there exists k such that ApO ~ m+k

u(Pm+!) (fPm+k AÜ } = (+, -, +) and, consequently, hPm+k IAü is an expanding cl' A~' Pm+k ""~ Pm+l

map.

3. If {f'pm, Ap* } = (-, +, -) or (-, -, -), then {hPm +!, Ap* } = (-, -, +), or 1\.00 m 1\,00 m+l

(+, +, +), or (+, -, +), or (-, +, +) and, hence, the case under consideration reduces to the already considered cases.

Both in the case where f~ (x) is symmetrie and in the case where it is asymmetrie,

we have shown that f!:~n' IA;rn' is expanding for some m' < 00. Thus, there exists m' <

m" < 00 sueh that

inf 1 DjPm" (x) 1 :?: .J2 xEAl'm"

and, therefore, A;m" contains no periodic intervals whose periods are greater than Pm'"

Hence, the number of vertices in the phase diagram of the map f~ : I ~ I is finite.

Thus, in a one-parameter family of piecewise smooth unimodal maps that are not

cl-smooth, the sequence of period doubling bifurcations ean be finite at most at one point.

Numerieal results demonstrate that the dynarnies of rnaps in the family JA = Af be

comes more eomplicated as the parameter A increases. Thus, it follows frorn Theorem

8.4 that, for the family x ~ Ax(l- x), the topological entropy and kneading invariant

are monotone funetions of the parameter. By the same theorem, if a family fA(x) =

Af(x) of eonvex maps with negative Sehwarzian is charaeterized by the property that,

for some Ao E A, the map f'Ao has a eycle of period m * 2k , k E N, then, for any A:?:

AO' the map JA also possesses a eycle of period m, i.e., bifurcations of eycles exhibit the property of monotonieity.

Section 3 Sequence of Period Doubling Bifurcations 211

It was conjectured that the families of unimodal maps with negative Schwarzian must be characterized by the property of monotonicity of bifurcations of cycles and by the monotone dependence of the topological entropy and the kneading invariant on the par

ameter. However, it was shown that if f is not a convex function, then, for the family

ft... = AI, it may happen that h (f,) and V (f,) are nonmonotone functions of A and no monotonicity of bifurcations of cycles is observed.

Theorem 8.9. There exists a unimodal map f with negative Schwarzian such that,

for the family ft... = AI, no monotonicity of bifurcations of cycles is observed and h(ftJ

and V (h) are nonmonotone functions of A.

Proof Let us construct a unimodal map f with negative Schwarzian such that the family fA = Af has the following properties: As the parameter A increases from ° to

some Ao > 0, one observes the appearance of cycles of all periods. As A increases fur

ther, aB cycles (except fixed points) first disappear (for some AI > 0) and then appear

again. The required map f is given by the equality

fex) =

where

c

a = 1 g~(a)l,

j g!!(x) = /l2x (l- x)(l- /lX(1- x», x E [0, xo], xo

g(x) = f (ax2 + bx + c)-2 dx + gll(xo), x> xo, x

/l E (3,4),

b = ~ a-3/ 2

2

1[ 3A2 -IJ -3/2 a=4 Y +2:l-'a a ,

Xo ( 3 ß2 -I) -3/2 - y+- a a 22'

1 xo = --0

2 ' ° < 0 < 2

We choose xo < 1/2 close to 1/2 and select constants a, b, and c such that f: IR + ~

IR + is a unimodal C3 -map with negative Schwarzian (see Fig. 46). Note that the last assertion can be readily verified by using the foBowing criterion of negativity for Schwarz-

ians: Sg (x) < ° in a given interval if and only if the function 1 g' (x) 1- 1/2 is concave in this interval.

For /l close to 4, the map x ~ /l2 x (l - x)( 1 - /lx (l - x» has cycles of all periods.

Moreover, for any /l ofthis sort, one can find sufficiently small 00 = 0o(/l) such that

(8.1)

x

Fig.46

We fix Il and Ö for whieh the map f: IR + ~ IR + has eycles of all periods and eondition (8.1) is satisfied. In this ease, eycles of all periods appear in the one-parameter family N' as A ehanges from 0 to 1. Further, by virtue of (8.1), for

the map Af possesses an attraeting fixed point other than the fixed point O. Note that SiAl< 0 and the map fiCI is unimodal. Therefore, by virtue of Theorem 5.3, this map

may have only fixed points. It is clear that, for the family of maps eonstrueted above, we observe not only the

violation of monotonicity of bifureations of eycles but also the nonmonotone dependenee of the entropy and kneading invariant on the parameter. Similarly, one ean construet a family of unimodal maps with negative Sehwarzian for whieh the topologie al entropy and kneading invariant regarded as funetions of the parameter may have arbitrarily many intervals of monotonicity.

Note that the family of maps eonstrueted above is defined for x E IR + and A E (0, 00). Clearly, it is possible to eonstruet a family of maps with the properties indieated in Theorem 8.9 but defined for XE [0, 1] and A E (0, 1).

Seetion 3 Sequence 01 Period Doubling Bifurcations 213

According to Theorem 8.5, the property of monotonicity of bifurcations for families of maps with negative Schwarzian is guaranteed by the convexity of maps from this families. The following assertion makes the result of Theorem 8.5 more precise:

Theorem 8.10. Let h.. = ÄI be a lamily 01 unimodal convex maps with negative

Schwarzian. Then

lim i --7 00

Ä[(2i-l)2n] - Ä[(2i+l)2n] Ä[(2i+l)2n] Ä[(2i+3)2n] = Yn(f) > 1.

Fig.47

Our proof of Theorem 8.10 is based on a hypothesis formulated somewhat later. First, we consider the case n = O. In the following lemma, we use the concept of cy

des of minimal type (or, simply, minimal cydes) introduced in Chapter 3:

Lemma 8.1. There exists a monotonically decreasing sequence {Äi, i ;:=: I} 01 va

lues 01 the parameter Ä such that the point c belongs to the minimal cycle 01 the map iN 01 period 2i + 1.

Proof. The arrangement of points of minimal cydes of periods 2i + 1, i;:=: 1, on the realline is known. In Fig. 47, we display the arrangement of points of these cydes

for i = 1, 2, 3 and the graph of the limit function 1'Ao (x), where Äo = lim Äi. 1---7~

In proving the lemma, we assurne that Ä > Ä[2]. Consider a point c+(Ä) E (c, 1)

such that A(c+(Ä)) = c. The restriction ofthe map 11 to the interval [c, c+(Ä)] is a

homeomorphism which covers [c, c+(Ä)]. Therefore, the map (h: [c, c+(Ä)] ~ [c,

c+(Ä)] such that

is weIl defined.

( 2)-1 ~~(x) = I~ (x)

Since h .. is a map with negative Schwarzian, <PA (x) is a monotone strictly increas

ing function and the fixed point x* = x* (A) of the map JA lying in the interval (c,

C+(A)) is a globally stable fixed point of <PA'

Considerfunctions C k( A), k ~ 0, defined on the interval [A [2], 1] by the equality

Ck(A) = <p~(c). Due to the monotonicity of <PA' we have COCA) < cI (A) < C2(A) < ... <Ck(A)<X*(A) for AE [A[2], 1]. Moreover, ck(A)~X*(A) as k~oo. Now let

Z(A) = Ji.(c). Then z(A[2]) > x*(A[2]) and z(1) = ° < c. Since the functions Z (A), x* (A), and C k( A), k ~ 0, are continuous, there exists a decreasing sequence AI>

A2 > A3 > ... such that Z(Ai) = ci_1 (Ai)' i = 1,2,3, ... , which is equivalent to the assertion of the lemma.

Denote lim Ai by AO' The following lemma establishes the geometrical rate of 1-7~

convergence of the sequence {Ai}'

Lemma 8.2.

A' ! - A· ( d ( )2 lim 1- I = - ft.. X*(AO)) . i-7~ Ai - Ai+! dx 0

Proof. By using the mean value theorem, we obtain

(8.2)

where f..l = AO + 8(Ai - AO)' 8 E (0,1). On the other hand,

where 81 E (0,1).

Denote Ci(A) -X*(A) by L\CA). Then

for large i. By using relations (8.2) and (8.3) and assurning that

we arrive at the required result. Indeed,

Section 3 Sequence of Period Doubling Bifurcations 215

D w c 8 All 1

Fig.48

The graph of the function f~ (x) is depicted in Fig. 48 together with the graph of the

function f~o (x).

Since the function f~ (x) is convex, one can show that the function f~ (x) is con

vex in the interval [~, 1.0 ]. Indeed,

Since any map with negative Schwarzian has at most one inflection point in each interval of monotonicity , it suffices to prove that f{~ (x) is negative at the points x = ~ and x =

1.0 ' For x =~, we can write

because the convexity of f~ implies the inequality Ro (~) < -1. Similarly, for x = 1.0 ,

To prove the theorem for n = 1, one must consider the map /1 in the interval [~, A], where ~ is the fixed point of the map JA other than O. In this case, it is necessary

to prove the inequality

d 6 d~ JA (e) = 0 for A = Al' where A] = lim A[2(2i + 1)].

"" I~OO

This problem is more complicated than the proof of the inequality from the hypothesis

considered above because, in this case, the dependence of the family FA = Jllr S, Al on

the parameter is not linear although the maps from this family are convex and their Schwarzians are negative. Thus, Theorem 8.10 remains true provided that the inequality

d 3·2" dA A (e) t:- 0

holds for proper values of the parameter An' n = I, 2, ....

4. Rate of Period Doubling Bifurcations

As already known, there exists an ordering of the set of natural numbers

1 <l 2 <l4 <l 8 <l ... <l 2k . 7 <l 2k • 5 <l 2k • 3 <l

<l 2· 7 <l 2· 5 <l 2· 3 <l ... <l7 <l 5 <l3

such that if a continuous map J: I ~ I (or J: IR ~ IR) has a cycle of period m, then it also has a cycle of period n for any n <l m. Hence, for any family JA (x) of continuous

maps of the realline into itself, the order of appearance of cycles is specified by the indicated ordering of natural numbers.

There are many families of maps for which one can observe not only single bifurcations but also infinite sequences ofbifurcations of cycles (as the parameter changes within a certain finite interval). Among maps of this sort, one can mention convex uni modal maps with negative Schwarzian and, in particular, quadratic maps. By analyzing the behavior of these maps, one can establish some "universal" properties of sequences of bifurcations of cycles.

In studying the family x ~ Ax(l-x), we observe an infinite sequence of period doubling bifurcations as the parameter A increases from A = 3 to A = 3.57 (as a result

Seetion 4 Rate of Period Doubling Bifurcations 217

of this sequence of bifurcations, the map has cyc1es of periods 2n, n = 0, I, 2, ... , for

A> 3.57). Note that it follows from the theorem on coexistence of periods of cyc1es that, in any family of smooth maps, the appearance of infinitely many cyc1es is a result of period doubling. Moreover, families of maps are characterized by the universal order of the appearance of cyc1es of new periods and, in addition, for a broad c1ass of families, the sequence of bifurcation values of the parameter converges with certain universal rate (for all families from a given c1ass, this rate is the same).

To clarify these observations, we consider the following family of quadratic maps:

gfl (x) = 1 - J.1x2, X E [-1, 1], J.1 E [0, 2]. The first period doubling bifurcation occurs at

J.1 = J.1o = 0.75: The fixed point ß! (0.75) = 2/3 generates a cyc1e of period two. The

subsequent bifurcation values corresponding to the appearance of cyc1es of periods 2n,

n = 2, 3, 4, ... , are equal to J.1! = 1.25, J.12 = 1.3681 ... , J.13 = 1.3940 ... , ... , respecti-

vely. As n ~ 00, the sequence J.1n approaches the value J.1~ = 1.40155 ... for which the

map ffl=: [-1, 1] ~ [-1, 1] has cyc1es of all periods equal to powers of two and has no

cycles of other periods. The ratio

° = J.1n - J.1n-l n

J.1 n+! - J.1n

takes values 0, = 4.23, 02 = 4.55, 03 = 4.65, 04 = 4.664, Os = 4.668, 06 = 4.669, '" .

As in the case of the family ft. .. (x) = Ax(l - x) (see Chapter 1), the limit of the sequence

on as n ~ 00 is equal to Ö = 4.6692 ....

The value of the quantity a, which characterizes the sizes of appearing cyc1es, also

coincides with the corresponding value for the family h.., i.e., if ß;n is the first point of

the cycle of period 2n (which appears for J.1 > J.1J to the right of x = 0 and

then

a = n

" 2"-1 ( , ) ß2n = gfl ß2n ,

ß2n - ß2n

ß2n+I - ßzn+l ~ a = 2.502 ... as n ~ 00.

The phenomenon of universality means that the sequences On and an determined

for different one-parameter families of maps (not only for quadratic maps but also for the

families A sin x, AX (1 - x)2, etc.) converge, for all these families, to the same values ° and a, respectively. This phenomenon was discovered and investigated by Feigenbaum in 1978 (see Feigenbaum [1, 2]); almost simultaneously, similar results were obtained by Grosmann and Thomae [1].

In order to explain the phenomenon of universality, we consider the set G formed by

unimodal maps \!f E Cl (I, 1), where 1= [-1, 1], such that \!f (0) = 1 and \!f (1) < 0.

For any 'Jf E G, we define

where a = 'Jf( 1). The nonlinear operator T: G ~ G is called the transfonnation of doubling.

Let Gm C G be the set of analytic functions from G. We want to determine fixed

points of the map T: Gm ~ Gm, i.e., the solutions of the functional equation T'Jf = 'Jf

in the set of analytic functions Gm. The degree of degeneracy of the critical point of the

function 'Jf E cm is invariant under the action of T. Therefore, the form of the solutions of the indicated functional equation depends on the degree of degeneracy. We require that the critical point of any function which is a fixed point of the operator T must be nondegenerate. By Theorem 8.14, this function must be even.

Proposition 8.4. There exists an even analytic function

'Jfo(x) = 1 - 1.52763 ... ·x2 + 0.104815 ... ·x4 - 0.0267057 ... ·x6 + ... ,

which is a fixed point of the operator of doubling T 'Jf(x) = 'Jf2(ax)/ a, where a = a(\jIo) = \jIo(1) = -lla = -0.3995 ....

Let Ji denote the Banach space of functions \jI(z) analytic and bounded in a certain

complex neighborhood of the interval land real-valued on the real axis. Let Jfo be the

subspace of Ji formed by the functions satisfying the conditions 'Jf(0) = 1, \jI' (0) = 0, and \jI"(O) "* O.

Proposition 8.5. There exists a neighborhood U ('Jf 0) of the point \jI a in Jia

such that TE C=(U(\jIo)' Jio). The operator D T(\jIo) is hyperbolic and possesses

a one-dimensional unstable subspace and a stable subspace of codimensionality one. The eigenvalue of DT(\jIo) in the unstable subspace is equal to Ö = 4.6692 ....

Let La C G be the "surface" formed by the maps \jI whose derivative at a fixed

point Xo = xo(\jI) E [0, 1] is equal to -1 and S'Jf{xo) < O.

Proposition 8.6. An unstable local "manifold" defined in a neighborhood of 'Jfo

can be extended to aglobai unstable "manifold" WU(\jIa) which transversally crosses

the surface La; the "manifold" WU(\jIo) consists ofmaps with negative Schwarzian.

At present, all known proofs of these propositions are computer-assisted (see Wul, Sinai, and Khanin [1]).

Section 4 Rate oJ Period Doubling Bifurcations 219

Fig.49

By using Propositions 8.4-8.6, one can explain the phenomenon of universality,

which is known as Feigenbaum universality. In the neighborhood U('Vo)' we have the

following picture (Fig. 49): If a one-parameter family ft. .. transversally crosses the stable

manifold Ws ('V 0)' then it transversally crosses the surfaces rnLo for sufficiently large

n. The points JA" of interseetion of ft. .. with T-nLo correspond to period doubling bi

furcations of cyc1es of periods 2n and the point JA.. corresponds to the accumulation

point of the set of bifurcation values, i.e., 'A~ = lim 'An" For large n, the distance be-n---7~

tween T-(n+l)Lo and WS('Vo) isabout Ö timeslessthanthedistancebetween T-nLo

and WS ('V 0)' Hence, the bifurcation values of the parameter 'A of the family h. satisfy the relation

where Co depends on the family of maps.

Let ßn be the point of the 2n-periodic cyc1e of the map JA" whose distance from the

point of extremum is minimal and let ßo be the fixed point of the map TnJt." from [0, 1]. Then we have

n

ßn = ßo rr ui' i;1


where CL.i is a renormalization constant for Tif'}". and the sequeoce {CL.J converges to

the value - I/CL. = - 0.3995 ....

Sullivan proved that the stable manifold WS ('" 0) contains a fairly broad class of functions (see, e.g., van Strien [2]):

Theorem 8.11. Let f be a unimodal map of I into itseif such thatf(-I) = f(I) = - l. Assume that f satisfies the following conditions:

(i) fis conjugate to "'0 (i.e., fis a map ofthe type 2=);

(ii) for the complex extension F of the map f, one can indicate a disk in the complex plane such that

(a) it contains the interval I,

(b) it contains the unique critical point ofthe map F, and

(c) under the map F, its boundary is mapped into the outside ofthe disko

Then f belongs to the stable manifold WS ('" 0)'

By using the properties of the operator T, one can construct the unstable manifold

Wu(", 0) numerically. Thus, a construction of this sort was suggested by Wul, Sinai, and Khanin [1].

As indicated above, Feigenbaum universality is observed for a broad class of oneparameter families of smooth unimodal maps. It is thus interesting to clarify the condi

tions under which an individual family fA. exhibits the phenomenon of universality.

First, for a given family ft.., it is desirable to establish simple conditions guaranteeing the monotonicity of the sequence of bifurcations similar to that observed for the family ofquadratic maps Ax(l-x). As follows from Theorems 8.5 and 8.7, forthe family Af, the required property is apparently guaranteed by the analyticity, convexity, and negativity of the Schwarzian of the map f Second, one can apply Theorem 8.11 to require

that the map A=f be similar to a quadratic map in a sense of Douady and Hubbard (i.e., that A=f satisfy the conditions ofTheorem 8.11).

Let 1). be the space of families with smooth dependence on the parameter A. We

now study the phenomenon of Feigenbaum universality for families of maps from the

space 1).. The following description of the doubling operator in the space of analytic

functions in the vicinity of its fixed point "'0 seems to be quite reasonable:

The stable manifold WS("'o) splits the neighborhood U("'o) into two parts (Fig. 49).

We define the fundamental domain V of the operator T-' as the domain bounded by

the surfaces L = {f1j2(0) = O} and r' L (Fig.50). The fundamental domain V' of

Section 4 Rate of Period Doubling Bifurcations 221

theoperator r 1 isboundedby L' = {flf 2 (O)=-I} and T-1L' (Fig.51). The

stable manifold WS(\jIül separates the maps with simple structure (i.e., with finite sets of

nonwandering points and topological entropy equal to zero) from the maps with complicated structure (with infinite sets of nonwandering points and positive topological entropy).

a b a b

Fig.50 Fig.51

If a family gt.. E '.F,." transversally crosses the manifold WS ('I' 0) then, for all suffici

ently large n, it transversally crosses the surfaces T-nL and, hence, we observe the

phenomenon of Feigenbaum universality. Moreover, for large n, the family gt.. trans-

versally crosses the surfaces T-nL'. The values of the parameter A~ corresponding to

the intersections of gt.. with T-nL' are points of bifurcations of creation of cycles of in

tervals of period 2n+ 1, and the accumulation point of the set of bifurcation values A= = lim A~ corresponds to the intersection of the family gt.. with WS ('I' 0)' Since the dis-

n-'>=

tance between T-nL' and WS (\jIo) is proportional to Ö-n, we have A~ - A= - cÖ-n.

The value A~ corresponds to the appearance of a trajectory homoclinic to the mini

mal cycle of period 2n in the farnily gt.. (i.e., to a cycle from the block N.). This means

that homoclinic trajectories appear with the same rate Ö as the corresponding cycles (but in the inverse order).

Consider another universal property of families of maps, which is a direct consequence of Feigenbaum universality. It characterizes bifurcations (creation) of cycles

whose periods are not powers of two. By Theorem 8.10, for families Af of unimodal convex maps with negative Schwarzian, each block Nk is characterized by a certain

asymptotic rate Y k = Yk(f) of creation of cycles of periods (2n + 1) 2k as n -t 00.

However, this theorem does not imply that the sequence {y k} converges as k -t 00. If

we assurne that Ai transversally crosses WS ('I' 0) for A = A= and use the fact that Af E

T- k V' for A = A[(2n + 1) 2k ], n ~ 1, then we arrive at the following conclusion:


For the family ft" = 'Af, we have Y k ~ Y as k ~ "", i.e., the rate of the process of

creation of cycles of periods (2n + 1) 2k (in the blocks Nk ) is asymptoticaBy constant

for large k. Numerical experiments corroborate these conclusions and give approximate values ofthe asymptotic rate. Thus, according to Kolyada and Sivak [1] and Geisel and

Nierwetberg [1], y = 2.9480 .... It should also be noted that the rate of creation of cy

cles of periods (2n + 1) 2k, where n is fixed and k ~ "" is equal to Ö.

Proposition 8.7. Let ft" = 'Af, where f is an analytie unimodal map with negative

Sehwarzian. f(O) = f(l) = 0, fee) = 1, 'A E [0,1], and j"(e) * O. Then

(i) the re exists

where Ö = 4.669201 '" ;

( ii) the re exists

where y = 2.94805 ... ;

(iii) there exists

where Ö = 4.669201 ... and

'A' [2n] = lim 'A[ (2m + 1)2n- 1], n ~ 1; m---t~

(iv) the family fA has no other bifureation values of the parameter in the interval

[0, 'A~), where

'A~ = lim 'A[ 2n ] n-;~

If a map f belongs to WS ('V 0)' then it has periodic points of aB periods 2n, n = 0,

1, 2, ... , and has no periodic points of other periods. The set of nonwandering points

Section 5 Universal Properties ofOne-Parameter Families 223

NW (f) is equal to Per (f) U K (f), where K (f) is a closed uncountable minimal set of

the map f (i.e., a 2~-type quasiattractor) (see Misiurewicz [2]).

Properties of the maps from WS (\jf 0) are characterized by the following theorem

proved by Paluba:

Theorem 8.12 (Paluba [1]). Assume that fand g belong to the stable "man i

fold" WS (\jfo)' Then the sets K(f) and K(g) are topologically conjugate and the

conjugating homeomorphism h belongs to CLip in a sense that, for any x E K (f) and y E [-1, 1], the re exists a Lipschitz constant 'Y 0 such that

Ih(x)-h(y)1 ~ 'Yolx-yl.

The fact that mes K (f) = 0 for any fE WS (\jf 0) is an important consequence of

Theorem 8.12 (because it is clear that mes K(\jfo) = 0).

5. Universal Properties of One-Parameter Families

Let ft. .. be a one-parameter farnily of smooth unimodal maps. For A = A *, we assume

that the central branch of the phase diagram of the map h! consists of infinitely many

vertices (see Chapter 5). Then the map ft. .. * possesses an infinite sequence of periodic

intervals 11 :::> 12 :::> ... :::> Im :::> ... , which contain the point of extremum. In this case,

for any m> 1, Pm-I is a divisor of Pm and the set

~ Pm

K = n n f~* (Im) m=l i=1

is a quasiattractor. The case where Pm + 1 / Pm = 2 for any m:2: 1 was studied in the pre

vious section. In particular, it was mentioned that lim I Im I/I Im+ d = 2.502 ... for m--->~

ft...* E WS (\jf 0)' where I Im I is the length of the interval Im'

It is natural to expect that universal properties are exhibited by the farnily h. not only

for Pm+l/pm =2 but also for Pm+l/Pm=k>2.

As in the previous section, we consider the set GO> of analytic unimodal maps f:

[-1, 1] ~ [-1, 1] such that f(O) = l. The set GO> can be decomposed into infinitely

many mutually disioint sub sets as folIows: GO> = U G(2i) where each G(2i) is form-J i~l '

ed by the maps satisfying the conditions

224 One-Parameter Families 01 Unimodal Maps

d r I (0) = 0 for r = 1, 2, ... , 2i - 1 dxr

and d 2il -2· (O)::F- O. dx'

An operator Tk, k ~ 2, is introduced by the formula

(TJ)(x) = -k/(UX), a =/(0), XE [-1,1].

Chapter 8

If, for a map IE Gm, we have m*(f) < 00, then, for some j ;::: 1, the operator Tk

is not weIl defined for the function (T/ I) (x) (T/ denotes the jth iteration of Tk ). At

the same time, one can easily give examples of maps from Gm for which aIl iterations of the operator Tk are well defined.

Let

for some subsequence m1 < m2 < m3 < ... }.

As above, it is not difficult to show that T/ I E Gm for all j;::: 0 if and only if I E

'E(Tk )·

We say that maps land g from the class Gm are of the same type if m * (f) = m*(g) and the types ofthe cycles of intervals A;m (f) and A;m (g) in the phase dia

grams of the maps land g coincide for all m (all relevant definitions can be found in Chapter 5). By using the concept of maps of the same type, we can decompose the set

'E( Tk ) into classes of maps of the same type. The class of maps from 'E( Tk ) that con

sists of maps of the same type as I E 'EJ,Tk ) is denoted by 'Ef(Tk ) (or simply 'Ef if this

does not lead to ambiguity).

Lemma 8.3. 11 k = 2 or k = 3, then 'E f(Tk ) = 'E(Tk ) lor any I E 'E(Tk )· 11

k = 4, then 'E( Tk ) splits into uncountably many classes 01 maps 01 the same type.

Proof. If apermutation 1t = (to' tl' ... , tn- 1 ) determines the type of a cycle of in

tervals of a unimodal map, then this permutation 1t is cyclic and the map 1t: {O, 1, ... , n - I} ~ {O, 1, ... , n - I} with 1t (i) = t i is unimodal. Permutations of this sort are

called U-permutations. If 1t is aU-permutation, then one can easily construct an example of a unimodal map with a cycle of intervals of the type 1t.

Suppose that a uni modal map I has a cycle of intervals A. If c is the point of maximum (minimum) ofthe map f, then the map 1t: {O, 1, ... , n - I} ~ {O, 1, ... , n - I}, defined by the permutation 1t (A) also has the maximum (minimum). The statement of

Section 5 Universal Properties of One-Parameter Families 225

the lemma for k = 2 and k = 3 follows from the fact that the only U-permutations of

lengths 2 and 3 with maximum are 1t2 = (1,0) and 1t3 = (1,2,0), respectively.

For k ~ 4, the situation is absolutely different. Thus, one can always find two dif

ferent U-permutations oflength k, e.g., 1tk = (1, 2, ... , k - 1, 0) and

1tk = (k/2,k-l,k-2, ... ,k/2+1,k/2-1,k/2-2, ... ,1,0)

for even k or

1tk = «k+ 1)/2, k-l, k- 2, ... , (k+ 1)/2 + 1, (k+ 1)/2-1, (k+ 1)/2-2, ... ,1,0)

for odd k. Note that 1tk and 1tk have no nontrivial divisors (for definitions, see Chapter 7).

It is dear that maps f, g E 'E( T k) have the same type if and onl y if

for all i ~ O. For any i ~ 0, the permutation 1t(A;2 (T~ f)) can be equal to any U-per

mutation of length k. In particular, it can be equal to ltk or ltk' Thus, for any se

quence {1t(O),1t(I), lt(2\ ... }, where lt(i) is equal to 1tk or ltk, i = 0, 1,2, ... , one can

find a map fE 'E( Tk) such that 1t (A;2 (T~ f)) = 1t(i).

Hence, the cardinality of the set of dasses of maps of the same type lying in 'E( Tk )

is not less than the cardinality of continuum because the set of infinite sequences over a two-letter alphabet has the cardinality of continuum.

For any map fE 'E(Tk), there are three possibilities, namely,

(b) Tlf'1. 'Ef for i = 1,2,3, ... ,n-l and TJ:fE 'Ef ;

In each of these cases, for the dass 'EI' we can, respectively, write


In case (a), it is natural to say that the class 'EI is a fixed class of the operator Tk. In

case (b), we say that this class is periodic with period n and, in case (c), we say that it is aperiodic.

Lemma 8.4. For k 2:: 4, the operator Tk has periodic classes of all periods.

Proof. Assume that a map fE 'E (Tk) is such that 1t (A;2 (TI f)) = 1t1e for i = nj

and 1t (A;2 (TI f)) = 1tk for i::F- nj, n > 0, j = 0, 1,2, ... , where 1t1e and 1tk are the

permutations defined in the proof of Lemma 8.3. By virtue of Theorem 2.6, this map

exists. Hence, 'EI is a periodic class of the operator Tk with period n.

Lemma 8.5. For any k 2:: 2, the number of fixed classes of the operator T k i s

finite.

The proof follows from the fact that, for any k 2:: 2, there are finitely many different

permutations of length k (including finitely many unimodal permutations of length k). The following assertion is an immediate consequence of the definition of periodic

classes of the operator Tk:

Lemma 8.6. If 'EI is an n-periodic class ofthe operator T k, then 'EI' Tk('E/ ),

... , T;:-l ('EI) are fixed classes of the operator Tkn-

Corollary 8.1. For any n 2:: 1, the operator T k has finitely many periodic classes

ofperiod n.

The theory of Feigenbaum universality is based, in particular, on the assumption that

the operator T2 : 'E(T2 ) ~ 'E(T2 ) possesses a unique fixed point f* E 'E(T2 ) globally

stable in the space 'E(T2 ) n d 2 ) and such that (f*)"(0) ::F- O. There are several known

methods for proving the existence of the fixed point of the operator T2 with the indicated properties. We also note that there are papers devoted to the investigation of the

spectrum of the operator D T2 (f*).

In what follows, unless otherwise stated, we always assume that fE d 2 ) and Tk :

d 2 ) ~ d 2 ) (in other words, we assume that f"(O) =#= 0 for all maps fE GOJ under

consideration). It is not difficult to show that, for any fE d 2 ), we have either Td E

d 2 ) or Tkf ~ GOJ• In order not to introduce new notation for the intersections of the in

dicated classes of maps of the same type with the space G( 2 ), we use the same notation both for these objects and for the original classes. It is worth noting that the reasoning presented below is also applicable to the investigation of the operator Tk in the spaces

d 2i ), i = 2, 3, ....

Section 5 Universal Properties of One-Parameter Families 227

Let k ~ 2 and let 'EI be a fixed dass of the operator Tk. Suppose that the operator

Tk has the following properties:

Property 1. The operator Tk possesses afixed point f* E 'EI and this point is

a glohally attracting fixed point of the operator Tk in 'EI = 'Er

Property 2. The operator D Tk(f*) has only one simple real eigenvalue 0 =

o(Tk,f*) which is greater than one; the other eigenvalues belong to the interior ofthe

unit disko

Consider the following "surfaces":

and

LI = {JE C(2)I!(-I) = f(l) = -I}.

Property 3. The unstahle manifold WU(Tk,f*) of the operator Tk which crosses

1* and corresponds to the eigenvalue 0 (Tk,f*) has dimensionality one (i.e., it is a

one-parameter family of maps from C(2)). This family transversally crosses the sur

faces 'Ej*(Tk) (i.e., the stahle "manifold" of Tk ), Lo' and LI'

Let F)..; A E [0, 1], be a farnily of maps from C(2) which is sufficiently dose to the

farnily WU(Tk,f*) and transversally crosses the surface 'Et*(Tk) as A = A=. Then, at

least for sufficiently large n, there exists a unique value An dose to A= and such that

FAn E Tk-n(LO) and a unique value ~n such that FIn E Ik-np:'l)' Without loss of

generality, we can assume that An < ~n' In this case, A= E (An, ~n)' If A E (An, ~n)' then for the phase diagram ofthe map FA; we have m * (F),) ~ n. Hence, the length of

the central branch of the phase diagram of maps from the family F'}., increases with n

and the values An and ~n correspond to bifurcations in the phase diagram.

Properties 1-3 of the operator Tk immediately imply that

This relation enables us to estimate the measure of the set of values of the parameter

for which the map F'}., possesses a cyde of intervals of period k n. Furthermore, the

constant a (Tk,f*) = (f* l (0) deterrnines the rate of decrease in the sizes of cydes of


intervals of periods k n as n increases. This rate is asymptotically equal to (r I .

It is convenient to represent the farnily WU(Tk,f*) corresponding to the eigenvalue

Ö(Tk,f*) of the operator Tk in the form WA,(x) = f*(x) + A\jI(X), where \jI(O) = 0

and \jI(l) = l. The following assertion indicates that the constants a (Tk,f*) and Ö (Tk,f*) sub

stantially depend on the behavior of the trajectory of the point 0 under the action of the map f* (i.e., under the action of the relevant fixed point of the operator Tk ; see Kolya

da and Sivak [2]):

Lemma 8.7. Thefollowing equalities hold:

where

LS = \jI((f*)s(a)) krt {[ (f*)'((f*l-i(a)] [(f*)'(U*)k-i(O)t} ! = I

for s=O, 1, ... ,k-2 and Lk_1 = \jI((f*)k-i(a)).

Proof. We have Td* = f*. By differentiating this identity two times, we obtain

(Td*)" (0) = (f*)" (0).

By using the chain rule for differentiation of composite functions and the assumption that (f*)" (0) *- 0, we immediately arrive at the first equality of the lemma.

To required relation for Ö (Tk,f*) can be established by using the fact that

for A = A=. Therefore,

Seetion 5 Universal Properties of One-Parameter Families 229

for some function 'P, and this equality can be regarded as a functional equation für finding the map \jI(x) provided that the function f* and the cünstant o(Tk,f*) are known.

Since 0 (Tk,f*) is a number, i.e., does not depend on x, the right-hand side of the indi

cated equality also does not depend on x. By setting x = 1 and using the representation for a (Tk,f*), we arrive at the second equality of Lemma 8.7.

Note that if a (Tk,f*) is sufficiently small (e.g., for large k), then the cünstants Ls

in the representation of o( Tk,f*) are dose to the values 'I'((f*}' (0».

Now assume that the fixed points f/ of a sequence of operators Tk , i ~ 1, converge I

to a map f:;' from the space d 2 ). Then, for large i, the constants a(Tk., fi*) and I

o (Tk , fi*) substantially depend on the values of the map f:;' at the points of the trajec-I

tory of the point O. As an illustration of this assertion, we consider the case where ki = i, i = 2, 3, ... ,

and fi* is a fixed point of the operator Ti for which the permutation 1t(A;z (fi*» is

equal to 1tk from Lemma 8.3. Fixed points of this sort are called minimal. (For odd

i > 1, cydes with permutations of the indicated type and the limiting function f:;' are depicted in Fig. 47.)

Theorem 8.13. Let fi* be minimal fixed points of the operators Ti' i = 2, 3, ... ,

respectively. Assume that

fi* ~ f:;' as i ~ 00 (8.4)

in the metric of CO. Then

(a)

where x* is the fixed point of the map f:;', and

(b)

Proof. The required equalities follow from the structure of trajectories of the point

o of the minimal fixed points fi* and from the representations of a and 0 in Lemma

8.7. Indeed, for large i, the right-hand sides of equalities for a(Ti , fi*) differ by the

number of multipliers dose to Yi = U:,,)' (x7)' where xi is the fixed point of the map

230 One-Parameter Families 01 Unimodal Maps Chapter 8

jj*. Hence, by using Lemma 8.7, we arrive at the equalities ofTheorem 8.13. A sirnilar result for another sequence of fixed points of the operators Ti was estab

lished by Eckmann, Epstein, and Wittwer [1]. They considered the sequence of maximal

fixed points ];* of the operators Ti (a fixed point ];* of the operator Ti is called maxi

mal if 1t(A;2' (];*)) = 1ti, where 1ti is defined in Lemma 8.3 and equal to (1,2,3, ... ,

i-I, 0)). Moreover, Eckmann, Epstein, and Wittwer [1] proved that, in this case, the

map 1 - 2x2 , XE [-1, 1], is the lirniting map ];*. By using their arguments, one can

show that condition (8.4) in Theorem 8.13 is satisfied and I:' = 1 - A=X2, where A=

is the value of the parameter A for which F ,Jx) = 1 - A x2 satisfies the conditions

F,Jl) < 0 and Ff(l) = x*, where x* is a fixed point ofthe map FA lying to the right

of the origin.

The following assertion seems to be true: If a sequence {1;*}:2 of fixed points of

the operators Ti has a limiting point I:' in the metric of Co, then I:' = 1 - AX2 for

some A E (0,2). This enables us to conclude that

for large i if we use parametrization introduced above. The equalities ofTheorem 8.13 are corroborated by the results of numerical calcula

tion of the relevant constants for farnilies of quadratic maps. Below, we present the cor-

responding results for the constants a, (Ti' 1;*) and Ö(Ti, 1;*). The numerical value of

the constant y is approximately equal to 1.71.

a,-I 8 a,-I Ö

3 9.27· 10° 5.52.101 4 6.26.10° 2.18.101

5 2.01. 101 2.55.102 6 2.09.101 2.18. 102

7 4.91.101 1.44.103 8 6.63.101 2.30· 103

9 1.29· 102 9.60.103 10 1.97· 102 2.10.104

11 3.52.102 7.00.104 12 5.68.102 1.77 . 105

13 9.78· 102 5.35. 105 14 1.61· 103 1.44.106

15 2.74· 103 4.18.106 16 4.57.103 1.15.107

17 7.70. 103 3.30· 107 18 1.28.104 9.24.107

19 2.21.104 2.56· 108

Seetion 5 Universal Properties of One-Parameter Families 231

For large i, computation becomes much more complicated because the sizes of cy

cles of intervals A;z (f/) rapidly decrease (the rate of this process is equal to y) and

the constants 0 (Ti' f/) rapidly increase as i increases. Computations were carried out

for the family of quadratic maps Ax(l - x), XE [0, 1], A E [0, 4], which is equivalent

tothefarnily I-h2, XE [0,1], AE [0,2].

y

Fig.52

Let us now study the problem of existence of solutions of the equation Td = f in the space of unimodal maps and investigate some properties of these solutions.

First, we describe a method for the construction of solutions of the functional equa

tion a-[ fk( ax) = f(x), where a is a non zero constant whose absolute value is less

than one. To avoid cumbersome explanations, we consider the case k = 3 as an exampIe. (Note that a method for the construction of even solutions in the case k = 2 was described by Cosnard and Eberhard [2]).

We choose a E (-1,0). Generally speaking, the choice of the sign of the constant a depends on the type of a solution to be constructed. Thus, for minimal fixed points

ft of the operators Ti' i = 2, 3, ... , the constants a (Ti' fi*) are positive for all even

i > 2 and negative for i = 2 and all odd i> 2. For maximal fixed points };*, the con

stants a (Ti' };*) are negative for aB i;::: 2.

Let k = 3 and let a E (-1, 0) be an arbitrary fixed number. We define Jo =

[a,-a], J[=[13,I], and J2 =[YI'Y2]' where -1<Y[<Y2<a<0<-a<!3<1.

We construct a map f* such that the intervals Jo' Jl' and J2 form a cycle of intervals

of period 3, f* (J [) = J 2' and f* (J2) = Jo' If we want to construct a unimodal map f*,

then it is necessary to require that f*(I) = Yp f*(ß) = Y2' f*(Yl) = a, and f*(Y2) = -a (Fig.52).

On the interval [-1, a], we define a continuous monotonically increasing function

fo- such that fO-(rl) = a, fÖ(Y2) = -<X, and fo-(a) > ß. Sirnilarly, on [-a, 1], we

define a continuous monotonically decreasing map fo+ such that fo+(-a) > ß, fo+(ß) =

Y2' and fo+(I) = YI' Let us now construct a solution which coincides with fo- and fo+

in the intervals [-1, a] and [-a, 1], respective1y, and reconstruct f*(x) in the inter

vals [a, _a2 ] and [a2, -a] according to the values of the initial functions fo- and

fo+. Denote

The solution f*(x) is reconstructed by using the following general recurrence relations:

Ji~l (x)

(8.5)

where {fO+)-1 0 (fo-rl is denoted by g. Then r x~O. f*(x) = Ji:(X), xEKt, i ~ 0, (8.6)

Ji (x), XE K;-, i ~ 0,

One can easily check that (8.6) is a solution of the equation T3f = f To guarantee

the continuity of the solution under consideration, it is necessary that fi- (a) = fo- (a)

and fi+(-a) = fo+(-a). As follows from the recurrence relations (8.5), for the initial

functions fo- and fo+, these equalities are equivalent to

(8.7)

Since conditions (8.7) establish the correspondence between fo- and fo+ only at finitely many points, one can always find initial functions satisfying these conditions. The following lemma demonstrates that conditions (8.7) do not guarantee the continuity of the map f*(x):

Lemma 8.8. If conditions (8.7) are satisfied for continuous initial functions, then

the map f* is continuous on [-1, 1]\ {O}. At the point 0, the map f* may be discontinuous.

Prooj. To prove that the map f* is continuous on [-1, 1] \ {O}, it suffices to show

that f/ (x) = fi~1 (x) for XE Kt n Ki: 1 and fi- (x) = fi~1 (x) for XE Ki- n Ki:-l'

It follows from (8.5) that the equality fi+ (x) = fi~l (x) is equivalent to the equality

Since the map g is monotone, the required equality for fi+ reduces to the analogous

equality for fi=1 and the equality for fi- reduces to the equality for fi~l' Hence, the required assertion can be obtained from conditions (8.7) by induction.

In order to prove that the map f* may be discontinuous at the point 0, we consider

the map ag: [+a, -al ~ [+a, -al Note that ag(a) = a and ag monotonically in

creases in the interval [+a, -al. It follows from (8.5) that

where fl and fo+ denote either fo+ or fo- (depending on i). These equalities im

ply that fi+ (x) and fi- (x) approach 1 if and only if the map ag has no fixed points

in the interval [a, -a] except a. It is clear that one can easily find initial functions

fo- and fo+ for which this condition is not satisfied. As follows from the proof of Lemma 8.8, for the solution f* (x) to be continuous, it

suffices to require that conditions (8.7) be satisfied and the initial functions fo- and fo+

be convex upward. In this case, ag has no fixed points other than a because it is convex downward and ag(-a) < -a. Note that the requirement of convexity of the initial

functions restricts the choice of YI' Y2' and ß but the construction of the required solu

tion is still possible.

Lemma 8.9. Assume that initial functions fo- and fo+ and the functions (fo-r l

and (fo+r l belong to C2 . Let fo- and fo+ be convex upward and satisfy condi

tions (8.7) and the conditions

difo- (a) dx'

ifi~ (a) dx'

234

and

over,

One-Parameter Families of Unimodal Maps

difo- (-a) dx'

if"+ (-a) dx'

Chapter 8

ProoJ. As in the proof of Lemma 8.8, by using the recurrence relations (8.5) and the conditions of Lemma 8.9, one can easily establish the existence and continuity of the corresponding derivatives everywhere in the interval [-1, 1] except 0 because, in this case, the proof of the existence and continuity of these derivatives can be reduced to the proof of their existence and continuity at the points a and - a and in the intervals [-1, a] and [-a, 1], i.e., to the conditions of the lemma.

Since initial functions are convex upward, we have f* (0) = 1. Let us show that

Cf*)'(O) = O. It follows from relations (8.5) that

Denoting Cf/),(x) by Zi, U;- )'(x) by Zi-I, and g'( a.t;:::I(~)) by €i, we obtain a

recurrence relation of t~e form Zi = €iZ; -I'

For large i, the argument of the function g' is dose to a (because f* (0) = 1 and the map f* is continuous). Hence, €i --+g'(a) as i --+ 00. Since the function g is con

vex upward and continuously differentiable, we have -1 < g' (a) < O. One can easily show that, in this case, all sequences satisfying the recurrence relation established above converge to zero independently of their initial values and, consequently, the derivative of the map f* at the point 0 exists and is continuous.

For the second derivative, relations (8.5) imply that

( +)" fi (x)

Seetion 5 Universal Properties ofOne-Parameter Families 235

In this case, we set

and denote the first term on the right-hand side of the first equality by Ei' This gives the

recurrence relation Zi = kiZ;_1 + Ei' As shown above, f* belongs to el and (f*)' (0) = O. Therefore, Ei ~ 0 as i ~ 00. The results established above also imply that k i ~

(e I g'(a) as i ~ 00. By the condition ofthe lemma, we have

and, consequently, 0 < a- I g' (a) < 1. It is easy to see that all sequences satisfying this recurrence relation converge to zero, whence it follows that (f*)" (0) = 0 and, there

fore, f* E e2 ([ -1, 1]). By using the proof of Lemma 8.9, one can easily show that if initial functions are

convex upward, belong to er, r ~ 1, together with their inverse functions, and satisfy conditions (8.7), the equalities

and

for all i::; r, and the inequality

ifo- (a) dx'

difo+ (-a) dx'

then f* E er ([ -1, 1]). In this case,

ifi.~ (a) dx'

ifi.+ (-a) dx'

if* (0) = 0 dx'

for all i::; r.

Thus, the equation Td = f may have infinitely many different unimodal solutions

with degenerate critical point from the dass er. If (fo+)'(1) = -00, then (under the corresponding conditions of continuity and compatibility imposed on the initial func-

tions) we can construct solutions of the equation Td = f from the dass L ([ -1, 1])

with infinitely degenerate critical point. Since the existence of solutions with nondegenerate critical point is usually postulated in the theory of universal behavior for families of one-dimensional maps, we do not present a detailed description of the procedure used to

construct C= -solutions. We also note that, in view of relations (8.5) and compatibility conditions imposed on the initial functions, the application of this procedure to the construction of solutions with nondegenerate critical point leads to a system of functional equations which is in no case simpler than the original equation.

Consider some properties of solutions with nondegenerate critical point.

Lemma 8.10. Let f be a C-unimodal map which solves the equation T d = f

dif* andlet (f*)"(0):f. O. Then --. (0) = 0 forallodd i5.r.

dx'

Proof. For simplicity, we consider only the case k = 3. In all other cases, the proof is similar.

First, we differentiate the identity (T3f*)(x) = f*(x) i times. This gives

L f*(x) dx'

(8.8)

where g = (f*)2. For i=2 and x=O, weobtain (f*)"(0) = ag'(l)(f*)"(O) (here, we have used the facts that f*(O) = 1 and (f*)'(0) = 0). Therefore, the inequality

(f*)" (0) :f. 0 implies that g' (1) = a- 1.

Assume that i is odd and greater than 2. Then, for all j > 0, we can write

This equality can be proved by induction on i. Actually, assume that d S f* (0) = 0 dxs

for all odd s < i and that i is odd. If i - j is odd, then :~;~J f* (ux) = 0 for x = O.

Otherwise, j is odd and then :J [g'(j*(UX»)] = 0 for x = 0 because

i.e., we arrive at an equality similar to (8.8) for j< i. Acting as above, we apply the assumption of induction to some terms and transform the other terms into an expression of

the form cog(S) (j*(ux» (f*)'(ax), where g(S) denotes the ~th derivative of the map


g. This expression is also equal to zero for x = 0 because (f*)' (0) = 0 and, hence, equality (8.8) can be rewritten in the form

a i - 2 if* (0). dx'

This equality implies the required assertion because a i - 2 ::;:. 1 for i> 2.

For analytic solutions of equations Td = f, Lemma 8.10 yields the following theorem:

Theorem 8.14. Assume that a map f* : [-1, 1] ~ [-1, 1] belongs to the class

d 2) andisafixedpointoftheoperator Tk, k~2. Thenf*(x) isanevenfunction,

i.e., f*(-x) = f*(x)forany XE [-1,1].

Note that if Properties 1-3 of the operator Tk in the vicinity of its fixed point f* were proved, then the assertion ofTheorem 8.14 would follow from the fact that any fix

ed c1ass 'E(Tk) of the operator Tk contains even functions (parallel with functions of

other types) and the evenness of maps is preserved under the action of the operator Tk.

Theorem 8.14 enables us to restrict the investigation of the operators T k to the sub

space of even functions G(2) of the space G( 2). By using the Schauder theorem, Lan

ford [1], [2] proved that the equation T2f = f is solvable in the space G(2). It is not

difficult to show that the existence of fixed points of various types for the operators Tk

with k> 2 can be proved by using the same method as in the indicated works. (Reeall that, according to Lemma 8.3, all fixed points of the operator T 2 are of the same type.)

Thus, we can regard the assertion that, for any k ~ 2, any fixed c1ass 'ElTk ) of the

operator Tk eontains a fixed point of this operator as an established fact.

It is worth noting that the result of Paluba mentioned in the previous seetion (Theorem 8.12; Paluba [1]) is an analog of the Hermann theorem on smoothness ofthe eonjugation of a diffeomorphism of a eirc1e with the eorresponding rotation. It seems likely

that Paluba's result ean be generalized to the ease of arbitrary stable manifolds 'Ej*(Tk ).

The results of numerieal simulation demonstrate that the eonstant I) = 4.6692 ... eharaeterizes the properties of period doubling bifurcations not only in one-dimensional dynamical systems. Sequenees of period doubling bifureations exhibiting the property of Feigenbaum universality were deteeted as a result of numerieal analysis of the Lorenz model, the Henon map, and some other systems. For these systems, we observe loeal expansion in one direction and eontraetion in all other directions. The dynamies of systems of this sort is fairly similar, in a eertain sense, to the dynamies of one-dimensional systems. In partieular, in this ease, we also observe an infinite sequenee of period doubling bifureations appearing at the same rate as in the one-dimensional ease.

For families of two-dimensional area-preserving maps, period doubling bifureations

are charaeterized by the same asymptotie law A~ - An - const I)-n but with I) = 8.72 ....

238 One-Parameter Families 0/ Unimodal Maps Chapter 8

In many-dimensional case, parallel with period doubling bifurcations, we observe period tripling bifurcations, period quadrupling bifurcations, etc. It is natural to expect that these bifurcations may also exhibit universal properties for some families of maps. Thus,

for period tripling bifurcations in a family /(z, J.1) of maps of a:: 1 into itself depending

on the complex parameter J.1, we have J.1oo - J.1n - const oin , where Ö:3 = 4.600 + 8.981 i (see Wul, Sinai, and Khanin [1]).

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SSR, Kishinev, 1970 (in Russian), English transi.: Noordhoff, Leiden, 1975.

Sinai, Va. G.

[1] Stochasticity of dynamical systems, Nonlinear Waves, Nauka, Moscow, 1979, pp. 192-221 (in Russian).

Singer, D.

[1] Stable orbits and bifurcations ofmaps on the interval, SIAM J. Appl. Math, 3S (1978),260-267.

Sivak, A. G.

[1] On a measure of attraction domain of semiattracting cycles of unimodal maps, Differential-Difference Equations and Their Applications, lust. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1985, pp. 57-59 (in Russian).

[2] Stability and bifurcations of periodic intervals of one-dimensional maps, Differential-Difference Equations and Their Applications, lust. Mat. Acad. Nauk Ukrain. SSR, Kiev, 1985, pp. 60-73 (in Russian).

[3] Limit behavior, stability, and bifurcations of one-dimensional maps, Preprint No. 86.51, Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1986 (in Russian).

Sivak, A. G. and Sharkovsky, A. N.

[1] Universal order and universal rate of bifurcations of solutions of differentialdifference equations, Rough and Qualitative Studies of Differential and Differential-Functional Equations, Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1983, pp. 98-106 (in Russian).

256 ReJerences

Smale, S.

[1] Differentiable dynamical systems, BuB. Amer. Math. Soc. 73 (1967), 747-817.

Smital, J.

[1] Chaotic Junctions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269-282.

Snoha, L.

[1] Characterization oJ potentially minimal periodic orbits oJ continuous mappings oJ an interval, Acta Math. Comenian. 52-53 (1987), 112-124.

Stefan, P.

[1] A theorem oJ Sharkovskii on the existence oJ periodic orbits oJ continuous endomorphisms oJthe realline, Commun. Math. Phys. 54 (1977), 237-248.

Strange Attractors (CoBection ofpapers), Mir, Moscow, 1981 (Russian translation).

Strien, S. J. van

[1] On the bifurcations creating horseshoes, Springer Lect. Notes in Math. 898 (1981),316-351.

[2] Smooth dynamics on the interval (with an emphasis on unimodal maps), Preprint No. 87.09, Delft University ofTechnology, Delft, 1987.

[3] Hyperbolicity and invariant measures Jor general C 2 interval maps satisfying the Misiurewicz condition, Preprint, Delft University of Technology, Delft, 1987.

Targonski, G.

[1] Topics in Iteration Theory, Vandenhoeck and Ruprecht, Göttingen, 1981.

Ulam,S.

[1] Unsolved Mathematical Problems, Nauka, Moscow, 1964 (Russian translation).

Vereikina, M. B. and Sharkovsky, A. N.

[1] Returnability in one-dimensional dynamical systems, Rough and Qualitative Studies of Differential and Differential-Functional Equations, Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1985, pp. 35-46 (in Russian).

References 257

[2] A set of almost-returning points of a dynamical system, Dokl. Akad. Nauk Ukrain. SSR, Sero A. (1984), No. 1,6-9 (in Russian).

Wul, E. B., Sinai, Va. G., and Khanin, K. M.

[1] Feigenbaum universality and thermodynamic formalism, Uspekhi Mat. Nauk 39 (1984), No. 3, 3-37; English transt.: Russian Math. Surveys39 (1984).

Xiong, J.

[1] A set of almost periodic points of continuous self-maps of an interval, Acta Math. Sinica 2 (1986), No. 1,73-77.

Ye,X.

[1] D-function of a minimal set and an extension of Sharkovskii's theorem to minimal sets, Ergod. Theory Dynam. Syst., 12 (1992), 365-376.

Y oung, L.-S.

[1] A closing lemma on the interval, Invent. Math. 54 (1979), 179-187.

SUBJECT InDEX

Attractor, 23 mixing, 22 strange, 22

Bifurcation, 201 mild, 202 of creation of cyde, 202 period doubling, 202 rigid, 202

Cycle, 2 attracting, 4 nonhyperbolic, 4 of minimal type, 65 repeIling, 4 simple, 74

Dynamical system, center of, 29 complex, 69 minimal, 8 simple, 69

Feigenbaum constants, 8,217

Hypergraphic property, 149

Kneading invariant, 41 Königs -Lamerey diagram, 2

Lexicographic ordering, 38 Lyapunovexponent, 14

Map, chaotic, 109 er -structurally stable, 186

259

Map, Lyapunov stable, 108 nonsingular, 166 of type 1;~, 74 simple, 74 structural O-stable, 196 S-unimodal, 142 unimodal, viii

Measure, absolutely continuous with respect

to the Lebesgue measure, 13, 166 invariant, 13

Mixing,22 Mixing repeller, 25 Multiplier,

ofacyde, 4 of a quasiattractor, 176

Permutation, cycIic, 57 minimal, 65 unimodal, 67

Phase diagram of a unimodal map, 123 Point,

address of, 39 almost periodic, 26 almost periodic in the sense ofBohr, 32 chain recurrent, 30 criticaI,

nondegenerate, 150 nonflat, 150

dynamical coordinate of, 41 nonwandering, 26

unilateral, 28 periodic, 2 recurrent, 26 regularly recurrent, 26 routeof, 39 CO-limit, 2

260

Quasiattractor, 25

Schwarzian, 140 Set,

of almost periodic points, 26 of almost periodic points in the sense

ofBohr, 32 of chain recurrent points, 30 of nonwandering points, 26 of periodic points, 25 ofrecurrent points, 26 of regularly recurrent points, 26 probabilistic limit, 36 statisticallimit, 21 co-limit, 2

Subject Index

Symbolic dynamies, 35

Theorem on coexistence of cyc1es, 57 Topological conjugation of

dynarnical systems, 11 Topological entropy, 15 Trajectory, 1

heteroc1inic, 22 homoc1inic, 19 periodic, 2

Transitivity, 22

Wandering interval, 119

F:l

~~j

1R(lRj

I

CoU, I)

2 x

A dA

intA

Uö(A)

mesA

fl A

fn

orb (x)

(0 (x)

n(f)

NW(f)

NW+(f), NW-(f)

CR(f)

Fix(f)

Per(f)

AP(f)

APB(f)

R(f)

RR(f)

C(f)

h(f)

vf

nOTRTlon

the set of natural numbers

the set of integer (nonnegative integer) numbers

the set of real (nonnegative real) numbers

a c10sed interval of IR

the space of continuous maps of I

the space of c10sed subsets of the space X

the c10sure of the set A

the boundary of the set A

internal points of the set A

the Ö-neighborhood of the set A

the Lebesgue measure of the set A

the restriction of the map f to the set A

the n th iteration of the map f the trajectory of the point x the set of ro-limit points of the trajectory of the point x

the set of ro-limit points

the set of nonwandering points

the sets of unilateral nonwandering points

the set of chain recurrent points

the set of fixed points

the set of periodic points

the set of almost periodic points

the set of almost periodic points in the sense of Bohr

the set of recurrent points

the set of regularly recurrent points

the center

topological entropy

kneading invariant

261

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