Topological Theory of Dynamical Systems, Volume 52: Recent Advances

Recent Advances

TOPOLOGICAL THEORY OF DYNAMICAL SYSTEMS Recent Advances

North-Holland Mathematical Library Board of Advisory Editors. M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, H.A. Lauwerier, W.A.J. Luxemburg, F.P. Peterson, I.M. Singer and A.C. Zaanen

VOLUME 52

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NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO

Topological Theory of Dynamical Systems Recent Advances

N . AOKI Department of Mathematics Tokyo Metropolitan University Minami-Ohsawa 1- I , Hachioji-shi Tokyo,Japan

K. HIRAIDE Department of Mathematics Faculty of Science Ehime University Matsuyama, Japan

1994 NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

L i b r a r y o f C o n g r e s s Cataloging-in-Publication D a t a

Aoki. Nobuo. 1927Topological theory of dynaaical s y s t e m s : r e c e n t a d v a n c e s / N. Aoki. K. Hiraide. p. cm. -- (North-Holland mathematical library) I n c l u d e s bibllographical r e f e r e n c e s and index. I S B N 0-444-89917-0 1. Differentiable dynamical systems. 2. T o p o l o g i c a l dynamics. I. Hiraide. K. ( ~ o i c h i ) 11. Title. 111. Series. QA614.8.A52 1994 514'.72--dc20 94-1 1 6 7 8 CIP

ISBN: 0 444 89917 0 O 1994 ELSEVIER SCIENCE B.V. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper Printed in The Netherlands

PREFACE

The history of theory of dynamics began with the work of Isaac Newton who formulated the law of motion and who stated the universal law of gravitation. In Newton's theory the motion of a dynamical system was described by a system of differential equations. The problem of applying the theory to planetary motion was challenging. People thought they would eventually succeed in explicitly integrating the differential equations. However, their hope did not come true. In the late nineteenth century, H.PoincarC created a new branch of mathematics by publishing his famous memoir "Les mCthodes nouvelles de la mkanique cC1estev on the qualitative theory of differential equations. The point of his idea was to relate the geometry of the phase space with the analysis. As a result of PoincarC's qualitative approach, the focus of the theory of dynamics shifted away from the differential equations. In G.D.Birkhoff's treatise on dynamical systems, he discussed many dynamical phenomena in the context of transformation groups acting on general metric spaces. Since then, many mathematicians have considered this theory in the environment of geometry. The purpose of this book is to provide an advanced account of some aspects of dynamical systems in the framework of general topology, and is intended for use by interested graduate students and working mathematicians. Even though some of the topics discussed here are relatively new, others are not. Therefore the book is not a collection of research papers, but it is a text book to present recent developments of the theory that could be the foundations for future developments. This book does not necessarily cover all aspects of modern topological dynamics, but we intend here to give a wider scope of various trends in recent developments of dynamical systems while avoiding overlap with other books in the concerned field. Nevertheless, the topics were selected from only the most significant results which may serve as the foundations for further developments of topological dynamics, and much effort has been done to make the subject matter accessible to a larger number of readers. This book contains a new theory developed by the authors to deal with the problems occurring in differentiable dynamics that are within the scope of general topology. To follow it, the book provides an adequate foundation for topological theory of dynamical systems, and contains tools which are sufficiently powerful through out the book. We assume that the reader has a rudimentary knowledge of algebra and analysis while a little Inore is presapposed from basic general topology, basic topological dynamics, and basic algebraic topology that may be obtained

vi

PREFACE

in the graduate course, e.g., the books, Modern General Topology by J. Nagata, Topics in General Topology by K. Morita and J. Nagata, and Algebraic Topology by E. Spanier. We believe that graduate students and some undergraduate students with sufficient knowledge of basic general topology, basic topological dynamics, and basic algebraic topology will find little difficulty in reading this book. We would be especially pleased if our book could help them in the preparation of their theses. The authors wish to express their gratitude to Professor Jun-iti Nagata and Professor Jacob Palis without whose help and encouragement would have made writing this book covering the significant area of mathematics impossible. The authors also wish to acknowledge that this book was written during their stay at IMPA in Rio de Janeiro. N. Aoki and K. Hiraide

CONTENTS

................15 CHAPTER 1 Some Properties of Anosov Systems $1.1Toral endomorphisms .......................................... 15 $1.2 Anosov differentiable systems .................................. 19 CHAPTER 2 Dynamics of Continuous Maps ........................31 $2.1 Self-covering maps ............................................. 31 $2.2 Expansivity ................................................... 36 $2.3 Pseudo orbit tracing property .................................. 78 92.4 Topological Anosov maps (TA-maps) ...........................88 CHAPTER 3 Nonwandering Sets ................................. 96 $3.1 Chain recurrent sets .......................................... 96 ......................................102 $3.2 Stable and unstable sets $3.3 Recurrent sets and Birkhoff centers ...........................107 $3.4 Nonwandering sets of TA-maps ............................... 117 ......................................... 122 $3.5 Inverse limit systems

CHAPTER 4 ~ ~ Partitions k ~ .................................. v 127 ............................... 127 $4.1 Markov partitions and subshifts ............................ -132 $4.2 Construction of Markov partitions $4.3 Symbolic dynamics ........................................... 140

........................... 147 CHAPTER 5 Local Product Structures .................................... 147 $5.1 Stable sets in strong sense $5.2 Local product structures for TA-covering maps ................ 155 $5.3 Expanding factors of TA-rnaps .............................. . -162 $5.4 Subclasses of the class of TA-maps ............................ 166 CHAPTER 6 TA-Covering Maps ..................................168 $6.1 Fundamental groups ..........................................168 $6.2 Universal covering spaces ..................................... 174 86.3 Covering transformation groups ............................... 182 $6.4 S-injectivity of TA-covering maps .............................. 189 $6.5 Structure groups for inverse limit systems ..................... 193 -203 $6.6 Lifting of local product structures .............. 211 $6.7 TA-covering maps of closed topological manifolds ..................... 219 $6.8 Classification of TA-covering maps on tori a...........................

viii

CONTENTS

.

CHAPTER 7 Solenoidal Groups and Self-covering Maps ......... -222 $7.1 Geometrical structures of solenoidal groups .................. 222 $7.2 Inverse limit systems of self-covering maps on tori ..............231

.

........................... .........

CHAPTER 8 TA-Covering Maps of Tori 241 241 $8.1 Toral endomorphisms homotopic to TA-covering maps $8.2 Construction of semi-conjugacy maps ......................... 244 $8.3 Nonwandering sets ...........................................250 $8.4 Injectivity of semi-conjugacy maps ............................257 .......................................268 $8.5 Proof of Theorem 6.8.1 .......................................269 $8.6 Proof of Theorem 6.8.2 $8.7 Remarks ...................................................~275 CHAPTER 9 Perturbations of Hyperbolic Toral Endomorphisms . 280 $9.1 TA-Cw regular maps that are not Anosov ..................... 280 $9.2 One-parameter families of homeomorphisms ................... 288 CHAPTER 10 Fixed Point Indices .............................. -304 $10.1 Chain complexes ............................................304 $10.2 Singular homology ..........................................310 320 $10.3 Euclidean neighborhood retracts (ENRs) $10.4 Fixed point indices .......................................... 326 $10.5 Lefschetz numbers .......................................... 330 $10.6 Orientability of manifolds ...................................332 ........................337 $10.7 Orientability of generalized foliations $10.8Fixedpointindicesofexpandingmaps .......................346 $10.9 Fixed point indices of TA-covering maps .....................348 .....a*..............

..................... 360 CHAPTER 11 Foundations of Ergodic Theory $11.1Measure theory ............................................. 360 $11.2 Measure preserving transformations ..........................363 $11-3 Ergodic theorems ........................................... 366 375 $11.4 Probability measures of compact metric spaces ........................ 387 $11.5 Applications to topological dynamics a..............

INTRODUCTION

Take in its broad sense, Dynamical Systems involves the branches of mathematics: differential equation, differential topology, general topology and ergodic theory. This book takes up the subjects in the narrow sense, omitting differential equation from the above list and minimizing differential topology and ergodic theory. The phase space of a dynamical system will be taken to be a differentiable manifold. A continuous surjection f: M -,N of metric spaces is a homeomorphism if it is injective and if the inverse map f : N -+ M is also continuous. A metric space M is called an n-dimensional topological manifold (or a manifold of class CO)if there exist open subsets Ui of M and homeomorphisms ai of Ui onto open subsets of the n-dimensional euclidean space Wn, such that {Ui) covers M. It follows that the composite map ai o a;' : aj(Ui n Uj) -t ai(Ui n Uj) is a homeomorphism whenever U; and Ui overlap. If, in addition, all the composite maps ai o a;' are T-times continuously differentiable, then M is called a differentiable manifold of class CT (1 5 T 5 00). A differentiable manifold of class C" is also said to be smooth. The local charts (Ui, a;) of an n-dimensional differentiable manifold M enable us to use the differential calculus in the neighborhood of any point x € M. Let M and N be n-dimensional differentiable manifolds and let f : M + N be a continuous map. If (U, a ) and (V,/3) are local charts of M and N respectively with f (U) C V, then we say that f is diflerentiable at x E U if /3 o f o a-' : a(U) -, P(V) is differentiable at a(%). This definition does not depend on the choice of local charts. A continuous map f : M -, N of a differentiable manifold is said to be a differentiable map of class CT if f is T-times continuously differentiable at all points of M. If f: M 4 N is a homeomorphism and if both f and f-' are T-times continuously differentiable, then the homeomorphism f: M -, N is a diffeomorphism of class CP. By a curve in a manifold M we mean the image into M of an open interval of the real line under a continuously differentiable map. If y is a curve, then the tangent vector to y at x E y can be defined. For x E M all such tangent vectors form a real n-dimensional vector space T,M, called the tangent space to M at x. The set of all pairs (3,v), where x E M and v E T, M , can be vested with the structure of a 2n-dimensional differentiable manifold T M , called the tangent bundle of M. If M is of class CP, then T M is of class CT-l.

-'

Dynamical Systems. We are now ready to define a dynamical system. A dynamical system with continuous time, or flow on a metric space X is a family {cpt : t E W) of homeomorphisms of X , such that the map (t, x) I+ cpt(x)

2

INTRODUCTION

is continuous and ~ t + ~ ( = x )cpt o cp,(x) for all x E X and all t, s E R . It follows that cpo is the identity map of X and cp-t is the inverse map of cpt. A family {cpt : t 2 0 ) of continuous maps of X is called a semi-flow if cpo is the identity map and if ( P ~ + ~ ( X=) cpt o cps(x) for all x E X and all t, s 0 , and (t, x ) w cpt(x) is continuous. The orbit of a point x E X under the flow or semi-flow is the set of points {cpt(x) : t E R ) or {cpt(x) : t 0 ) . The point x is said to be a fixed point if cpt(x) = x for all t, and t o be a periodic point if cp,(x) = x for some p > 0 . This implies cpt+,(x) = cpt(x) for all t . A dynamical system with discrete time on a metric space X is a family { f n : n E Z) of homeomorphisms of X , or a family { f : n 0 ) of continuous maps of X with f O the identity map, such that f n+m(x) = f o f m ( x ) for all x E X and all n,m. It follows that fn = f o o f (n-times) for n > 0 , where f = f l , and if f is a homeomorphism then f-" = ( f n ) - ' for n > 0 . Conversely, the iterations of a homeomorphism or a continuous map f : X + X form a discrete dynamical system. A point x is said to be a fixed point if f ( x ) = x and be a periodic point if fP(x) = x for some p > 0. A dynamical system with continuous time is obtained as solutions of differential equation of the form

>

>

>

.

where x belongs to a domain D of Rn and F : D + Rn is a map on the phase space D. In the theory of differential equations a fundamental theorem says that if F is continuously differentiable then, for any given point xo, the equation has a unique solution cpt(xo) which is defined for It1 sufficiently small and satisfies the initial condition cpo(xo) = 30. The time-dependence together with uniqueness ensures that c p t + a ( x ~= ) cpt(cpa(xo)).The restriction to small value of It1 is essential. In many problems the phase space will be a domain of the real n-dimensional space Rn. However, other spaces also arise quite naturally. For example, if the differential equation can be restricted on a compact m-dimensional manifold M contained in the domain D (i.e. for every x E M the value F ( x ) is a tangent vector of M at x ) , for any given point xo E M the equation has then a solution cpt(xo) E M for all t E R . Therefore we shall have a flow {cpt) on the phase space M. There are two ways to relate such an equation d x = F ( x ) to a dynamical dt system with discrete time. The first method is to discretize time, i.e. for any choice of small to > 0 this leads to the equation

Then the iterations { f n ) of the map f becomes a dynamical system with discrete time on the space. The other method can be applied if the equation has a periodic solution, i.e. cp,(xo) = xo for some xo and some T > 0. Then we consider a hypersurface S transverse to the curve t I+ cpt(xo), and in this hypersurface a neighborhood U of s o . Then a map g: U -, S is induced by

INTRODUCTION

3

associating to y E U the next intersection with S of the trajectory {cpt(y)). If the first such intersection occurs at y', then we define g(y) = y', and g is a map U --t S which is reduced from the flow. Thus we have a dynamical system with discrete time modelled on the differential equation. In this book we shall discuss mostly dynamical systems with discrete time on a compact metric space X. As a special case, we shall regard X as a differentiable manifold and f as a differentiable map of X. The general case may be called a topological dynarnical system and the special case a differentiable dynamical system. We shall be primarily interested in differentiable maps, or diffeomorphisms of a differentiable manifold. However, such maps may have an invariant set which is not a manifold. The topological theory of dynamical systems is useful to investigate the behaviour of the differentiable map on the invariant set. Our book is written from this point of view. We give an important example of a discrete dynamical system with a topological structure. Let k 1 be an integer and let Yf denote the set of all biinfinite sequences x = (. ,x-1, 5 0 , 21, ) where z, E Yk = {0,1, ,k - 1). The set Yf becomes a compact metric space if we define the distance between two points x, y by d(z, y) = 0 if x = y and d(x, y) = 2-" if x # y where m is the largest integer such that xn = y, for all n with In1 < m. The set Yf is homeomorphic to the Cantor set. The shift map u defined by u(x) = y where x = (x,), y = (y,) and y, = x,+I, n E Z, is a homeomorphism of Yf. Thus the family of homeomorphisms u n , n E Z, is a discrete dynamical system, called a symbolic dynamics. The periodic points of this system are dense in Yf and there exists a point x E whose orbit 0,(x) = {un(z) : n E Z) is dense in Yf. If S is a closed subset of Y? and u-invariant (i.e. a(S) = S), then the restriction of u to S is called a subshift, denoted by 01s. The subshift u p defines a discrete dynamical systems on S. As an important class of subshifts we give the following class. Let A = ( a i j ) be a k x k matrix of 0's and 1's and let SAdenote the set of all x = (2,) E ~f with aZnBn+l= 1 for all n E Z. Then SAis closed in ~f and u(SA) = SA.A subshift of this type is called a Markov subshift. A continuous surjection f : X + X of a metric space is said to be topologically transitive if for any nonempty open sets U, V there is n > 0 such that f n ( U ) n V is nonempty. It is said to be topologically mixing if for any nonempty open sets U,V there is N > 0 such that f "(U) fl V is nonempty for all n 2 N. A Markov subshift up, : SA-' SAis topologically transitive if and only if the matrix A is irreducible, and it is topologically mixing if and only if every element of Am is positive for some m > 0. It may seem that Markov subshifts have no relevance to physical systems. But the fact is that they are basic components of some physical systems. For example, Smale's theorem says that in a neighborhood of the orbit through

> .

---

.

YE

INTRODUCTION

4

any transversal homoclinic point, associated to a hyperbolic fixed point of a diffeomorphism, there is an invariant set such that some iterations of the diffeomorphism on the invariant set act as the shift map of : Y for some k > 0. This theorem applies to the restricted three-body problem.

1-dimensional dynamics. The iterations of maps of an interval into itself certainly present an example of nonlinear dynamical systems. The dynamics on an interval has many applications to physical systems. Let f : [O,1] + [O, 11 be a continuous map of the interval. The iterations of the form x, H xn+l = f(xn) can be viewed as a discrete time version of a topological dynamical system. Here n plays the role of time variable. In natural sciences, the dynamical system often depends on quantities, called parameters. Therefore it is important to study a one-parameter family of maps which is the most basic system. For example, we consider a continuous map f,(x) = rx(1 - x) from [O,1] to itself, which is called the logistic map. F i x r with 0 5 r 5 4 and define a sequence {x,} by xn+l = rz,(l - 2,) for n 1 0. It can be shown that the behaviours of these maps, with the parameter r, where 0 5 r 5 4, has the following properties : (1) If 0 5 r < 1 then the sequence {x,) is decreasing and x, + 0 as n+m. (2) If 1 5 r 5 2 then the sequence {x,) is increasing and x, + 1- l l r as n -+ 00. (3) If 2 < r 5 3 then the sequence {x,) converges to 1 - llr. (4) If 3 < r 5 1 & then the sequence {x,) converges to some periodic orbit with period 2. (5) If 1 & < r 5 4 then the behaviour of {x,) is very complicated, and when r varies increasingly in the interval [1+&,41 there exists a sequence of parameters r, in [1+&,41 such that f,, has 2,-periodic points. On the set of natural numbers we can define an ordering, which is called the 2?harkovskii ordering, as follows.

+

+

The ~harkovskii'stheorem says that if f : [O,1] -t [O, 11 is continuous and has a periodic point with period p, then it has a periodic point with period q for every q < p with respect to the above ordering.

INTRODUCTION

5

From this result it follows that if the continuous map has a 3-periodic point then it has periodic points of all periods. Concerning the question whether the behaviour of the continuous map is chaotic, Li-Yorke showed that the &period implies chaos, i.e. if f: [ O , l ] + [O, 11 is a unimordal map with a periodic point of period 3, then there is an uncountable set S of points and E > 0 such that for every x, y E S , x # y

liminf Ifn(x) - fn(y)I = 0. n-+w

It can be shown that the logistic map f, has a &periodic point if r 2 3.83. Thus Li-Yorke's theorem holds for such a map and its behaviour is chaotic. For the case r = 4 in particular, the logistic map f4 is topologically conjugate to a continuous map g: [O, 11 -, [O, 11, called the tent map, i.e. g(x) = 22 for 0 5 x 5 112 and g(x) = 2(1 - x) for 112 5 x 5 1. It can be shown that the map g is topologically semi-conjugate to a symbolic dynamics. Let X and Y be metric spaces. If continuous maps f: X -+ X and g: Y -,Y satisfy the relation f o h = h o g for some homeomorphism h: Y -+ X , then we say that f is topologically conjugate to g. When the relation f o h = h o g holds for some continuous surjection h: Y -,X, we say that f is topologically semi-conjugate to g. The dynamics of maps of an interval is quite interesting. However, we do not discuss the subject in this book. Higher dimensional dynamics. A 2-dimensional torus T2 = R2/Z2 is obtained from the 2-dimensional euclidean space R2 by identifying the points (21, yl) and (22, y2) whenever 2 2 - xl and y2 - yl are integers, i.e. (x2 X I , y2 - yl) E Z2. The torus is a compact smooth manifold. A linear map of the plane with matrix

induces a map f = f A of the torus to itself if a, b, c, d are integers. The map f: T2 t T2 is a differentiable map. If the determinant ad - bc of A is non-zero, then f is called a toral endomorphism. If ad - bc = f1 then f is bijective, and we call it a tom1 automorphism. The toral automorphism (endomorphism) f is said to be hyperbolic if A has no eigenvalues on the unit circle. Then the behaviour of f is quite complicated. It follows that a point p is periodic if and only if p has a coordinate consisting of rational numbers, and that the periodic points are dense in the torus. Moreover, there exists a point p whose orbit Of(p) is dense on the torus. It may be asked whether the behaviour of f is chaotic. However, the question is somewhat vague because there is no widely accepted definition of chaotic behaviour of higher dimension. Nevertheless, the toral automorphism f is

INTRODUCTION

6

chaotic according to reasonable definitions which are not equivalent in general.

Anosov Systems. In a similar way we can define an n-dimensional torus

Tn = Rn/Zn from the n-dimensional euclidean space Rn. An automorphism (endomorphism) f = f A of Tn is induced by an n x n matrix A of integers. If the determinant of A is non-zero and A has no eigenvalues on the unit circle, then Wn is the direct sum of two subspaces Eg and Eu, invariant under A, such that every eigenvalue of the restriction of A to Ed,respectively Eu has an absolute value less than, respectively greater than one. Thus the behaviour of f is similar to that in the 2-dimensional case. To generalize the situation further, we consider a compact smooth manifold M . A diffeomorphism f : M -+ M is said to be an Anosov difeomorphism if (1) for each x E M the tangent space T,M is the direct sum of two subspaces T, M = Ei $ E," and

(2) there exist C > 0 and 0 < X

< 1 such that for all x E M

and all n

>0

By use of (1) and (2) it can be shown that the subspaces Ei and E," vary continuously with respect to x. For a differentiable map f : M -+ M of a compact smooth manifold a point x E M is called a regular point if the differential D,f : T,M -+ Tf(,)M is surjective, and a singular point otherwise. If x is a regular point, by the inverse function theorem some open neighborhood of x is mapped diffeomorphically onto an open set of M by f . Hence the set S ( f ) of all singular points of f is closed in M. If S ( f ) is empty, then f is called a regular map. The set A( f ) = f n(M) is the maximal closed f -invariant set, i.e. f (A(f )) = A( f ). If A is a closed invariant set (f (A) = A) then A is a subset of A( f ), and fla : A -+A is a local homeomorphism whenever A n S ( f ) = 0. The set

nn20

M f = {(xi) : xi E A( f ) and f (xi) =

i E Z}

nE-,

is a closed subset of the product topological space MZ = Mi where each M iis a replica of M , which can be regarded as the set of all (two sided) orbits of f . A closed invariant set A is called a hyperbolic set of f if A n S(f ) is empty and there exist C > 0 and 0 < X < 1 such that for every x = (xi) E Af = {(xi) : xi E A and f (xi) = xi+l,i E Z} there is a splitting

INTRODUCTION

so that for all i E ;Z

(1) DZif (E,Oi) = (2) for all n 2 0

where a = s,u,

If, in particular, T,M = Ui Eti for all x = (xi) E Af, then flA : A + A is said to be expanding. If the entire space M is a hyperbolic set of f , then f : M + M is called an Anosov diflerentiable map. It follows that if an Anosov differentiable map is injective then it is an Anosov diffeomorphism. From the definitions of Anosov diffeomorphism and Anosov differentiable map, deduced are the following two important general properties : (A) Whenever f is an Anosov diffeomorphism, f is expansive, i.e. there exists e > 0 such that if x and y are any two distinct points of M then d(fn(x), fn(y)) > e for some integer n. Whenever f is an Anosov differentiable map, f is c-expansive, i.e. there exists e > 0 such that if (xi) and (yi) are any two distinct points of M f then d(xn, y,) > e for some integer n. Whenever f is an expanding differentiable map, f is positively expansive, i.e. there exists e > 0 such that if x # y then d( f n(x), f n(y)) > e for some n 2 0. (B) Whenever f is an Anosov diffeomorphism, for each 6 > 0 there is a corresponding E > 0 such that if d(x, y) < E then there exists z E M such that d(fn(x), f n ( r ) )< 6 and d(f-n(y), f-n(z)) < 6 for all n 2 0. Whenever f is an Anosov differentiable map or an expanding differentiable map, for each 6 > 0 there is E > 0 such that for (xi), (yi) E Mf if d(xo, yo) < E then there is (zi) E Mf with the property that for all i 2 0, d ( ~ i~, i <) 6 and d(y-i, z-i) < 6. It follows from (A) that if 6 < e/2 then the points z and (zi) in (B) are uniquely determined by the points x, y and (xi), (yi) respectively. For a differentiable map f : M -r M it can be shown that if A is a hyperbolic set of f and if (xi) E M f implies (xi) E Af whenever all xi are in some neighborhood of A, then the restriction of f to A satisfies the same properties as in (A) and (B). The properties (A) and (B) make sense for any metric space and any continuous map, or any homeomorphism from the space to itself. We shall study more generally continuous surjections of a compact metric space with the properties (A) and (B). This approach will turn out to be a reasonably clear and coherent one. It can be shown that (A) and (B) are equivalent to (A) and (B)' f has the pseudo orbit tracing property (POTP).

INTRODUCTION

8

A sequence of points, {x,}, is said to be a 6-pseudo orbit if d( f (x,), x,+~) 5 6 for n. This is quite a natural notion since on account of rounding errors a computer will actually calculate a pseudo orbit, rather than an orbit. A 6pseudo orbit {x,) is said to be E-traced by an orbit { f ,(x)) if d(f n(x), x,) 5 E for all n. Thus the pseudo orbit is uniformly approximated by a genuine orbit. We say that f has POTP if for each e > 0 there is 6 > 0 such that every 6-pseudo orbit can be e-traced by some orbit. It can be deduced that if a homeomorphism f : X + X of a compact metric space has (A) and (B)' then it is topologically stable, i.e. for every E > 0 there is 6 > 0 such that if g: X + X is any homeomorphism with d(g(x), f (x)) < 6 for all x E X, then there is a continuous map h: X + X with h o g = f o h and d(h(x), x) < e for all x E X. The same definition applies to a continuous surjection. From topological stability follows the structural stability theorem of Anosov diffeomorphisms of a compact smooth manifold. Let M be a compact smooth manifold and Diffl(M) the set of all diffeomorphisms endowed with the C1 topology. The structural stability theorem says that there is a C1neighborhood U of an Anosov diffeomorphism f such that for every g E U, there exists a homeomorphism h: M + M for which g o h = h o f holds, i.e. g is topologically conjugate to f . The structural stability means that even if the dynamics is very complicated, the perturbed systems behave in the same way. By the work of Robbin and Robinson it was shown that if a C 1 diffeomorphism f: M -t M of a compact smooth manifold is Axiom A and satisfies strong transversality (see the following definitions) then f is structurally stable. Finally MaiiC (1988) has shown that the converse is true. A point x belonging to M is said to be a nonwandering point if for any neighborhood U of x there is an integer n # 0 such that U n fn(U) is nonempty. The set R(f) of all nonwandering points is called the nonwandering set. Clearly R(f) is closed in M and invariant under f . A diffeomorphism f : M -t M is said to be Adorn A if R(f) has a subset of all periodic points which is dense in R(f) and R(f) is a hyperbolic set of f . Topological decomposition theorem says that if f is Axiom A then R(f) can be written as the finite disjoint union R(f) = Al U U A, of closed invariant sets Ai such that each fini is topologically transitive. Such a set Ai is called a basic set. For x E M the stable and unstable manifolds are defined by

---

W"(X) = {YE M : d(fn(x), fn(y)) -t 0 as n + oo}, w U ( x )= {Y E M : d(fFn(x), f-n(y))

-t

O as n

+ oo).

The stable manifold theorem ensures that if f is Axiom A then each of WS(x) and Wu(x) is actually a manifold which is an injective immersion of some euclidean space, and M = UxEn(f)Wu(x) for B = s,u. We write Wu(Ai) = UxEAiWu(x), o = s,u, for basic sets Ai. A relation between the basic sets hi, 1 5 i 5 s, is defined as Ai > Aj if (Wd(Ai)\hi) n Wu(Aj) # 0, that is, there is

INTRODUCTION

a point x which comes from Ai and goes to Aj by negative iterations o f f . We say that f has no cycles with respect to O(f) if Aio > Ail > > hi, > Aio cannot happen between the basic sets. An Axiom A diffeomorphiim f: M -+ M is said to satisfy strong transversality if, for x E M , the stable manifold W8(x) and the unstable manifold Wu(x) are transversal, i.e. T,M = T,WS(x) T, WU(x). Fkom Smale's theorem it follows that f has no cycles with respect to R(f) if it satisfies strong transversality. The structural stability of differentiable maps is defined in a form analogous to that of diffeomorphisms. However, it is known (by Przytycki) that if the manifold M has dimension 2 2 then every Anosov differentiable map which is neither injective nor expanding does not satisfy structural stability. This indicates that the behaviour of an Anosov differentiable map is more complicated than that of an Anosov diffeomorphism. We can also give the notion of Axiom A for a differentiable map in a similar fashion. Then every differentiable map satisfying Axiom A yields the topological decomposition theorem. Thus we can take of no cycles condition for differentiable maps. However we can find in the literatures no definition of strong transversality for differentiable map. This seems likely due to the obstruction caused by singularities. In particular let M be the unit circle and R(M) the set of regular maps of M endowed with the C1 topology. Then it can be shown that the set of maps in R(M) statisfying structural stability coincides with the set of maps in R(M) satisfying Axiom A and no cycles condition.

..-

+

Dynamical Systems i n t h e General Situation. We now return to the general situation to think of a continuous surjection, or a homeomorphism of a compact metric space. A 6-pseudo orbit {x,} is said to be periodic if there is q > 0 such that x,+, = x, for all n. A point x E X is said to be chain recurrent if for 6 > 0 there is a periodic 6-pseudo orbit which passes through x. The set C R of all chain recurrent points is a closed set and f (CR) C C R (f ( CR) = CR for a homeomorphism). Chain recurrence is useful in certain aspects. It can be shown that if a continuous surjection f of a compact metric space has the properties (A) and (B)' then every chain recurrent point is a limit point of periodic points and thus f (CR) = CR. Moreover, the chain recurrent set C R is the union of finitely many disjoint closed subsets R1,. ,R, invariant under f , and such that the restriction of f to each Ri is topologically transitive. The sets R1,. ,R, are called basic sets. If, in particular, s = 1 then we have C R = X. Each set R; is the union of a finite number mi of disjoint closed subsets, which are permuted cyclically by f in such a way that the restriction of f mito each of subsets is topologically mixing. This topological decomposition theorem gives us a concept of how the continuous surjection behaves on the chain recurrent set CR. A more precise description of the behaviour on the chain recurrent set can be given. This is useful only in case f is a homeomorphism. It can be shown that for each set

--

..

INTRODUCTION

10

Ri there is a Markov subshift (EAi,a ) and a continuous map xi of E A ~ onto Ri, with xi o a = f o xi, such that each point of Riis the image under xi of at most d points of CAi,for a certain positive integer d. This implies that the homeomorphism f restricted to Ri behaves as the shift map a restricted to EAi does. Even if a continuous surjection has the properties (A) and (B)', the restriction of the map to a basic set can not necessarily be realized by a Markov subshift under at most d points to one continuous map. More precise descriptions of the behaviours of dynamics having POTP and of expansive dynamics will be found in the following chapters.

Topological Anosov Maps. We return to the case of a continuous surjection of the torus. To see how the dynamics is complicated, we raise the question whether a continuous surjection with the properties (A) and (B)' of an n-dimensional torus is Co-perturbed to a certain hyperbolic toral automorphism (endomorphism). To deal with this problem, let C(Tn) denote the set of all covering maps of an n-dimensional torus Tn. A continuous surjection f:Tn 4 Tn is a covering map if f is a local homeomorphism. Let f = fA : Tn 4 Tn be a hyperbolic differentiable map induced by an n x n matrix A. Then f is a differentiable map belonging to one of the following three types : (I) f is a toral automorphism and A has eigenvalues of modulus < 1 and modulus > 1, (11) f is a toral endomorphism and all eigenvalues of A are larger than one, when f is called an eqvanding toral endomorphism, (111) f is a toral endomorphism which is not injective, and A has eigenvalues of modulus < 1 and modulus > 1. We say that a continuous surjection f:W 4 Tn is a topological Anosov map if f is c-expansive and has POTP. A continuous surjection f is said to be a special topological Anosov map if f satisfies the conditions : (1) f is a topological Anosov map, (2) for every (xi), (y;) E (Tn)f with xo = yo

where WZ"((zi))= {YO E En : there is (yi) E (Tn)f such that d ( ~ iy,) , -+ O as i -+ -w) (this set is called an unstable set). There exist continuous surjections that satisfy some condition stronger than the one satisfied by special topological

INTRODUCTION

11

Anosov maps. A continuous surjection f : 'Ir" + 'IP' is said to be a strongly special topological Anosov map if f satisfies the conditions : (1) f is a special topological Anosov map which is not injective, (2) for each x E X define the stable set Wd(x) = { y E Tn : d( f n(x), f n(y)) --t 0 as n , 00); then the path connected component of x in Wd(x) is dense in Tn. The complicated behaviours of topological Anosov maps belonging to C(Tn) can be classified by use of topological methods. Among the main results to be obtained in this book are the following statements: (i) Every topological Anosov homeomorphism belonging to C('Ir") is topologically conjugate to a toral automorphism of type (I). (ii) Every positively expansive map belonging to C(Tn) is topologically conjugate to a toral endomorphism of type (11). (iii) Every strongly special topological Anosov map belonging to C(Tn) is topologically conjugate to a toral endomorphism of type (111). (iv) If a topological Anosov map which is injective is neither positively expansive nor strongly special, then its inverse limit system is topologically conjugate to a solenoidal group automorphism, in other words, to the inverse limit system of a toral endomorphism of type (111).

Ergodic Theory. This book contains arguments on measure-theoretical aspects of continuous surjections of a compact metric space. At the origins of ergodic theory were problems in statistical mechanics, related to Hamiltonian flows, one-parameter groups of diffeomorphisms preserving Liouville measure, geodesic flows etc. However, many theorems in ergodic theory are formulated in a much simpler setting, i.e. the main objects of the (ergodic) theory are measure preserving transformations of measure spaces. In the last chapter of this book we shall discuss some results (described below) of ergodic theory to apply them to topological dynamics. Let X be a set. A u-algebra of subsets of X is a collection B of subsets of X such that (1) B contains the entire space X , (2) X \ B E B when B E B, (3) Ui2, B; E B when each Bi contains in B.

A measure space is a triple (X,B,m) where m is a function m : B --t R+ satisfying m(U;=, B,) = C;==,m(Bn) if {B,) is a pairwise disjoint sequence of elements of B. If m(X) = 1 then (X, 23, m) is called a probability space. We say that a map f : X + X is a measure preserving transformation if E E B implies f -'(E) E B and m(f -'(E)) = m(E). Here m is said to be f -invariant. Let f : X 4 X be a measure preserving transformation of a probability space (X, B, m). For a function ( : X 4 R such that Jx I(ldm < oo we define <(fi(x)) and the space mean of 6 to the time mean of ( to be lim,,,

~yzl

INTRODUCTION

12

Sx

be ((x)dm. The first major result in ergodic theory is the ergodic theorem which was obtained in 1931 by G.D.Birkhoff. The ergodic theorem says that these means are almost everywhere equal if and only if f: X -, X is ergodic, i.e. if f ( E ) = E and E E L? then m ( E ) = 0 or m(E) = 1. Such a measure m is called an ergodic measure. Let X be a compact metric space and 23 the family of Bore1 sets, i.e. the smallest a-algebra containing all open subsets of X. If C(X) is the Banach space of continuous real-valued functions of X with the sup norm, then every probability measure p on X induces a non-negative linear functional on C(X) by the map ( I+ S t d p . Conversely, the Riesz representation theorem says that to any non-negative linear functional J on C(X) with J ( l ) = 1, there corresponds a unique probability measure p on X such that J ( t ) = 1, (dp for € E C(X). The space M ( X ) of all probability measures on X is obviously a convex set. We may define a topology in M ( X ) by taking as a basis of open neighborhoods for p E M ( X ) the sets

with ~j > 0 and tj E C(X). This topology is called the weak topology of M(X). With this topology M ( X ) becomes a compact metrizable space. A sequence pn E M ( X ) converges to p if and only if one of the following equivalent conditions holds : (1) limn,, J (dpn = 1(dp for all t E C(X), (2) liminf,,, pn(U) 2 p(U) for all open U c X , (3) limn,, pn(A) = p(A) for all A E 23 with p(3A) = 0. Here 8A denotes the boundary of the set A. The support Supp(p) of a probabiity measure p is the smallest closed set C with p(C) = 1. Equivalently, Supp(p) is the set of all x E X with the property that p(U) > 0 for any open U containing x. Let f : X -, X be a continuous surjection of a compact metric space. It can be shown that the set M ( f ) of all f-invariant probability measures on X is nonempty and a compact convex subset of M(X). As before a : Y : -, Y : is the shift map for a fixed k 2 1. A probability measure p on Y : is a-invariant if for all cyclinders [ae, . ,am],p satisfies p([ae, ,a,]) = p ( [ ~ e + ~ , ,am+l]). In the set of a-invariant probability measures, Bernoulli measures are important. The k-tuple .~r= (pl, - . ,pk) with p j 2: 0 and p j = 1 defines a Bernoulli measure p by

..

x:=,

..

-

This is the simplest type of probability measures on .Y : is called a Bernoulli shift if p is a Bernoulli measure.

The system (Yt, p, a)

INTRODUCTION

13

Probability spaces ( X , B , p ) and ( X ' , Br,p') are said to be isomorphic if there are M E B with p ( M ) = 1, M' E B' with pl(M') = 1 and a bijection h: M -+ M' such that h and h-' are measurable and p(h-'(A')) = p1(A') for all A' in the restriction of B' to MI. If a measurable map h: M + M' is surjective and satisfies p(h-'(A')) = p(At) for all A' in the restriction of B' to MI, then we say that ( X ' , B', p') is a factor of ( X ,B,p). Let f: X + X and f' : X' -+ X' be measure preserving continuous maps of compact metric spaces. Then ( X ,B, p, f ) and ( X I ,B', p', f ' ) are said to be measure theoretically conjugate if h is a measurable bijection and satisfies h o f = f' o h. When h is a measurable surjection and satisfies h o f = f' o h, ( X ' , B', p', f ') is called a factor of ( X ,B, p, f ). A system ( X ,B, p, f ) is called a Bernoulli shift if the system is measure theoretically conjugate to a Bernoulli shift (Y:, p, u). A probability measure p is said to be absolutely continuous with respect to v, written p << v , if every set of v measure zero has also p measure zero. In this case, the Radon-Nikodym theorem says that there exists a v-integrable function ( such that p ( B ) = SB(dv for all B E 8. I f p << Y and Y << p then p and v are said to be equivalent. It can be shown that if p and v are ergodic measures in M ( f ) and p << v, then p = v, and that ergodic measures in M (f ) are exactly the extremal points of M ( f ). Here we meen by an extremal point one that can not be written in the form apl ( 1 - a l p 2 with p1, p2 E M ( f ), p1 # p2 and 0 < a < 1. The Krein-Milman theorem says that every invariant probability measure is a limit of convex combinations of ergodic measures. A measure p E M ( f ) is said to be strongly mixing if

+

and weakly mixing if

1 n-l lim l p ( A n f - j ( B ) ) - p(A)p(B)l = 0 n

n-a,

j=O

for all A , B E B. By definition a strongly mixing measure is weakly mixing, and a weakly mixing measure is ergodic. It can be shown that a factor of an ergodic system ( X ,23, p, f ) is ergodic, and that the product of two ergodic systems need not be ergodic, and moreover that the product of an ergodic system with a weakly mixing system is ergodic. A system is weakly mixing if and only if the product with itself is ergodic. To the measure theoretical notions of mixing and ergodicity correspond similar concepts in the topological framework. A continuous surjection f : X -+ X of a compact metric space is said to be minimal if the orbit of f for every x E X is dense in X , equivalently if there are no non-trivial closed f-invariant subsets of X . A point x E X is said to be recurrent if, for every neighborhood

14

INTRODUCTION

E U \ { x ) . The set R of all recurrent points is a Bore1 set and p(R) = 1 for all p E M ( f ) . This is, in essence, PoincarC's recurrence theorem which is one of the oldest results in ergodic theory. Let p E M ( f ) be positive on all nonempty open sets of X. It can be shown that f is topologically transitive if p is ergodic, and topologically mixing if p is strongly mixing. However, there exits an example of topologically transitive homeomorphism such that some measures in M ( f ) are positive on all nonempty open sets but none of them is ergodic.

U of x, there exist infinitely many n with fn(x)

The hyperbolicity considered in this book is uniform hyperbolicity. The study of non-uniformly hyperbolic dynamical systems was initiated by Pesin. In recent years it has become increasingly important in the chaotic theory. Nevertheless this book does not treat the theory of non-uniformly hyperbolic dynamical systems. A useful survery of the theory is given by Pesin [Pe], Douady [Do], and Katok and Streleyn [K-S].

CHAPTER 1 Some Properties of Anosov Systems

We shall present in this chapter basic properties of differentiable dynamics, which will be needed for understanding later chapters, and provide an adequate foundation for the topological theory of dynamical systems. The readers may skip this chapter if they like.

51.1 Toral endomorphisms

.

Let X be a set. For x, y E X denote a product of x and y by x y. We say that X is a group if this operation satisfies the following conditions : (1) if x, y, z are any three elements in X , then x (y z) = ( 2 . y) z, (2) there exists the identity element e in X with the property that x .e = e x = x for every x in X , (3) to each x in X there corresponds another element in X , called the inverse of x and written x-', with the property that 3.2-' = x-' .x = e. A group X is called an abelian group if x y = ye x for every x, y in X . If X is abelian, then we shall use additive operation as the operation of X. A group X is said to be a topological group if X is a topological space and for x, y E X , the operations (2, y) H x y and x H x-I are continuous. Let X be a topological group and H a subgroup of X. If H is closed, then H, endowed with the topology induced from X , is called a topological subgroup (simply a subgroup). The fact that the group operations are separately continuous in H is easy to see. Moreover, if H is a normal subgroup of X (i.e. sHx-' = H for x E X), then X I H , the collection of all distinct cosets {xH : x E X), forms a group called a factor group of X . Let .rr : X -+ X / H denote the natural projection. As usual, we define a quotient topology on X / H as follows. A set w in X / H is open if and only if T-'(w) is an open subset of X. It is easy to see that with this topology X / H is a topological group. Let n be a natural number. We denote as Rn the n-dimensional euclidean space and as Zn the lattice of points with integer coordinates. With usual operations Rn is a (vector) group and Zn is a subgroup of Rn. The factor group Rn/Zn = {a: Zn : x E Rn) is called an n-torus and we write Tn = Rn/Zn. A metric for Tn is defined by

-

.

.

+

.

CHAPTER 1

16

where 2 denotes the euclidean metric for Rn, and then the metric d is compatible with the quotient topology of 11'". Let 2 = (aij) be an n x n matrix where each aij is an integer. Since 2% E Zn for all x E Zn, we have 2 ( .Z n .) c Zn. In particular, if the determinant of 2 equals f1, then 2 ( Z n ) = Zn because is a matrix with integer elements. 2 induces a linear map fx : Rn -+ Rn, which is defined by fx(x) = Obviously Ax for x E Rn. To avoid complication we identify fx with 2. Since x ( x Zn) c 2 ( x ) Zn for x E Rn, we obtain a continuous map A : Tn -t T" by

z-'

+

+

A(x

+ Zn) = 2 ( x ) + Zn

for x E Rn,

which preserves the group operation and satisfies a relation A o ?r = ?r o 2 where ?r : Rn -t T" is the natural projection, namely, the following diagram commutes. A R" R"

-

The linear map is called a lift of A by ?r and sometimes said to cover A. A homeomorphism of a topological abelian group is an automorphism if it preserves the algebraic operation (i.e. f (x y) = f (x) f (y)). When a continuous surjection satisfies the same condition, it here is called an endomorphism. Observe that every endomorphism of an n-torus is induced from an n x n integer matrix with the determinant non-zero. A linear automorphism of Rn is said to be hyperbolic if it has no eigenvalues of modulus one. Let A : Tn -t Tn be an endomorphism and denote as : Rn -+ Rn the linear map which covers A. If 2 is hyperbolic, then A is called a hyperbolic toral endomorphism. In this case A is one of the following types. (I) A is an automorphism. An endomorphism of this type is especially called a hyperbolic tom1 automorphism. (11) A is an endomorphism and 2 is ezpanding, i.e. all eigenvalues of 2 have absolute values greater than one. An endomorphism of this type is especially called an expanding toral endomorphism. (111) A is an endomorphism which is not injective nor of type (11), i.e. there is at least one eigenvalue of 51 with absolute value less than one. An endomorphism of this type is especially called a hyperbolic tom1 endomorphism of type (III). The following matrices give simple examples of toral endomorphisms of types (I), (11) and (111) respectively.

+

+

$1.1Toral endomorphisms

17

A toral automorphism of type (I) and a toral endomorphism of type (11) have the strong property of being structurally stable (see $1.2). However, a toral endomorphism of type (111) does not have such a property (see a result of Przytycki [Pr] stated in $1.2). Let A : 'lr" -+ Tn be an endomorphism and 'll : Rn -+ Rn the linear map which covers A. Since A is surjective, 2 is bijective and 'll(zn) C Zn. Let G be the character group of Tn (see Chapter 7 for definition).

Remark 1.1.1. Let K be a subgroup of T such that the factor group T n / K is n-dimensional. Then K is a finite group. This shows that the set of all periodic points, Per(A), is dense in Tn. Indeed, let p be a prime number greater than the absolute value of the determinant of 'll and for m a natural number let G, = {pmy : " IG}. Then G1 3 G2 3 > 00 Gm = (0). Define a map A : G -+ G by (X(y))(x) = y(A(z)) for y E G and a: E 'P. Then X(G,) c Gm for all m. Let K , be the annihilator of G, in Tn (K, = ann(Tn, G,)). Since the rank of G, equals that of G, the factor group F l K , is n-dimensional. Thus K, is a finite group and so A(Km) C Km. By the choice of p it follows that A(K,) = K,, and so each of K m is a set of periodic points. Since cl(U&, K,) = cl(ann(Tn, G,)) = Tn, it follows that UZzl K, is dense in T. Here cl(E) denotes the closure of E in Tn.

nm=l

nZZ1

Let p be a Lebesgue measure of Tn with p(Tn) = 1 (probability Lebesgue measure). The details for the ergodic theory will be prepared in Chapter 11.

Remark 1.1.2. An endomorphism A : Tn -t Tn preserves the measure p (i.e. p ( E ) = p(A-l(E)) for all Borel sets E ). Such a map is called a measure preserving tmnsfonncrtion. Indeed, let m(E) = p(A-l(E)) for Borel sets E. Then m is a probability Borel measure and

Since A is surjective, by the uniqueness property of probability Lebesgue measure we have p = m. Let (X, B,p) be a probability space and f : X -+ X a measure preserving transformation. We say that f is ergodic if for B E B one has p(B) = 0 or p(B) = 1 whenever f - l ( B ) = B.

Remark 1.1.3. An endomorphism A : Tn -+ Tn is ergodic with respect to a Lebesgue measure p if and only if for y E G one has y = 1 whenever yo =7 for some l > 0. Suppose that if y E G and y o Ae = y for some t! > 0 then we have y = 1, and that 5 E L2(p) and ( o A = (. Here L2(p) denotes the Hilbert space consisting of square integrable functions. Since ( has the Fourier expansion

CHAPTER 1

18

= C a,? where C la,12 < oo, it follows that C a,y(A(x)) = C a,y(x). If are all distinct, then their corresponding coefficients are y, y o A, y o A2, equal and therefore zero. Thus, if a, # 0 then y o Ae = y for some 1 > 0. Then y = 1 by assumption and so ( is constant p-a.e., which implies that A is ergodic. Conversely, let A be ergodic and y o = y for some 1 > 0. If 1 is the least such integer, then ( = y y o A . . 7 o A ~ - ' is A-invariant and by ergodicity we have y = 1.

+

+ .+

Remark 1.1.4. An endomorphism A : Tn -+ Tn is ergodic (with respect to the probability Lebesgue measure p) if and only if 2 has no roots of unity as eigenvalues. We can identify with G = Zn the character group G of Tn. Then ';;IE = -

'x denotes the transposed matrix of 2. If A is not ergodic then 2' n' -z = m for some m # 0 and 1 > 0. Then 'ze,and so -e A , has 1 as an eigenvalue. Thus 2 has an Gth root of unity as an eigenvalue. Conversely, if '2 has an t-th root of unity as an eigenvalue, then 'zehas 1 as an eigenvalue and so 'ze- I induces a singular linear map of Rn where I denotes the identity matrix. Thus 'ze- I induces a many-to-one map from A(m) for m =

t En. Here

(:I)

m,

Zn into En. Therefore there is 0 # m E Zn such that

'zem = 114.

Remark 1.1.5. An endomorphism A : Tn + Tn is ergodic if and only if there is xo E Tn such that the orbit OA(xO)= { A i ( x O ) ) ~isOdense in Tn. Suppose A is ergodic. Let { U k } & b e a countable base for P. Then we A-~(T" \ Uk) (see $52 and have that {x E Tn : cl(OA(x)) # Tn) = 4 of Chapter 2). Since A-i(Tn \ Uk) is A-invariant (p-a.e.), by ergodicity we have measure 0 or 1. Notice that the Lebesgue measure is a probability Bore1 measure of Tn which gives a positive measure to every nonempty open set. Since Uk is open, we have p ( n z o A-~(T" \ Uk)) = 0 and so p({x E F : c1(OA(x)) # F ) ) = 0. Therefore, p({x E Tn : c l ( O ~ ( x ) = ) Tn)) = 1. Conversely, suppose OA(xO)is dense in Y. Since Tn is connected, OAi(xo) is dense in Tn for all i > 0. If y o = y for some 1 > 0 and a character y, then we have y = constant. Therefore A is ergodic by Remark 1.1.3.

nzo

UEl

Remark 1.1.6 (Kroneker). Let Z[x] denote the ring of polynomials in x with integer coefficients. If all the roots X of p(x) E Z[x] has [XI = 1, then each root X is a root of unity. Remark 1.1.7.

Let A : P -t P be an automorphism corresponding to

$1.2 Anosov differentiable systems

Then the eigenvalues of the matrix are (3 -

Thus A :

JS * i

+ T4 is

R e m a r k 1.1.8. matrix

J G ) / 4

( both absolute value = I),

ergodic. We leave the proof to the readers.

Let

2 : W2

+

R2 be an linear map corresponding to the

Then the eigenvalues of the matrix are (3 f 4i)/5 ( absolute values = 1). Denote as G the smallest subgroup generated by Urn d ( Z 2 ) , then G is an A-invariant subgroup and Z2 C G = X(G) C W2. Suppose G is endowed with the discrete topology. Then the character group of G, S2, is a solenoidal group (see Chapter 7 for details). Since Z2 # G, S2is not a torus. As described in Chapter 7 an automorphism A : S2 -, S2 is induced by the transposed matrix t?l. Then A preserves the probability Haar measure. By applying Remark 1.1.3 we can check that A is ergodic. However, notice that Remark 1.1.4 does not hold. Though ergodic automorphisms are not necessarily hyperbolic, it was proved (Katznelson [Ka]) that every ergodic toral automorphism is a Bernoulli shift. After this the same result was checked for solenoidal group case ([Ao~],[Li], [Mi-TI], [Mi-TZ], [Mi-T31). Aoki's proof [A061 is much simpler than the other extant proof. $1.2 Anosov differentiable systems

The survey presented in this section is written in view of the differentiable dynamics, and to give the reader an understanding of some of the basic properties in Anosov differentiable systems. Let M be a closed smooth manifold, i.e. a compact connected smooth manifold without boundary. A C1 diffeomorphism f : M + M is called an Anosov diffeomorphism if there is a splitting of the tangent bundle TM = Ed@EU which is preserved by the differential D f off and if there are constants C > 0, 0 < p < 1 and a Riemannian metric 11 11 on TM such that for all n 2 0 (1)

IIDfn(v)(l I CpnllvII

(2)

(IDfn(v)ll

> C-lp-"llvII

for v E E8, for v E EU.

20

CHAPTER 1

A typical example of such diffeomorphisms is the toral automorphisms of type (I). The definition of Anosov diffeomorphism does not depend on the choice of Riemannian metrics because M is compact. For x E M define the following subsets : W"(X) = {Y E M : d(fn(x), fn(y)) -t 0 as n -t oo), W,d(x)= { Y E M : d(fn(x), fn(y)) 5 E for all n 2 0), WU(x)= {Y E M : d(f-n(z), f-n(y)) + 0 as n + m), w t ( ~= ) {y E M : d(f-"(x), f-"(y)) 5 E for all n 2 0) where d is the metric for M induced by a Riemannian metric on T M . Then the behaviour of Anosov diffeomorphisms is characterized as follows. For E > 0 small (A) W,"(x) and W,"(x) are C1 disks, T,Wl(x) = Ei and T,W,U(x) = E">I (B) there exist constants c > 0, 0 < X < 1 such that for all n 0

>

(C) W,d(x) and W,U(x) vary continuously with respect to x. See Hirsch and Pugh [H-PI. From (B) we have W,"(x) c WU(x) for x E M (a= s,u) and then

Because W,d(x) intersects W,"(x) transversally and W,d(x) n W,"(x) = {x), we have that for E > 0 there iis 6 > 0 such that WF(x) n W,b(y) consists of a single point, say [x,y], whenever d(x, y) < 6. Then the map

is continuous. For the interesting properties of Anosov diffeomorphisms, the readers may refer to Anosov [Anl], Smale [Sml, Sm21, Franks [Fl, F2], Manning [ManS], and Brin and Manning [B-MI. Let C1(M, M ) be the set of all C1 maps of a closed smooth manifold M endowed with the C1 topology. A map f E C'(M, M) is said to be expanding if there are constants C > 0, 0 < p < 1 and a Riemannian metric )I 11 on T M such that for all n 2 0 and allu~TM IlDf n(v)ll 2 CP-~IIVII.

51.2 Anosov differentiable systems

21

A typical example of such maps is the toral endomorphisms of type (11). If f : M -+ M is expanding, then there exist constants 60 > 0, 0 < X < 1 such that for x, y E M under some Riemannian distance. From (D) it is checked that every expanding differentiable map has fixed points (see the proof of Theorem 2.2.14 of the next chapter). However it is unknown for Anosov diffeomorphisms. For the properties of expanding differentiable maps, the readers may refer t o [F2], Shub [Shl], Gromov [Gr], and Shub and Sullivan [S-S]. The above two classes of dynamical systems were extended to a wider class by Przytycki [Pr] as follows. A map f E C1(M, M ) is called an Anosov diferentiable map if f is a C1 regular map and if there exist constants C > 0, 0 < p < 1 and a Riemannian metric 11 11 on T M such that for a sequence (x,) of points in M satisfying f (x,) = xn+l for every integer n, there is a splitting

which is preserved by the differential D f , and conditions (I), (2) mentioned above are satisfied. Note that E,", depends on the sequence (x,). Thus it may happen that E:o # E;o if x0 = yo but (x,) # (y,). Such a phenomenon is impossible for E& , that is, it depends only on xo. Let (xi) be a sequence of points in M satisfying f (xi) = xi+l for i E Z and define a set

where d is the metric for M induced by a Riemannian metric on T M . The behaviour of Anosov differentiable maps was characterized in [Pr] as follows. For E > 0 small (E) W,"(x) and W,U((xi)) are C1 disks, T,W,"(x) = Ei and Tz,W,U((xi)) = Ego9 (F) there exist constants c > 0, 0 < X < 1 such that for all n 2 0 d(fn(x), fn(y)) 5 cXnd(x,y) for Y E W,6(z), d(~-,, Y-,) cXnd(~o,yo) for some (yi) with yo E W,U((xi)),

<

(G) W,"(x) varies continuously with respect to x, and if ( x l ) converges to (xi) as n + oo then W,U((xq)) varies continuously to W,U((xi)). If a sequence (xi) satisfies xo = x, then W,d(x) intersects W,U((xi)) transversally and W,"(x)n W,"((xi)) = {x). Thus we have that for E > 0 there is S > 0

CHAPTER 1

22

such that W,"(x)nW,"((yi)) is exactly one point [x, (yi)] whenever d(x, yo) < 6. Then the map

is continuous. Here Mf = {(xi) : xi E M and f (xi) = zi+l,i E Z) is a closed set of the product topological space M Z = {(xi) : xi E M , i E Z). A map f E C1(M, M) is called a special Anosov differentiable map if f is an Anosov differentiable map and E," does not depend on the sequence (x,) with XC, = x. Thus every special Anosov differentiable map has a splitting of the tangent bundle as well as diffeomorphisms. A typical example of such a map is the toral endomorphisms of type (111). The class of special differentiable Anosov maps contains Anosov diffeomorphisms, expanding differentiable maps and the products of them. For the properties of Anosov differentiable maps and special Anosov differentiable maps, the readers may refer to Maii6 and Pugh [M-PI and Przytycki [Prl. A diffeomorphism f : M + M is said to be expansive if there is a constant e > 0, called an expansive constant, such that if x, y E M and x # y then

for some n E Z.We say that a differentiable map f: M -, M is positively expansive if there is e > 0 such that x # y implies d(fn(x), fn(y)) > e for some n > 0. When there is e > 0 such that for (xi), (yi) E Mf if d(xi, y;) 5 e for all i E Z then (xi) = (yi), the differentiable map f : M M is said to be c-expansive. A sequence of points {xi E M : a < i < b) is c d e d a 6pseudo orbit of f if d(f (xi),xi+l) < 6 for i E (a, b - 1). For given E > 0 a &pseudo orbit {xi) is said to be €-traced by x E M if d(fi(x),xi) < E for i E (a, b). Here a and b are taken as -00 a < b 5 ca if f is bijective, and as 0 a < b ca if f is not bijective. We say that f has the pseudo orbit tracing property (abbreviated POTP) if for E > 0 there is 6 > 0 such that every &pseudo orbit of f can be €-traced by some point of M. For Anosov systems we have the following

<

<

<

Theorem 1.2.1. (1) every Anosov diffeomorphism has expansivity and POTP, (2) every expanding diflerentiable map has positive expansivity and POTP, (3) every Anosov diflerentiable map has c-ezpansivity and POTP. Proof. Expansivity of (1) and (3) follows from the facts that W,"(x) n W,U(x) = {x) and Wi(x) n W,U((xi)) = {x) respectively, and that of (2) follows from (*). We give the proof of P O T P of (I), (2) and (3). It is clear that if N > 0 and E > 0 then there is a > 0 such that each finite a-pseudo orbit {xi : 0 5 i 5 N ) satisfies d(f i(xo),xi) < E (0 i 5 N).

<

$1.2 Anosov differentiable systems

23

We first show POTP of ( 1 ) by using the proof due to Bowen [Boll. It is enough to see that for P > 0 there is a > 0 such that each a-pseudo orbit {xi : 0 i 5 b ) is P-traced by some point. Indeed, let {zi : i E Z) be an a-pseudo orbit of f . For k > 0 we set xi = zi-k (0 5 i 2k). Then { x i : 0 5 i 5 212) is an a-pseudo orbit o f f , and so it is P-traced by some point wk E M . It is easily checked that an accumulation point of { f k ( w k ) )P-traces

<

{~i).

Let

E

<

> 0 be a number to be determined later and 6 > 0 such that

Choose N so large that C X ~ E< 612 and then there is a > 0 such that if {yi : 0 i 5 N) is an a-pseudo orbit then d ( f i ( y o )yi) , < 612 for 0 i 5 N . Let r > 0 and consider an a-pseudo orbit {xi : 0 5 i 5 r N ) . Put xkN = X,N and then d ( f N ( x ( , - l ) ~ ) , x :<~ 6. ) Take yiN E W , " ( f N ( x ( , - l ) ~ )n) W,"(zkN) and put = f-N(y:N). Then we have

<

<

~i,-~)~

) 6. By induction {xiN : by the choice of a. Thus d(f N ( ~ ( , - 2 ) N ) , ~ l ( , - ~ ) ~ < 0 5 i 5 r ) is constructed. Now set x = xo and for 0 5 i 5 r N choose s such that sN 5 i < ( s l ) N . Then we have

+

since f N ( xIt N )E W : ( X ' ( ~ - ~ )On ~ )the . other hand, since f N ( x b N ) E w;(fN ( x d N ) )we , have d(f i-dN(x:N), f i - ~( x~ , ~ ) 5 ) E and by the choice of a, d ( fi - d N ( x d ~x )i ), < 612. Thus

This is less than P if E is small. ( 1 ) was proved. Since (F) holds, the proof of POTP of (3) is very similar to that of (1). To show POTP of ( 2 ) , let and X be as in (D). For 0 < E < So choose 6 with 0 < 6 / ( 1 - A ) < E such that Usl(l-x)(f( x ) )C f(U,(x)) for x E M. Here U,(x) denotes an open ball with radius E centered at x. Let { x , : n 0 ) be a 6-pseudo orbit of f . Fix n 2 0. For x(O) E Ua(f(3,)) there is x(') E U,(x,)

>

CHAPTER 1

24

such that x(O) = f ( x ( ' ) ) . Since d(x('),x,) A-'d(x('), 2,). Therefore

< E,

we have d ( f ( x ( ' ) ) f, ( x n ) ) 2

from which there is x ( ~ E) U,(X,-~) such that x(') = f(x(')). Since d ( ~ ( ~ ) , x,-') < E , we have d(x,-', x ( ~ )<) 6(X2 A) and obviously d ( ~ ( f~(x,-2)) ), < 6(X2+ X 1). In this fashion we have that for x(') E Ua(xi+...+x+l)( f (x,-i)) there is x('+') E U,(X,-~) such that

+

+

If i = n then d(x(,+'), xo) < E , and for 0 < i

5n

Since n is arbitrary, the conclusion is easily obtained. Theorem 1.2.2. Let A : T 2 -+ T 2 be a toral endomorphism. Then there exists a Cm regular map g: T 2 + T 2 near to A under C0 topology such that (a) i f A is a hyperbolic automorphism then g is an expansive Cm difleomorphism with POTP that fails to be Anosov, (b) if A is a hyperbolic endomorphism of type (111) then g is a c-expansive Cm regular map with POTP that fails to be Anosov.

For the proof we need some preparations and so we shall give it in Chapter 9.

.

Remark 1.2.3 (Lewowicz [Ll]) An example of expansive diffeomorphism that fails to be Anosov is constructed as follows. We define a diffeomorphism g: T 2 + T 2 by

Then g is non-hyperbolic at x = y = 0 because of

But g is expansive. This is followed by defining a Lyapunov function V ( ( x l ,yl), ( ~ 2 3 ~ 2 by ) ) v = -(Y2 - Yl)(Y2 - 3 2 - ( Y l - 2 1 ) ) .

51.2 Anosov differentiable systems

25

In fact, V : T2 x T2 4 R is continuous, V(x1,x') = 0 for x' E T2 and V(g(x1), g(y1)) -V(xt, y') > 0 whenever 0 < d(xt, y') 5 a for a > 0 small enough. Then g is expansive. This is easily checked as follows. Let 0 < e < a. Then e is an expansive constant for g. Indeed, suppose x # y and d(gn(x),gn(y)) 5 e for all n E %. If we have V(x, y) 2 0, then V(g(x),g(y)) 2 E for some E > 0 since V(g(x), g(y)) > V(x, y). Since T2 is compact, there exists 6 > 0 such that for u, v E T2 d(u, v) < 6 =$ V(S(U),g(v)) < E, from which we have d(x, y) 6. Since V(g2(x),g2(y)) > V ( g ( ~ ) , g ( ~ ) ) E , we have d(g(x), g(y)) 2 6 and by induction

>

>

Define Y

= min{V(g(u), g(v)) - V(U,v) : (u, v ) E T2 satisfying 6 5 d(u, v) Ie}.

Then we have

and thus V(gn(s), gn(y))

> nu + V(x, Y) L nv,

n L 0.

However, we have max{V(u,v) : (u,v) E T2}< 00,thus contradicting. For the case V(x, y) 5 0 we obtain the conclusion by the same way. Let X and Y be compact metric spaces and let f: X -, X and g: Y -, Y be continuous surjections. Then f is said to be topologically conjugate to g if there exists a homeomorphism cp : Y 4 X such that f o cp = cp o g. If cp is a continuous surjection, then f is said to be toplogically semi-conjugate to g, in other words, f is called a factor of g. When f is topologically conjugate to g, let us define a relation f g. Then the relation is an equivalence relation on the class of all continuous surjections. If f g, then dynamics of f and g are regarded as the same in topological sense because there is a homeomorphic correspondence between orbits of f and orbits of g. A homeomorphism f : X -, X is topologically transitive if there exists xo E X such that the orbit {f n(xo) : n E Z) is dense in X . If, for all x E X , the orbit { fn(x) : n E Z) is dense in X , then f is called a minimal homeomorphism.

-

-

N

Remark 1.2.4.

Let f : X 4 X and g:Y 4 Y be homeomorphisms and suppose f is topologically conjugate to g. Then f is expansive if and only if so is g, and f has POTP if and only if so does g. Also, f is minimal if and only if so is g, and f is topologically transitive if and only if so is g. It is known that every expansive homeomorphism g:T2 4 T2 is topologically conjugate to a hyperbolic toral automorphism. See Hiraide [Hi51and Lewowicz [L2].

CHAPTER 1

26

Remark 1.2.5 (Coven and Reddy [C-R]). As an example of a positively expansive Cm regular map on the circle that fails to be expanding, we have the following: Let 7 : R + R be a C"O regular map satisfying (1) 7(0) =o, ?(I) = 2, (2) T ( 0 ) = j l ( l ) = 1, ~ ~ (>21)(0 < x < I), (3) f ( x ) = 2 n + f ( x - n ) ( n ~ x < n + l , n € Z ) . Let f:T1 --+ T1 be a Cm regular map of the torus induced by f . Then f is positively expansive but not expanding. This is checked as follows. Since Dfn(u) = u for n 2 1 and all tangent vectors u at x = 0, f is not expanding. To show positive expansivity o f f we first show that there is 6 > 0 such that if [?(s) - fi(y)l < 6 for all i 2 0 then x = y, that is, f : R -t R is positively expansive. After that we see that f is positively expansive by applying Lemma 2.2.34 of the next chapter. Choose 0 < 6 < 112 such that

and X

> 1 such that

Then we have the following : If f3(x) infinitely many i 2 0 such that

9

Z for all j 2 0, then there exist

for all n E Z. Indeed, if this is false and k is the largest integer with 1 7 ~ ( x-) nl 2 26 for all n E Z, then there exists n E Z such that either

Without loss of generality we suppose the former. -k+l (2) - n. By induction it is easily checked that 0 < ji(y) Let y = f for all i 2 0. The mean value theorem derives that

< 26

where 0 < ti < ?(y). Since f i(y) < 1, we have TJ)(t,)> 1. Therefore {?(y)) is increasing. Since it is bounded, ?(y) + z < 1 and then j ( z ) = z. But f (z) = fl(t)z where 0 < t < 1, and so j(z) # z. This is a contradiction.

51.2 Anosov differentiable systems

i

We now prove that 6 is an expansive constant for f. For x, y with x i+ j E Z for some i, j 0, then f (x), fi+j(y) E Z and so

f (x), f J ( y)

>

27

> y if

Suppose fi(x) $ Z for all j 2 0. The sequence {I?(x) - f(y)l) is nondecreasing. This follows from the fact that

f (f

for some ti with ?(y) < ti < (x). Since ~ ' ( 2 )# Z for all j E Z, there are infinitely many i > 0 such that (x) - nl 2 26 for all n E Z. For any such i and any n E Zwe have that

7(t;)>

>

and thus A. If ~ f ( x )- fi(y)l < 6 for all i 0, then 17~+l(?)if + l (y)~>A~~(~)-~~(~)~forinfinitelymanyi>0.Therefore{f(x)-~(~)) is unbounded, thus a contradiction. Therefore 7 is positively expansive. A map f E C1(M, M ) is said to be structurally stable if there is an open neighborhood N ( f ) o f f in C1(M, M ) such that g E N ( f ) implies that f and g are topologically conjugate. Anosov [Anl] proved that every Anosov diffeomorphism is structually stable, and Shub [Shl] showed the same result for expanding differentiable maps. Przytycki proved in [Pr] that Anosov differentiable maps do not be structurally stable; more precisely this is mentioned as follows, every non-empty open subset of the set of all Anosov differentiable maps of class C1, which are not diffeomorphisms nor expanding, contains an uncountable subset such that, if f # g are elements of its, then there exist no continuous surjections cp : M + M which make the commutative diagram

M-M 9

We denote as Diff (M) the set of all C' diffeomorphisms of a closed smooth manifold endowed with the C ' topology where 1 5 T 5 oo. For f E Diffl(M), MaiiC proved recently in [Ma31 that every structurally stable diffeomorphism of a closed smooth manifold satisfies Axiom A, and Palis showed in [PI that every a-stable diffeomorphism satisfies Axiom A. A diffeomorphism satisfying the conditions (1) all periodic points of f are hyperbolic, and (2) given all the pair (x, y) of periodic points, WS(x) and Wu(y) meet transversally,

28

CHAPTER 1

is called Kupka-Smale. It is well known that the set of all Kupka-Smale diffeomorphisms is a residual subset of Diff(M) for all T 2 1. Denote as P1(M) the set of f E Diffl(M) such that all the periodic points of f are hyperbolic. Obviously P1(M) is a residual subset of Diffl(M). In [A051 Aoki proved that every diffeomorphism belonging to the interior of p l ( M ) satisfies Axiom A and no cycles, and that every diffeomorphism belonging to the C1 interior of the set of Kupka-Smale diffeomorphisms satisfies Axiom A and strong transversality. Maiik showed in [Ma41 that every diffeomorphism belonging to the C' interior of the set of all expansive diffeomorphisms satisfies Axiom A and T,Wd(x) n T,Wu(x) = (0) for x E M , which is called a quasi-Anosov difleomorphism. It was proved in Sakai [S3] that each diffeomorphism belonging to the C1 interior of the set of all diffeomorphisms having POTP satisfies Axiom A and strong transversality. If the manifold M is a torus, then it is well known that every Anosov diffeomorphism is topologically conjugate to a toral automorphism of type (I) (Manning [ManS]) and every expanding differentiable map is topologically conjugate to a toral endomorphism of type (11) (Shub [Shl]). However the following problem is unsolved.

Problem 1. Is every special Anosov diflerentiable map of a torus topologically conjugate to a hyperbolic toral endomorphism ? One of aims of this book is to deal with the problem in topological setting. Moreover this book will present topological developments that could be the foundation for forward dealing with the following problem for the interesting class of topological Anosov maps. Let M be a closed topological manifold, i.e. a compact connected topological manifold without boundary. If a continuous surjection f : M + M is c-expansive and has POTP, then f is called a topological Anosov map (abbreviated TA-map). By Theorem 1.2.1 every Anosov differentiable map is topological Anosov. We say that f : M + M is a special topological Anosov map (abbrev. special TA-map) if f satisfies the conditions : ( 1 ) f is a topological Anosov map, (2) WU((xi))= WU((yj))for (xi), (yi) € Mf with xo = yo. It follows that every special Anosov differentiable map is special topological Anosov.

Problem 2. Is every special topological Anosov covering maps of an arbitrary closed topological manifold topologically conjugate to a hyperbolic infranil-endomorphism of an infm-nil-manifold ? Let L be a connected simply connected nilpotent Lie group. Let C be a compact group of automorphisms of L, and let G = L C be the Lie group obtained by considering L as acting on itself by left translation and taking the semi-direct product of L and C in Diff(L). Let I' be a uniform discrete

91.2 Anosov differentiable systems

29

subgroup of G. Then I'nL is a uniform discrete subgroup of L and I'/(l?nL) is finite (see L.Auslander [Au]). If I' is torsion free, then the orbit space X = LlI' is a compact manifold. We call it an infm-nil-manifold. Let A: L -+ L be an automorphism of L such that by conjugating I' by A in Diff(L), AI'A-' c I'. Then A projects to a differetiable map of X. Such a map is called an infranil-endomorphism. Manning [Man31 proved that every Anosov diffeomorphism of an infra-nilmanifold is topologically conjugate to a hyperbolic infra-nil-automorphism. After this Brin and Manning [B-M] proved, by using a result related to growth of finitely generated groups of Gromov [Gr], that every Anosov diffeomorphism of an arbitrary closed smooth manifold satisfying a certain spectral condition is topologically conjugate to a hyperbolic infra-nil-automorphism. In [Gr] Gromov showed that every expanding map of an arbitrary closed smooth manifold is topologically conjugate to an expanding infra-nil-endomorphism. After this Hiraide proved in [Hi41 the same result for positively expansive maps. On the direction, he obtained that if a continuous map of a compact connected locally connected semilocally 1-connected metric space is open and positively expansive, then the space must be homeomorphic to an infra-nil-manifold and the map is topologically conjugate to an expanding infra-nil-endomorphism ([Hi6]). A topological space X is said to be semilocally 1-connected if for x E X there is a neighborhood U of x such that every loop contained in U with a base point x (i.e. a continuous map u : [O, 11 -+ U satisfying u(0) = u(1) = x) can be deformed continuously in X to one point. We refer the reader to Smale [Sml, Sm2], Bowen [Boll, Irwin [I], Palis and de Melo [P-MI, Shub [Sh2], and MaiiC [Ma21 for background information of Anosov Systems. In Halmos and von Neuman's treatise on ergodic theory, they discussed algebraic models which indicate a characteristic of some subclass of homeomorphisms. We can find in their paper [Hal-N] the following theorem.

Theorem 1.2.6. Let f : X -+ X be a homeomorphism of a compact metric space. I f f is topologically transitive and an isometry under some metric for X , then f is topologically conjugate to a minimal rotation on a compact metric abelian group. Proof. Let d be an isometry metric for X. By assumption there exists xo E X such that Of(xo) is dense in X. Define a multiplication * in Of (xo) by

for n,m E Z. Then we have

30

CHAPTER 1

from which the map * : Of( x o ) x Of( 2 0 ) 4 Of( x o ) is uniformly continuous and can therefore be extended uniquely to a continuous map * : X x X -t X. since d ( f - n ( x ~ ) f, - m ( x ~ ) = ) d ( f m ( x ~ )f , the map : Of("0) 4 Of( 2 0 ) (inverse) is uniformly continuous and can be uniquely extended to a continuous map of X . Thus X is a topological group and is abelian since Of( s o is ) a dense abelian group of X. Since

for all n E Z,we have f ( x ) = f ( x o ) * a: for all x E X , and therefore f is the rotation by f ( 2 0 ) .

CHAPTER 2 Dynamics of Continuous M a p s

It is sometimes said that the theory of dynamical systems is the study of the behaviours of orbits of maps. Among many ways of the study of dynamical systems, we select the path of general topology. We use expansivity and pseudo orbit tracing property, related to Anosov and Axiom A systems, as tools of study. This chapter provides these tools which are sufficiently powerful through out this book.

52.1 Self-covering m a p s Let X and Y be metric spaces. A continuous surjection f : X + Y is called a covering map if for y E Y there exists an open neighborhood V , of y in Y such that f-'(v.)= ( i # i ' i UinUi. = 0 )

UU; i

where each of Ui is open in X and flui : U; + V , is a homeomorphism. A covering map f : X + Y is especially called a self-covering map if X = Y. We say that a continuous surjection f : X + Y is a local homeomorphism if for x E X there is an open neighborhood U, of x in X such that f(U,) is open in Y and flu= : U, -+ f (U,) is a homeomorphism. It is clear that every covering map is a local homeomorphism. Conversely, if X is compact, then a local homeomorphism f : X -,Y is a covering map. This is stated precisely as follows: T h e o r e m 2.1.1 (Eilenberg [El). Let X be compact and f : X -t Y a continuous sujection. If f is a local homeomorphism, then them exist two positive numbers X and 8 such that each subset D of Y with diameter less than X determines a decomposition of the set f - ' ( D ) with the following properties: ( 1 ) f -'(D) = Dl U D2 U U Dk, ( 2 ) f maps each Di homeomorphically onto D , ( 3 ) if i # j then no point of Di is closer than 28 to a point of D j , ( 4 ) for every 11 > 0 them is 0 < E < X such that if the diameter of D is less than E then each diameter of D j is less than 7 . If, in addition, Y is connected, then there is a constant k > 0 such that f : X + Y is a 12-to-one map.

..

The numbers ( A l e ) in Theorem 2.1.1 are called Eilenberg's constants for the map f : X -, Y . For the proof we need the following lemma.

CHAPTER 2

32

Lemma 2.1.2. Let X be a compact metric space and a a finite open cover of X. Then there exists 6 = 6(a) > 0 such that for any subset A if the diameter of A is less than 6 then A c B for some B E a. Such a number 6 is called a Lebesgue number of a. Proof. If this is false, for any n > 0 there exists a subset A, such that diam(A,) = sup{d(a, b) : a, b E A,) 5 l / n and A, (t B for all B E a. Take x, E A, for n 2 1. Since X is compact, there is a subsequence {x,, ) such that x,, -t x as i -+ oo. Then x E B for some B E a. Since X \ B is compact, we have a = d(x,X \ B) = inf{d(x, b) : b E X \ B ) > 0. Take sufficiently large ni satisfying ni > 2/a and d(x,,, x) < a/2. Then, for y E A,, we have d(yl X) I d(y1 xni ) d(zni 3) Il / n i a/2 < a

+

+

and thus y E B. Since y is arbitrary in A,,, we have A,, dicting.

c B , thus contra-

Proof of Theorem 2.1.1. We first prove that there exists 8 > 0 such that if x # y and f (2) = f (y) then d(x, y) > 48. If this is false, then for any n > 0 there exist x,, y, 6 X with x, # y, and f (x,) = f (y,) such that d(x,, y,) 5 l / n . Without loss of generality we suppose that x, -+ x and y, 4 y as n -, oo. Then x = y. Since f: X + Y is a local homeomorphism, there exists an open neighborhood U, of x such that flu=: U, -t f (U,) is a homeomorphism. Thus x,, y, E U, for sufficiently large n and so f (x,) = f (y,), thus contradicting. Let k > 0 and define a set Yk by Yk={y E Y : gf-l(y) = k). Then we have Yk n Ykl = 0 for integers k # k'. Next, we prove that there is K > 0 such that Y = Yl U U YK and each of Yk is open. Since X is compact, by the above fact it follows that Yk = 0 for integers k larger than a certain K > 0. Thus Y = Yl U U YK.To show = ($1 , ,xk). We openness of Yk (1 5 k I K) take y E Yk. Then f can choose open neighborhoods U,, of xi such that flu=,is a homeomorphism and U,, n UZj = 0 if i # j. Clearly

... ...

nf=l

.

f (U,,) contains a neighborhood U(y) of y such that U(y) C We claim that Yk. Indeed, if this is false, for n > 0 there is yn E f (U,,) such that

$2.1 Self-covering maps

33

Iff-'(yn) = {x;L,---, x g , ~ ~ + ~ , - . . ) a n-td x i~( a s n + oo)forl 5 i 5 k+l, then we have that f (xi) = y and d ( ~ ixj) , > 48 for 1 5 i 5 k 1. Therefore, y E Yk+' U U YK. This is a contradiction. We are now ready to prove Theorem 2.1.1. Let 8 > 0 and K > 0 be as above. Since X is compact and f is a local homeomorphism, we can choose a finite open cover a = {Ul , ,Urn) satisfying (a) f (Ui) is open in Y, (b) f: cl(Ui) -t f (cl(Ui)) is injective, (c) diam(cl(Ui)) < 8 for 1 5 i 5 m. Here cl(E) denotes the closure of E. Let y E Y, then y E Yk for some k (1 5 k 5 K) and so define Q(y) = Yk n (n{f(Ui) : y E f (Ui))). Then Q(y) is an open neighborhood of y. Since y is arbitrary in Y, we can take yl,. ,ye such that U = {Q(yl), ,Q(ye)) is a finite open cover of Y. Let X > 0 be a Lebesgue number of U and let D be a subset satisfying diam(D) < A. Then D C Q(y) for some y E {yl,. ,ye). If y E Yk for some k, then f-'(y) = {xl, . ,xk) and xi E Uni for some Uni E a. Notice that f -'(y) n Uni = {xi) for 1 5 i 5 k. We now define Di = Uni n f -'(D) for 1 5 i 5 k. Then we have f(Di) = D for 1 5 i 5 k. Indeed, since Q(y) = Yk n f(Uni)), if x E D then x = f(z) for some z E U,,,. Thus z E Uni n f-'(2) c Uni n f-'(D) = Di and therefore x = f(z) E f(Di), i.e. D c f (Di). Since f (Di) = f (Uni n f -'(D)) c f (Uni) n D = D, we have D = f(Di) for 1 5 i 5 k. (1) was proved. Since DiC Uni and D C f (Uni), flDi maps each Di homeomorphically onto D. Thus (2) was proved. (3) is computed as follows. If i # j, then d(Di, D,) 2 d(Uni, Unj) 2 d(xi, xj) - diam(Uni) - diam(Unj) > 48 - 8 - 8 = 28. Here d(A, B) is defined by d(A, B) = inf{d(a, b) : a E A, b E B). To show (4) let q > 0. For 1 5 i 5 m there is 0 < ~i < X such that if d( f (x), f (z)) < ~i (f (x), f (z)E f (cl(Ui))) then d(x, z) < q. This follows from the fact that f : cl(Ui) -t f (cl(Ui)) is injective. Put E = min{ci : 1 5 i 5 m ) . If diam(D) < E, then we have diam(Di) < q for 1 5 i 5 k. Since Y = Yk and each Yk is open and closed, when Y is connected, we have Y = Yk for some k, i.e. f is k-to-one.

+

...

...

-

.

.

..

(n;=,

uFZl

In the remainder of this section we describe well known theorems that will be used in the sequel. Let X be a topological space. A path in X is a continuous map from the unit interval [0, 11 to X. If any two points in X are joined by a path, then X is said to be path connected. In general, a connected space need not be path connected. An arc in X is an injective continuous map from [ O , l ] to X. We say that X is arcwise connected if any two points in X are joined by an arc. It is clear that if X is arcwise connected then it is path connected.

Theorem 2.1.3. A Hausdorfl space X is path connected if and only if X is arcwise connected.

CHAPTER 2

34

Proof. Let f : [O,1] -t X be any path with f (0) # f (1). We now construct an arc h in X joining f (0) and f (1). Choose a closed subinterval Il = [allbl] C [O, 11 whose length 0 5 e(I1) = bl - al < 1 is maximal, satisfying the condition that f (al) = f (bl). The existence of such a subinterval is ensured by the fact that X is Hausdorff. Next, among all subintervals of [O, 11 which are disjoint from Illchoose an interval I2= [az,bz] of maximal length satisfying f (az) = f (bz). Continue this process inductively, we can construct disjoint subintervals of maximal lengths !(Il) 2 !(I2) 2 . 2 0 satisfying the condition that f is constant on the boundary of each Ij.

-

arc

path

Figure 1 Let a : [O, 11 + X be the unique map which is constant on each closed interval Ij and which coincides with f outside of these subintervals. Then a ( t ) coincides with f (t) for t E dIj. It is easily checked that a is continuous, and that for each point x E a([O, I]), a-'(2) c [O,1] is either one point or a closed interval. From now on we show that these conditions imply that A = a([O, 11) is the image of some arc joining a(0) and a(1). To do so choose a countable dense subset {tl, t z , . } C [0, 11. Define a total ordering of A by a(s) < a ( t ) if a(s) # a(t) for s < t. Then an order preserving homeomorphism h: [O,1] -+ A can be constructed inductively as follows. Set h(0) = a(0) and h(1) = a(1). For each dyadic fraction 0 < m/2k < 1 with m odd, suppose that h((m - 1 ) / 2 ~ )and h((m 1)/2" have already been defined. Then there exists the smallest index j such that

-

+

and define h(m/2'") = a(tj). It is not difficult to check that the h constructed in this manner on dyadic fractions extends uniquely to an order preserving map from [O,1] to A, which is a homeomorphism. By definition a topological space X is locally connected if each point in X has an arbitrary small connected neighborhood. There exists a corresponding concept of locally path connected, or locally arcwise connected.

$2.2 Expansivity

35

T h e o r e m 2.1.4. If a compact metric space X is locally connected, then X is locally path connected. Moreover every connected component of X is path connected.

Proof. Let E > 0. Since X is locally connected, there exists a sequence of numbers 6, > 0 such that any two points with d ( x ,X I ) < 6, are contained in a connected set of diameter less than €12,. We show that any two points x(0) and x(1) with d(x(O),x(l))< 60 can be . do this choose a sequence of jointed by a path of diameter at most 4 ~ To , eack of which divides the next. For denominators 1 = Ito < k1 < k2 < each fraction of the form i l k , between 0 and 1 we choose an intermediate point %(ilk,) satisfying two conditions. ( 1 ) Any two consecutive fractions i l k , and (i l ) / k , correspond to points x(i/k,) and x ( ( i l ) / k , ) which have distance less than 6,. Thus any two such points are contained in a connected set C ( i ,k,) which has diameter less than ~ 1 2 The ~ . second condition is the following: ( 2 ) Each point of the form ~ ( j / k , + ~ where ), j/k,+l lies between i l k , and ( i l ) / k n , belongs to the set C ( i ,k,), and thus

.

+

+

+

By induction on n the construction of such denominators 1, and intermediate points ~ ( i l k , )is straightforward. Thus we may suppose that z ( r ) has been defined for a dense set of rational numbers r in [0,11. Next we prove that the correspondence r H x ( r ) is uniformly continuous. Let r and r' be any two rational numbers such that x ( r ) and x ( r l ) are defined. Suppose that ( r - T I ( < l/k,. Then we can choose i l k , as close as possible to both r and r'. Then we have

and so

d ( x ( r ) ,x ( r l ) )< 4 ~ / k , . This proves uniform continuity. Thus there is a unique continuous extension t H x ( t ) which is defined for all t E [ O , l ] . We have constructed the required path of diameter 5 4~ from x ( 0 ) to ~ ( 1 )Therefore . X is locally path connected. The rest of the argument is straightforward. T h e o r e m 2.1.5. Let X and Y be Hausdorff spaces and f : X + Y a continuous sujection. If X is arcwise connected, then Y is arcwise connected.

Proof. Take arbitrary points yo, yl in Y . Since f is surjective, there exist xo, 2 1 in X such that f ( x o ) = yo, f ( x l ) = yl. Since X is arcwise connected, we can find a path u joining xo and X I . Then f o u : I + Y is a path joining 90 and yl. This shows that Y is arcwise connected.

CHAPTER 2

52.2 Expansivity Let X be a metric space with metric d. A homeomorphism f : X -t X is expansive if there is a constant e > 0 such that x # y ( x ,y E X ) implies

for some integer n. Such a constant e is called an expansive constant for f . This property (although not e ) is independent of the choice of metrics for X when X is compact, but not so for noncompact spaces. As one of notions that are weaker than expansivity we capture the notion which is called sensitive dependence on initial conditions. This is defined by the property that if there is 6 > 0 such that for each x E X and each neighborhood U of x there exist y E U and n E Z such that d(f n ( x ) ,f n ( y ) )> 6. From definition it follows that X has no isolated points. A homeomorphism f : X -, X is topologically transitive if there exists xo E X such that the orbit O f ( $ 0 ) = { f n ( x o ): n E Z ) is dense in X .

Remark 2.2.1. Let f : X + X be a homeomorphism of a compact metric space. Then f is topologically transitive if and only if for U, V nonempty open sets there is n E Z such that fn(U) n V # 8. Suppose O f ( x o )is dense in X and U,V are nonempty open sets. Then f " ( ~ 0 E) U and f m ( x o ) E V for some n and m. Thus f ' ( U ) )n V # 0 where k=n-m. To see the converse let {Ui : i > 0 ) be a countable base for X . For x E X we have

O f (x) is not dense in X O f (x)n Un = 0 for some n f m ( x ) E X \ Un for each m E Z 00

u

xE

0

f-m(X\Un)

for somen

m=-oo

Since U;=-, f-m(U,) is dense in X by the assumption, nz=-,(X \ f-m (Un))is nowhere dense and thus Uz=P=, f-m(X \ Un) is a set of first category. Since X is a compact metric space, { x E X : O f ( x ) is dense in X ) is dense in X .

nz=-oo

Remark 2.2.2 ([B-S]). Let f: X -, X be a homeomorphism of a compact metric space. Suppose X is an infinite set. If f is topologically transitive and

$2.2 Expansivity

37

Per(f) = {x E X : fn(x) = x for some n > 0) is dense in X , then f has sensitive dependence on initial conditions. This is easily checked as follows. We first show that Per(f) # X. For n > 0 let Fn = {x E X : fn(x) = x). Then Fn is closed in X and Fn = Per(f). Suppose Per(f) = X. Then there exists n > 0 such that int(Fn) # 0. Since f is topologically transitive, it follows that Fn = X , and so Fn is a finite set, thus contradicting. Notice that there is 6 > 0 such that for x E X there exists p E Per(f) such that d(x, Of(p)) > 6. Indeed, if this is false, then for n > 0 there is xn E X such that d(xn, Of(9)) 5 l l n for all q E Per(f). Since X is compact, without loss of generality suppose x, + x as n + oo. Then d(x, Of(9)) = 0 and thus x E Of(q) for all q E Per(f). This can not happen. Let A = 614. For x E X let U(x) be an arbitrary neighborhood of x in X . Then U = U(x) n Ux(x) is a neighborhood of x where Ux(x) = {y E X : d(x, y) < A). Take p E Per(f) n U with f n(p) = p for some n > 0 and choose q, E Per(f) satisfying d(x,Of(q,)) > 6. Since V = r);=-, f - j ( ~ x ( f j ( q ~ ) ) ) is a nonempty open set and f is topologically transitive, there is k E jZ such that fk(V) n U # 0. Notice that Ikl is chosen sufficiently large. Let j' > 0 be the integer such that [kl= nj' r where 0 <_ r < n and put j = j' 1. Then 0 < n j - lkl 5 n. Take y E f k ( V ) n U . Since k = - n j l - r when k < 0, we have

Urzl

+

since -n 5 n j

+

+ k < n. When k > 0 we have k = nj' + r and so

since -n 5 -nj

+ k 5 n. Since fnj(p) = p, when (1) holds,

If (2) holds, then we have d(f-nj(y), f-"j(p)) > 612. In any case we have d(f *nj(x), f f n j ( ~ ) )> A or d(ff "j(x), f fnj(p)) > A, i.e. f has sensitive dependence on initial conditions. Let f : X -+ X be a homeomorphism of a compact metric space. A finite open cover a of X is a generator (weak generator) for f if for every f-"(cl(An)) is at most one point bisequence {A,) of members of a, (nr=-, f-n(An) is at most one point). Here cl(E) denotes the closure of a subset E. These concepts are due to Keynes and Robertson [K-R].

CHAPTER 2

38

If a, /3 are open covers of X , then their joint a V /3 is given by

An open cover p is a refinement of an open cover a (written a 5 p ) if every member of p is a subset of some member of a. It is clear that a 5 a V p and p 5 a v p. If f : X -t X is a continuous surjection, then f-'(a) = {f-'(A) : A E a) is an open cover of X. It is easily checked that f-'(aVP) = f-l(a)Vf-l(P), f

(a)

f

( )

if

a

I P.

Theorem 2.2.3. Let f : X + X be a homeomorphism of a compact metric space. Then the following are equivalent. (1) f is expansive, (2) f has a generator, (3) f has a weak generator. Proof. (2) + (3) is clear (3) + (2) : Let /3 = {B1,.. ,B,) be a weak generator for f and let 6 > 0 be a Lebesgue number for P. Let a be a finite open cover consisting of sets Ai with diam(cl(A;)) 5 6. If {Ai,) is a bisequence in a, for every n there is j , such that cl(Ajn) C Bjn, and so f-"(cl( Aj,)) C f-"(Bj,). Therefore a is a generator. (1) + (2) : Let 6 > 0 be an expansive constant for f and let a00be a finite cover consisting of open balls of radius 612. Suppose x, y E f -,(cl( A,)) where A, E a. Then d( f ,(x), f ,(y)) 5 6 for every n, and by the assumption x = y. (3) =+ (1) : Suppose a is a weak generator. Let 6 > 0 be a Lebesgue number for a. If d(fn(x), fn(y)) < 6 for n E Z,then for n there is A, E a such that f "(x), f ,(y) E A, and so x, y E f-"(A,) which is at most one point.

nz=-,

Or=-,

nn=-,

n:=-,

Theorem 2.2.4. Let f : X -, X be a homeomorphism of a compact metric space and let k # 0 be an integer. Then f is expansive if and only if f k as expansive.

~2;'

Proof. This follows from the facts that if a is a generator for f then f -i ( a ) = a V f -l ( a ) V . . V f lkl-'(a) is a generator for f ', and that if a is a generator for f k then a is also a generator for f .

.

The following theorem is easily checked. Thus we leave the proof to the readers.

92.2 Expansivity

Theorem 2.2.5. ( 1 ) I f f : X -,X is expansive and Y is a closed subset of X with f ( Y )= Y , then f l y : Y -+ Y is expansive. ( 2 ) If fi : Xi -t X i (i = 1,2) are expansive, then the homeomorphism fl x f2 : X I x X 2 -+ X 1 x X 2 defined by

is expansive. Every finite direct product of expansive homeomorphisms is expansive. ( 3 ) If X is compact and f : X + X is expansive, then h o f o h-' : Y + Y is expansive where h: X + Y is a homeomorphism.

..

Let k be a fixed natural number and let Y k = {0,1, ,k - 1). Put the discrete topology on Yk and define the product space Y: = equipped with the product topology. Then a homeomorphism u : + Y, is defined ) )xi+l, ~ i E Z, for x = ( x i ) . u is called the shift map. by ( ( ~ ( 2 =

nrmYk

YE

Theorem 2.2.6. Let k 2 2. Then the shift map o : Y:

-+

Y: is expansive.

..

Proof, For 0 5 i 5 12-1 define Ai = {(x,) : xo = i ) . Then a = {Ao,. ,Ak-1) is an open and closed cover of Y:. If x E u-"(Ain) where Ain E a, then x = (. ,i - 1 , io,il , ) and thus a is a generator.

..

n= :-,

..

Theorem 2.2.7. Let f :X -t X be an expansive homeomorphism of a compact metric space. Then there exist k > 0 and a closed subset E of Y, such that Z is u-invariant ( u ( E ) = C ) , and moreover there is a continuous sujection ?F : E -, X satisfying ?r o u = f o ?F, i.e. the diagmm

TI X - X

commutes. f

Proof. Let 26 > 0 be an expansive constant for f . Choose a finite cover (Y = {Ao,. . ,Ak-1) by closed sets such that diam(Ai) < 6 for 0 5 i 5 k - 1, and such that Ai intersect only in their boundaries. Let 8 denote the union of the boundaries of Ai. Then 8 , = U =,:fn(8) is a set of first category and so X\a, is dense in X . For x E X\8, we can assign uniquely a member of y ,by x (an)

G

whenever f n ( x ) E Aan. Let A denote the collection of all sequences arising in this way. Then A C Y, and

CHAPTER 2

40

If cp is uniformly continuous, then cp : A -+ X can be uniquely extended to a continuous map T : C -, X , satisfying T o ~ ( ( x , ) )= f o ~ ( ( x , ) )for (x,) E C, where C = cl(A). It only remains to see uniform continuity of cp. Let E > 0. Since a is a generator for f , there is N > 0 such that each member of V> f n ( a ) has a diameter less than E . If this is false, then there is E > 0 such that for every j > 0 there are xj, y j E X with d(xj, yj) > E and Aj,; E a ( - j 5 i 5 j ) with xj, yj E ni=-j f -'(A~,;). Suppose x j -+ x and yj + y. Then x # y. Since a is finite, there is A. E {Aj,o : j E Z}such that xj, yj E A. for infinitely many j. Thus, x, y E cl(Ao). Similarly, for each n infinitely many elements of {Aj,, : j E Z} coincide and we have A, E a with x, y E f-"(cl(A,)). Therefore, x, y E f -,(cl(A,)). This is a contradiction. If (a,), (b,) E A and a, = b, for In1 5 N , then cp((a,)) and cp((b,)) are in the same member of f "(a) and thus d(cp((a,)), cp((b,))) < E . Therefore cp is uniformly continuous.

n;=-,

,v!

Let X be a metric space. A continuous surjection f : X -+ X is positively exjmnsive if there is a constant e > 0 such that if x # y then d(fn(x), fn(y)) > e for some non-negative integer n (e is called an expansive constant for f ). For compact spaces, this is independent of the compatible metrics used, although not the expansive constants.

Theorem 2.2.8. Suppose X is compact. Let k positively expansive if and only if so is f k .

> 0 be an integer. Then f is

Proof. Let f : X -+ X be positively expansive and e > 0 an expansive constant. Since f is uniformly continuous, there is 6 > 0 such that for 1 5 i 5 k

which implies that 6 is an expansive constant for f k . Similarly, the converse is proved. The following theorem is easily checked. Thus we leave the proof to the readers.

Theorem 2.2.9. (1) I f f : X -+ X is positively expansive and Y is a closed subset of X with f (Y) = Y, then f l y : Y -,Y is positively expansive. (2) If fi : Xi -+ Xi (i = 1,2) are positively expansive, then the continuous sujection fl x f2 : X1 x X2 -,X1 x X 2 defined by

is positively expansive. Every finite direct product of positively expansive maps is positively expansive. (3) If X is compact and f : X -+ X is positively expansive, then h o f o h-I : Y -+ Y is positively expansive where h: X -+ Y is a homeomo~phism.

$2.2 Expansivity

41

Theorem 2.2.10 (Reddy [Rl]). If X is compact and f: X 4 X is a positively expansive map, then there exist a compatible metric D and constants 5 > 0, X > 1 such that for x, y E X

For the proof we need the following result due to Frink [Fr]. Lemma 2.2.11. Let Vn be a decmasing sequence of open symmetric neighborhoods of the diagonal set A = {(x, x) : x E X ) of X x X such that

and Vn+l

0

vn+l

0

vn+l C

vn

for all n 2 1. Then there is a metric D compatible with the topology of X such that for n 1 1 1 Vn C {(x, Y) : D(x,Y) < -) C Vn-12n Here a subset A of X x X is said to be symmetric if A satisfies the property that (x,y) E A H (y,x) E A , a n d A o B is defined b y A o B = { ( x , ~ ): 3z E X s.t. (x, z) E A, (2, y) E B). Proof. Define a function ( : X x X

4

R by

...

, x,, and for ( x ,y) E X x X denote by c(x, y) a finite sequence x = XO, X I , xn+l = y. We write C(x, y) the collection of all such sequences c(x, y). Let us put D(X, Y) = €(xi, : ~ ( 5Y) , E C(X,Y)}. i

It is easily checked that for any points x, y, z in X (i) (ii) (iii) Since D(x, y) 5 ((x, y)

< 1/2n for (x, y) E Vn, we have

CHAPTER 2

If we hold that for n

21

nr=o

then x = y whenever D(x, y) = 0, since (2, y) E Vn = A. From (ii) and (iii) together with this fact, it follows that D is a metric compatible with the topology of X. To obtain (iv) it sufficies to show that for each finite sequence x = xo, XI,"', xn+1 = Y

For, if D(x, y)

< 1/2",

by (v) we have

and thus (x, y) E Vn-l. The remainder of the proof is only to show that (v) is true. If n = 0 then ((xo, xl) 5 2((xo, xl). Suppose (v) holds for 0 I k I n - 1, i.e. ((20, xk) I 2 ( ( X ~ , X ~ +Letting ~ ) . a = C~L:((xi, xi+l), we have either

~bzl

For the case (vi) denote as k

Then we have

((xi,

> 1 the greatest integer among {kt)such that

< a/2

and by hypothesis

$2.2 Expansivity

43

Choose m > 0 such that 1/2m 5 a < 1/2~-'. Then (xo,xk), (xk,xk+l), (xktl,xn) E Vmdl, from which ( x o , ~ , ) E Vm-2 (we consider Vm-2 = l4j if m - 2 5 0). Thus we have

For the case (vii) we have

and by hypothesis

Since [(xo,xl) 5 a

< 1/2"-',

we have ( x o , ~ , ) E

Therefore

In any case we have (v). 0 Proof of Theorem 2.2.10. To show the existence of a compatible metric D that is our requirement, let e > 0 be an expansive constant for f . Write Wo = X x X and for n 2 1 let

Then it is easily checked that {W,) is a decreasing sequence of open symmetric neighborhoods of diagonal set A such that Wn = A, and letting g = f xf :XxX+XxX,

nz=o

Since Wl is open, there is 6 > 0 such that

Since X is compact and

nrz0W, = A, there is N > 1such that

CHAPTER 2

44

0W~+N 0 WI+N C Wl. Thus we have W ~ + N Now define a new sequence {V,) by

V0 = W0, Vn = W l + ( n - l ) ~ , 2 1. The sequence Vn are open symmetric neighborhoods of A. If we establish the relation Vn+l o Vn+l o Vn+l c Vn for n 2 0, by Lemma 2.2.11 there is a metric D for X such that Vn c {(x,Y): D(x, y)

n 2 1.

< 1/2n) c Vn-1,

>

To use Lemma 2.2.11 we first prove that the relation is true for n 2. Take 2. Then there are a,b E X such that (x, y) E Vn+1 0 Vn+1 o Vn+1 for n (3, a), (a, b), (b, y) E Vn+1, and so for 0 5 j 5 ( n - l ) N gj(x,y) = ( f j ( 4 , f j ( a > )0 ( f i b ) , fj(b)) 0 (fj(b), fj(y))

E g j ( ~ n + l0) gj (Vn+l) ogj(vn+l) C Wl+nN-j 0 Wl+nN-j 0 W l + n ~ - j

c vz 0 vz 0 vz c Vl. Therefore, d(fj(x), fj(y)) < e for 0 I

j

I (n

-

Wl+(n-1)N = %. To obtain the conclusion, suppose 0 < D(x, y) n -1 such that (x, y) E Vn+l - Vn+z and thus

l ) N and thus (z,y) E

< 1/32. Then there is

1 1 2n+3 5 D(x, y) 5 min{-32 ' -1. 2"+'

This shows n 2 3. Since (x, y) E W I + ~ N - W l + ( n + l ) ~there , is j such that nN

< j 5 (n + 1)N,

d(fJ(x), fj(y)) 2 e.

Then we have Let (z, w) = ( f3N(x), Os(n-3)N< j-3NI(n-2)N, d( f 3-3N ( ~ ) , f j - ~ ~ ( w = )d(fi(x),fj(y)) ) 2 e. Thus (z, w) @ Wl+(n-2)N = Vn-I and

Put y = (1/2)& and define a metric Dl by 3N-1

D'(x, Y) =

C r i ~ ('(31, f f "~1)). i=O

Then Dl has the desired property.

52.2 Expansivity

45

Theorem 2.2.12. If a homeomorphism of a compact metric space is positively expansive, then the space is a set consisting of finite points. Proof. Since f is positively expansive, there are a compatible metric D, constants 6 > 0, 0 < X < 1 such that D(x, y) 6 implies D(fW1(x),f-l(y)) 5 X D(x ). Thus @- = {f-' : i 2 0) is uniformly equi-continuous. Put a+ = {$ : i 2 0). By a metric p defined by p(f ,g) = max{D(f (x), g(x)) : x E X), @- is totally bounded. Define a map G : $- -+ @+ by G(fW') = f i for i >_ 0 Then G is pisometric. Therefore @+ is totally bounded. Since X is compact, @+ is uniformly equi-continuous and so there is v > 0 such that D(x, y) < v implies D( f i(x), f i(y)) 5 e for all i 2 0 (e is an expansive constant). This shows x = y and therefore x is an isolated point.

<

.

Theorem 2.2.13. The closed interval I = [O, 11 does not admit positively expansive maps. Proof. Suppose f: I -t I is a positively expansive map. Let 0 < E < 1 and denote as I, (E) the closed subinterval [x, x c] where x E [O,1- E] By positive expansivity of f there exists 0 < E < 1 such that flI-(,) : I,(€) + f (I, ( E ) ) is injective for all x E [O,1 - €1. The point a E Io(c/2) which gives the greatest value of flro(e12) is one of the end points of Io(c/2). Thus, a = 0 or a = ~ / 2 . Indeed, if a is an interior point of Io(c/2), then we have that for some v > 0

+

f (0)

.

) f (a) - v < f (a), < f (a) - v < f (a), f ( ~ 1 2 <

so that there exist X I , xz E I o ( ~ / 2with ) xl # 2 2 such that f (XI)= f (zz). This cannot happen. If a = €12 gives the greatest values of f on Io(~/2),then the point E must be a point of the greatest value of f on I,l2(e/2). Continue this argument, then we have that f: I -t I is a homeomorphism, which contradicts Theorem 2.2.12.

Theorem 2.2.14. Let f : X + X be a positively expansive map of a compact connected metric space. I f f is an open map, then f has fixed points in X. Proof. Notice that f is a local homeomorphism. Since X is compact, f is a covering map. Positive expansivity ensures that there exists a compatible metric D and constants 6 > O,X > 1 such that D(x, y) < 6 implies D( f (x), f (y)) 2 XD(x, y). If V is an open set of diameter less than 6 and if q E fP1(V), then there is a unique open set containing q of diameter less than (1/X)6, which is mapped homeomorphically onto V. This enables us to lift chains of small open sets. Let {V, : 1 5 i 5 n) be a fixed open cover of X so that the diameter of each is less than 6. Take xo E X with f(xo) # xo. Since X is connected, there is a chain of open sets chosen from among {X)from f (xo) to xo. Lift this chain to a chain of open sets from xo to x l such that f (XI)= xo. Further, lift this chain to a chain of open sets from x l to 2 2 such that f (x2) = X I , and continue this inductively. Since the original chain had length at most n6, the

CHAPTER 2

46

length of the first lifted chain is at most (l/X)nS and in general the length of the t th lifted chain is at most (1/X)~nn6. If j < t then

Thus { x j ) is a Cauchy sequence, which therefore converges to y E X . Since we have f ( y ) = f ( l i m x j ) = lim f ( x j ) = limzj-1 = y. f ( x j )= For f : X + X a homeomorphism of a compact metric space, define a local stable set W,d(x,d ) and a local unstable set W:(x, d ) by

The symbols W f ( x ,d ) ( a = s, u ) will be changed by W,Q(x)later.

L e m m a 2.2.15. f : X -+ X is an expansive homeomorphism with expansive constant e if and only if there exists e > 0 such that for all y > 0 there is N > 0 such that for all x E X and all n 2 N

Proof. If this is false, then we can find sequences x,, y, E X , rn, > 0 such that y, E W,S(xn,d ) , limn,, m, = oo and d ( f ""(x,), f mn(y,)) > y. Since Yn E W , J ( x n d, ) , we have d(f

f mn ( x n ) ,f

0

f

""

(y,)) 5 e

for all i

> -m,.

If fmn(xn) -+ x and fmn(yn) + y a s n t m, then d ( f i ( x ) ,f i ( ~ ) )5 e for all i E Z. On the other hand, we have

thus contradicting. Conversely if d ( f n ( x ) ,f n ( y ) ) 5 e for all n E Z , for any y y E W,9(x,d). Since y is arbtrary, we have x = y.

> 0 we have

For x E X the stable set and unstable set of a homeomorphism f are defined by

W d ( x , d )= { y E X : lim d ( f n ( x ) ,f n ( y ) ) = 01, n-oo

W U ( x , d )= { y E X : n+-oo lim d ( f n ( x ) ,f n ( y ) ) = 0).

$2.2 Expansivity

Lemma 2.2.16. If f : X + X is an expansive homeomorphism, for E number less than an expansive constant

Proof, If y E W,"(fn(x), d).

47

>0

,U ,, f-nW,d(fn(x),d), then there is n >_ 0 with fn(y) T ~ Gfor , y > 0 there is N 2 0 such that for all m 2 N

a

E

and so d(f "+"(y), f m+n(x)) 5 7 for m 2 N. Therefore, y E Wd(x,d). Conversely, if t E Wd(x, d) and E > 0, then we can find N 2 0 such that d(fn(x), fn(y)) 5 c for n >_ N. Thus d ( f j o fN(x), f j o fN(y)) 5 6 for all j 2 0. Therefore, f (y) E W:(f N(x), d) and so

For expansive homeomorphisms of a compact metric space we have the following theorem (compare to Theorem 2.2.10). Theorem 2.2.17 (Reddy [R2]). If f: X + X is an expansive homeomorphism, then there exist a compatible metric D and constants E > 0, c > 0 and 0 < X < 1 such that

for all n

> 0.

We say that the metric D as in Theorems 2.2.10 and 2.2.17 is a hyperbolic metric for the map f: X --, X. Proof, Define Wo = X x X and for n 2 1

where e is an expansive constant. Then {W,) is a decreasing sequence of open symmetric neighborhoods of the diagonal set A. Since f is expansive, if

CHAPTER 2

48

(x, y) 4 A, there is n > 0 with (x, y) 4 Wn. This implies there are E > 0 and N 2 1 such that

nn,, Wn = A. Thus -

We now define a new sequence of open symmetric neighborhoods of A by

Vn = A. TO use Lemma 2.2.11 we must prove that

Obviously,

It is clear that Vl o Vl o Vl c Vo and by the choice of E and N, V2 o Vz o V2 C Vl . Let k 2 2 and take (x, y) E Vk+l oVk+l Then we prove that (x, y) E Vk. Let z, w E X be points such that (x, w), (w, z), (2, y) E Vk+1. If (p, q) is any of these three points, then we have

If lil

< 1+ (12 - l ) N and lkl < 1+ N, obviously

and so

f 0 f i(q)) = d(f "YP), f k+i(q)) < e. Therefore, (f i(p), f i(q)) E Vz and thus d(f

0

f

fi(~)l

from which d(f '(x), f i(y)) < e,

+

lil < 1 ( b - l)N.

Therefore, (x, y) E Vk. Let D be a compatible metric satisfying

Let A C X x X and denote by xA the subset {y E X : (x, y) E A}. Then we have for n 2 1

$2.2 Expansivity

Indeed, the first equality is computed as

f ( w , " ( ~ , dn) zWn) = f {Y : d ( f i ( z ) ,f " ~ ) ) < ee, i 2 - n ) = { z : d ( f i o f ( x ) ,fi o f ( z ) )< e, i 2 -(n = w,"(f ( ~ ) , dn)f (z)Wn+l. Similarly the second equality is computed. Let N be as above. By induction we have for n

+ 1))

21

FYom now on we prove that D is a hyperbolic metric. Since there is q > 0 such that D ( x , y) < q implies d(x,y) < e, we have W,d(x,D ) c W,d(x,d)for x E X. Put v = min{q, 1/41 and take y E W,"(x,D). We first show that

Since ( x ,y) E Vn and ( x ,y)

4 Vn+l > Nll2n+a(A,D ) for some n , we have

and thus n 2 1. Since y E W,d(x,d ) n xVn, we have

and so

Let

1 1 1 ~ ( f ~ ~ f3N(v)) ( x ) .< 9 _ - -2. -2"+2

1

<~

( x , Y ) .

L = 3N. It follows from induction that

Define X = (1/2)*. To obtain the conclusion we show that if y E W,d(x,D ) then D ( f r n ( x )f,r n ( y ) )5 8XrnD(x,y), m L 0. Put m = kL+ j for k 2 0 and 0 5 j for some n 1, we have

>

< L. Since ( f " ( x ) , f k L ( y ) ) E Vn-Vn+l

CHAPTER 2

and thus fkL+j(y) E w;(fkL+j(x), d) n fkL+'(z)w1+(n-l)~+j, from which (fkL+j(x),fkL+j(y)) E Wl+(n-l)N+j C Wl+(n-l)N

= Vn C Nl12n(A,

Since (fkL(x),fkL(y))4 Vn+l 3 Ni12*+a(A,D), we have 1 D(fkL(x),fkL(y))t Z"tl and thus

Since m = bL

+ j, we have < ~ A " + ~ D ( xy), = 8XmD(x,y).

By replacing f by W,U(x,D).

f-l,

we can establish the required property for y E

We denote as Hn the half space of Rn, i.e. H n = {(xl, x2, - . . ,x,) E Rn : xn 2 0). Then Hn > Rn-'. A metric space M is called an n-dimensional topological manifold if for each p E M there is a neighborhood U(p) such that U(p) is homeomorphic to an open subset V of H n . Let q, denote the point in V corresponding to p. We write a M = {p E M : qp E Rn-') and OM is called the boundary of M . If 8 M = 0, we can take Rn instead of Hn. In this case M is called a topological manifold without boundary. We here say that a topological manifold M is closed if it has no boundary and is compact, connected. If U is an open subset of M and $ is a homeomorphism from U onto an open subset of Rn, then (U, $) is called a chart (or a coordinate neighborhood) of M . For p E U we can write $(p) = ( x ' ( ~ ) , ,xn(p)). Then x l , . ,zn is continuous functions on U. We say that (xl, -. ,xn) is a local coordinate system of M at the chart (U, $), and that ( x ' ( ~ ) , ,xn(p)) is a local coordinate of a point p in U. Let S = {(U,, $), : a E A) be a family of charts of M . If {U, : a E A) is an open cover of M , then S is called an atlas (or a system of coordinate neighborhoods) of M. A continuous surjection f : X + X of a metric space is said to be ezpanding if f is positively expansive and it is an open map. If X is compact, then an expanding map f : X + X is a local homeomorphism, and so f is a selfcovering map by Theorem 2.1.1. In the case when X is a closed topological manifold, we have that every positively expansive map of X is expanding in our sense because it is a local homeomorphism by the following theorem (cf. [SPI1.

... -

$2.2 Expansivity

51

Theorem 2.2.18 (Brouwer theorem). If U and V are homeomorphic subsets of Rn and U is open in Rn,then V is open in Rn. The following theorem is a wider result which contains Theorem 2.2.13.

Theorem 2.2.19 (Hiraide [Hi7]). No compact connected topological manifold with boundary admits a positively expansive map. For the proof we need the following lemma.

Lemma 2.2.20. Let X be a compact connected locally connected metric space and f : X + X a positively expansive map. If a closed proper subset K satisfy the conditions : (1) (2)

f(X\K)cX\K, f : X \ K -+ X \ K is an open map,

then K = 0. Let M be a compact connected topological manifold and 8 M denote the boundary of M . If M admits a positively expansive map f , then f: M + M is locally injective. By applying Theorem 2.2.18 we have that f ( M \ 8 M ) C M \ 8 M ) and f i j q B M is an open map. Therefore BM = 0 by Lemma 2.2.20.

Proof of Lemma 2.2.20. Since f: X + X is positively expansive, there exists a metric D satisfying all the properties of Theorem 2.2.10. For any x E X define U,(x) = { y E X : D ( x , y) < E ) and denote as C c ( x )the connected component of x in Uc(x). Then C,(x) is open in X. Since X is connected and locally connected, X is locally arcwise connected by Theorems 2.1.3 and 2.1.4. Let 6 > 0 be as in Theorem 2.2.10 and take E with 0 < E < 612. We first show that for x E X \ K

where X > 0 is as in Theorem 2.2.10. Suppose y E Cxc(f ( x ) )\ f (C,(x)). Since Cx,(f ( x ) ) is arcwise connected, there is an arc w : [O, 11 + Cx,(x) such that w(0) = f ( x ) and w(1) = y. Since C,(x) c X \ K and C c ( x )is open in X , f ( C c ( x ) )is open by (2). Since w(0) = f ( x ) E f ( C c ( x ) ) ,there is 0 < to < 1 satisfying w([0,to)) C f ( C c ( x ) ) and w(to)# f (C,(x)). Then we have

) , the choice of 6 we have that flcl(ce(+)) : cl(Cc(x)) Since cl(Cc(x))C U 6 / 2 ( ~by a homeomorphism. Since w([O,t o ] )c cl(f (C,(x))), there is an arc G : [0,to] + cl(Cc(x))such that f o lij = w. Since f o G([O,t o ) ) = w([O,to))c f ( C , ( x ) ) , we have G([O,to))c C,(x) and since w(t0) $ f ( C c ( x ) ) , obviously G(to) # C c ( x ) . + cl(f ( C c ( x ) ) is )

CHAPTER 2

52

On the other hand, since w(t0) E Cxc(f (x)), we have

and so s ( t o ) E Uc(x). Thus to) E Cc(x) since x E ~j([O,to))C Cc(x) C Uc(x). This is a contradiction. Therefore, Cxc(f (x)) C f (Cc(x)). Let us define X(E) = {x E X

\ K : Cc(x) C X \ K).

Since K # X , there is 0 < €0 < 612 such that X(eo) # 0. To obtain the conclusion of the lemma suppose K # 0. Take x E X(eO). Then we have Cxc0(f ( 2 ) )C f (CcO(x)).Thus, f ( X ( ~ 0 ) C ) ~(AEo). It is easily checked that X(AEO)c X(peo) c X(eO)for 1 < p < A. We show that cl(X(k0)) C X(pe0). To do this let {xi) be a sequence of X ( k o ) and suppose xi + x as i + oo. Since CEO (x) is open in X , we have xi E CEO (x) for sufficiently large i. Thus Cco(x) C Cpc0(xi) c Cxco(xi) which implies x E X(pco). Therefore cl(X(Aeo)) c X(peO). We have shown that

n:=o

Thus Y = f " c l ( ~ ( e ~ ) )is) a nonempty closed set and f (Y) = Y. Since Y # X and X is connected, Y is clearly not open in X . For A c X and a > 0 define Na(A) = UaEACa(a). Then N(,-l)co(Y) c X(EO).Indeed, let z E Y and x E C~,-l),o(z). Then Cco(x) c Cpc0(z). Since z E Y c X ( ~ E O we ) , have Cco(x)c Cpc0(z) C X \ K and so x E X(EO). Since f (Y) = Y, we have

nzo

fi(~(,-l)co(~)). and thus Y = Since C(,-l)co(z) c Cpc0(z)c X \ K for z E Y, we have f (C(,-l)co(z)) G ( , - l ) c o ( f (2)) and thus

>

Therefore, Y = N(,-l)co(Y) and Y is open in X. This is a contradiction.

Remark 2.2.21. We can construct an example of positively expansive map which is not open (an example due to Rosenholtz ([R])).Consider the subset X in the plane defined by

52.2 Expansivity

53

Give X the arclength metric. Then a map is defined as follows : stretch each of the small circles onto the big circle, stretch each of the upper and under semicircles of the big circle first around a small circle, and then across the other semicircle and finally around the other smaller circle. More precisely we describe the map f by

Then it is easily checked that f is positively expansive, but not open.

I

Figure 2

If f : X + X is a homeomorphism of a compact metric space and x is a point in X , then a-limit set of x, denoted as a(x), consists of those points y E X such that y = limj,, fnj(x) for some strictly decreasing sequence of integers nj. The w-limit set of x, denoted as w(x), is similarly defined for strictly increasing sequence. The set a(x) (w(x)) is closed, nonempty and f invariant, i.e. f(a(x)) = a ( ~ for ) x E X. If, for some point x, a ( ~ and ) w(x) each consists of a single point,then we say that x has converging semi-orbits under f .

Theorem 2.2.22 (Reddy [R3]). If f : X -+ X is an expansive homeomorphism of a compact metric space, the set of points having converging semiorbits under f is a countable set. Proof. Since f is expansive, it follows that f has at most finitely many fixed points, say 91,. ,qk. Let A denote the set of points having converging semiorbits under f . Suppose A is uncountable.

.

54

CHAPTER 2

If x E A, both a(x) and w(x) are fixed points. Let A(i, j ) be the set of points satisfying a(x) = qi and w(x) = q,. Then A is the union of the finitely many sets A(i, j), so that one of these, say B , is uncountable. For each N > 0 let B ( N ) be the collection of points x E B such that, for n 2 N, fn(x) is e/2-close to w(x) and f-"(2) is e/2-close to a(x). Since B is the union of the sets B(N), one of them (say B(M)) must be infinite. Since X is compact, there exist distinct points y and z of B(M) with d(y,z) < e/2 so small that fn(z) is e-close to fn(y) if In1 5 M , where e denotes an expansive constant for f . Thus, from the definition of B(M), we have that d(fn(y), fn(z)) < e for all n E Z, thus contradicting the choice of e. Therefore, A is countable.

Theorem 2.2.23 (Jacobsen and Utz [J-U]). There exists no expansive homeomorphism of a closed arc. Proof. If f : [ O , l ] -t [O, 11 is a homeomorphism, then f has either f (0) = 0 and f (1) = 1, or f (0) = 1 and f (1) = 0. In both cases f a induces a homeomorphism satisfying f 2(0) = 0 and f a ( l ) = 1. Put Fix(f 2, = {x € [0,1] : f 2(x) = x). Then ~ i x (2,f is closed. If Fix(f2) = [O, 11, then all points of [O,1] have converging semi-orbits under f 2 , and so f is not expansive by Theorem 2.2.22.

Figure 3 If Fix(f2) # [O, 11, then U = [0, 1]\Fix(f2) is a nonempty open set. Thus U is the union of the countable intervals Ij where Ij's are mutually disjoint. For x E U there is Ij such that x E Ij. Then the extremal points of Ij are or fixed points of f 2 . Thus we have that for x E Ij,x > f2(x) > f4(x) > x < f2(x) < f4(x) < --..In any case {f2j(x) : j 2 0) converges to a fixed point of f 2 . Since Ij is uncountable, f is not expansive. m e - ,

Let f: X 4 X be a homeomorphism of a compact metric space. If, for all x E X , the orbit Of(x) = {fn(x),n E Z) is dense in X, then f is called a minimal homeomorphism. Obviously every minimal homeomorphism is topologically transitive.

$2.2 Expansivity

55

Theorem 2.2.24. A homeomorphism f : X -, X is minimal if and only if E = 0 or X whenever f ( E ) = E and E is closed. Proof. Suppose f is minimal. If E is closed and f ( E ) = E # 0, then E > Of (x) for x E E and cl(Of(x)) C E. Thus we have E = X. Conversely, for x E X,cl(Of (3)) is a closed f-invariant set and thus cl(Of(x)) = X.

Figure 4

A closed subset E which is f-invariant is called a minimal set with respect to f if f l E is minimal.

Theorem 2.2.25. Every homeomorphism f : X

-, X

has a minimal set.

Proof. Let 0 denote the collection of all closed nonempty f-invariant subsets of X. Clearly 0 # 0 because of X E 0.0 is a partially ordered set under the inclusion. Every linearly ordered set of 0 has the least element. Thus 0 has a minimum element by Zorn's lemma. This element is a minimal set with respect to f .

Theorem 2.2.26. There exists no expansive homeomorphism of the unit circle S' . Proof, Suppose f : S' -, S1 is expansive. By Theorems 2.2.22 and 2.2.23, f has no periodic points. Indeed, suppose f has distinct periodic points x and y. Then fk(x) = x and fk((y = y for some L > 0. For simplicity put g = f k . There exists a closed arc C in S1 such that g(C) = C, or g2(C) = C. If g(C) = C, then g p : C -, C is expansive. This is impossible by Theorem 2.2.23. If f has only one fixed point, for any point x E S1 the semi-orbit {x, f (x), f (x), ) converges to the fixed point, and also {x, f -'(x), f -2 (x), ) converges to the fixed point. Thus f : S1-, S1is not expansive by Theorem 2.2.22. Let M be a minimal set of f in S1.Since f : M -r M is expansive and minimal, we see that M is totally disconnected (see Theorem 2.2.44 below). Notice that M is an infinite set because f has no periodic points.

---

CHAPTER 2

56

We claim that isolated points are not in M. Indeed, if x E M is an isolated point, then Ua(x) n M = {x) for some 6 > 0 and so {x} is open in M. Since M is minimal, the closure of the orbit of x equals M and thus fn(x) E Ua(x) n M = {x) for some n E Z, a contradiction. Since S1\ M is open, S1\ M is expressed as countable disjoint union S1\M = Uj Ij of open arcs I, in S1. Obviously diam(Ij) + 0 as j -+ oo and for fixed j,fn(Ij) # Ij for all n # 0. Thus, diam(fn(Ij)) + 0 as In1 + oo. Take

-

-

. .

. .

a, b E Ij and let f nO(ab)be the arc with the longest length in { f "(ab) : n E Z). Let E > 0. Then we can choose Ij and a, b E Ij such that the length between h

fno(a) and fnO(b) is less than E . Then, diam(fn(ab)) < E for all n E Z and therefore f : S1 -+ S1 is not expansive. This is a contradiction. The following theorem was proved in Hiraide [Hi51 and Lewowicz [L2].

Theorem 2.2.27. S2.

There exists no expansive homeomorphism of the bsphere

We omit the proof. If the theorem is of interest, the reader should see [Hi51 and [L2]. Let X be a compact metric space with metric d and xZdenote the product topological space xZ= {(?i) : x; E X , i E Z). Then xZis compact. We define a compatible metric d for xZby

A homeomorphism a : xZ-+ XZ, which is defined by

xi)) = (yi) and yi = ~ i + for l all i E Z, is called the shift map. For f : X

-+

X a continuous surjection, we let

Xf = {(zi) : xi E X and f ( ~ i = ) ~ i + l ,Ei Z). Then Xf is a closed subset of xZ.Moreover we have a((xi)) = (f (xi)) for all (xi) E X f , and so X f is a-invariant, i.e. a ( X f ) = Xf. The space X is called the inverse limit space constructed by f and we sometimes write X f = lim(X, f). The restriction a = alx, : X f + Xf is called the shift map t determined by f . Let i E Z and denote as pi : Xf -+ X the projection defined by (xi) H xi. Then pi o a = f o pi holds, i.e. the diagram

Pi

IP~

1

X - X

f

commutes,

52.2 Expansivity

57

and po o a' = f o po for all i 2 0. If, in particular, f is bijective, then it is clear that each pi is bijective. In the case where X is a torus and f: X -, X is a toral endomorphism, we can show that the inverse limit space X j has a structure of compact connected finite dimensional abelian group, which is called the solenoidal group. The properties of solenoidal groups will be &scussed in Chapter 7. It is easily checked that X j is homeomorphic to the space X; = {(xi)? : xi E X and f

= xi,i

2 0)

equipped with the product topology, and the shift map a is topologically conjugate to a homeomorphism o' defined by al((xi)?) = (f (xi))? for (xi)? E X;, i.e. there exists a homeomorphiim h: X j -+ X; such that the diagram

1.

1

commutes.

A continuous surjection f : X -+ X is c-expansive if there is a constant e > 0 (called an ezpansive constant) such that for (xi), (yi) E X j if d(xi, yi) 5 e for all i E Z then (xi) = (yi). T h i is independent of the compatible metrics used. The notion of c-expansivity is weaker than that of positive expansivity. For homeomorphisms c-expansivity implies expansivity. Theorem 2.2.28. Let k and only if so is f '.

>0

be an integer. Then f: X + X is c-expansive if

Proof. We notice that f is uniformly continuous. Let e constant for f . Then we have

> 0 be an expansive

for some 6 > 0. This 6 is an expansive constant for f k . In fact, for (xi), (yi) E X j suppose d(xki, yki) < 6 for all i E Z. Then we have d(xi, yi) < e for i E Z. By expansivity, xi = yi for i E Z, from which xki = yki for i E Z. The converse is proved by the same way. For the relation between continuous surjections and their inverse limit systems we have the following

Theorem 2.2.29. Let X be a compact metric space. A continuous surjection f : X 4 X is c-expansive if and only if a : X j -+ X j is eapnsive. Proof. If (xi) # (yi) then there is k E Z such that d(xk,yk) > e by c) ) yk) > e, where d((zi), (yi)) = expansivity. Then J(ak((xi)),~ ~ ( ( ~ i2) d(xk, 00 d(xi, yi)/21il. Therefore a is expansive.

xi=-,,

CHAPTER 2

58

Conversely, let e > 0 be an expansive constant for a and let d(xi, yi) < e/4 (i E Z) for (xi), (yi) E Xf. Then we have d(an((xi)), un((yi))) 5 e for n E Z, and hence (xi) = (yi) by expansivity of a. Therefore e/4 is an expansive constant for f: X + X . The following theorem is easily checked. Thus we leave the proof to the readers.

Theorem 2.2.30. (1) Iff: X + X is c-expansive and Y is a closed subset of X with f (Y) = Y, then f l y : Y + Y is c-expansive. (2) If fi : Xi + Xi (i = 1,2) are c-expansive, then the continuous sujection fl x f2 : XI x Xz + XI x X2 defined by

is c-expansive. Every finite direct product of c-expansive maps is c-expansive. (3) If X is compact and f: X + X is c-expansive, then h o f o h-l : Y + Y is c-expansive where h: X -r Y is a homeomorphism.

Theorem 2.2.31. The closed interval does not admit c-expansive maps. Proof. Let I denote the closed interval [O, 11. For f : I + I a continuous surjection, its inverse limit system (If, a ) is defined by

Suppose that a2 : If + If is expansive. It is easily checked that ( I f , a 2 ) is topologically conjugate to (If=,u). Since f is surjective, there exist a, b E I with 0 5 a < b 5 1 such that either (1) f (a) = 0 and f (b) = 1

or

(2) f (a) = 1 and f (b) = 0.

If we have (2), then we see that f2(a') = 0 and f2(b') = 1 for some a', b' E I with 0 5 a' < b' 5 1. Thus, to obtain the conclusion it is enough to prove (1). Take two points u,v E I ( u < v) such that

This is ensured by the facts that f has at least two fixed points and the set of fixed points is not dense in I. (3) is divided into the following cases, (4) f (x) > x for u

<x
We first check the case (4).

or

(5) f (x) < x for u

< x < v.

82.2 Expansivity

59

If fllU,+] : [u, V] -+ [u, v] is bijective, then the inverse limit system of ([u, v], fllu,,l) is a subsystem of (If,o). Thus the system is expansive. But it contradicts Theorem 2.2.23. Therefore we can find two points x1 and x2 such that u < 2' < x2 < v, f ( x l ) = f ( 2 2 ) . Now we define = f ( x O )= m a x { f ( x ) : u

< x 5 x2).

Then, u < xO 5 x2 5 yo. Since the graph of f on [u,v] is over the diagonal set of I x I, there exists a sequence x: > x! > . in [u,v] with

.

Since 20, -+ u as n

for some N

> 0.

-, oo,

Take 6

for 0 < E

< x2 - xO we have

> 0 so small that for x, y E I

Write c1 = x& - 6 and c2 = x&

+ 6. Then

and thus f N ( c i ) < x0 + E < x2. By the definition of yo we have fN+'(ci) 5 yo for i = 1,2. If f N+l(ci) = yo for some i, then we can find a sequence

such that ci = ci and f ( c i ) = cjdl for j 2 1. Notice that {ci : j 2 0 ) is strictly decreasing. Define a sequence { b j ) in [u, v] by

Since cI"

xkl

< 26 for i = 1,2, we have

CHAPTER 2

60

>

and thus Ibj - x!l < e for j N. Therefore, 121, - x;l < e for all j 2 0. Since fN+'(ci) = f(xO) = yo, we have fj(bo) = fj(x:) for j > 0. Therefore, for j > 0, j-th components of an(bo,bl,...)

and

an(x~,x~,..-)

are closer than E: for all n E Z. Let x3 = max{fN+'(ci) : i = 1,2). If z3 <

then there exist c3, c4 with

such that f N+'(C3)

= fN+1(c4) = c4.

By (4) we can find sequences {cj : j 2 0), i = 3,4, in [v,v] such that ci = c' and f (c;) = ci-, for j 2 1. They are strictly decreasing sequences. By the similar argument we have that for j 2 0, j-th components of

are closer than E: for all n E Z. Therefore a : If + If is not expansive. The case (5) is obtained in the same argument. For toral endomorphisms we obtain the following theorem from Lemmas 2.2.33 and 2.2.34 below.

Theorem 2.2.32. (1) A toml automorphism is expansive if and only if it is an automorphism of type (1). (2) A toml endomorphism is positively expansive if and only if it is an endomorphism of type (11). (3) A toml endomorphism which is not injective is c-expansive but not positively expansive if and only if it is an endomorphism of type (III). Lemma 2.2.33. Let f be a linear map of the euclidean space Bn, where n 2 1, and let d be the euclidean metric for Rn. Then f is expansive under d if and only iff is hyperbolic, i.e. it ha8 no eigenvalues of modulus 1. Proof. Suppose all eigenvalues of f are off the unit circle. Then Rn splits into the direct sum Bn = Ed$ Eu of subspaces Ed and Eu such that f (Ed) = Ed and f ( E u ) = Eu, and such that there are c > 1 , O < X < 1 so that (1) (2)

Ilfn(v)ll

s cA"llvll,

Ilf -n(v)ll 5 cXnllvll,

v E Ed v E EU

$2.2 Expansivity

for n 2 0. Therefore, f is expansive under d. Conversely, if f has an eigenvalue of modulus 1, then Rn splits into the direct sum of subspaces as follows : (i) Rn = EC$En $ Eu, (ii) f (Eu) = Eu(u = C ,s , u), (iii) there are c > 0 and 0 < X < 1 such that (1) and (2) hold, and (iv) f l E c is linearly conjugate to a linear map coresponding to Jordan form (in the real field)

where either

for 1 5 j 5 k. Here I is a 2 x 2 identity matrix and

By (iv) we can find a subspace F of Ec such that f l F is an isometry under d. Therefore, f is not expansive. Let

X

:

-+

and X be metric spaces with metrics 2 and d respectively, and let X be a continuous surjection. Then ?r is called a locally isometric covering map if for each z E X there exists a neighborhood U(z) of a: such that n - ' ( ~ ( x )= ) U, (a# a' + U, n U,. = 0) ?r

U 0

where U, is open and .lup : U,

-+

U(x) is an isometry.

x

Lemma 2.2.34. Let f : -+ and g: X -+ X be continuous sujections and ?r : -+ X a locally isometric covering map. Suppose ?r o f = g o n and X

x

is compact, and suppose there exists So > 0 such that for each x E 0 < 6 5 b0 the open ball Ua(z) of x with radius 6 is connected and

is an isometry. Then

X

and

CHAPTER 2

62

( 1 ) f is ezpansive (under 2 ) i f and only if g is expansive, when f and g are homeomorphisms, ( 2 ) f is positively expansive if and only if g is positively expansive, ( 3 ) f is c-expansive if g is c-expansive, and i f f is c-expansive and satisfies the following condition ( C ) : for one has - E > 0 there ezists N > 0 such that for (pi),(9;) E d(po,qo) E whenever a(p;,qi) 5 e for all i with lil 5 N where e > 0 is an expansive constant for f , then g is c-expansive.

xf

<

Proof. From the assumptions it follows that for each x E X and 0

<55

and if p, q E n - l ( x ) and p # q, then U6(p)n U6(q)= 0. Indeed, it is clear that T - ' ( u ~ ( x ) )> U,ET-l(xl U ~ ( P ) .Let a E T - ~ ( U ~ ( X )Since ). n1u6(,,) : Us(.) -r U6(7r(a))is bijective and x E U 6 ( n ( a ) ) we , have a E Ua(p) for some p E n-'(x), and so (i) holds. If b E Ua(p)n U6(q)# 0, then p = q because n ~ ~ ~ :cU6(b) b ) -r Ua(a(b))is bijective. To obtain the lemma, we first show that f is uniformly continuous. Let 0<~<5~andtake0<~<6~suchthatforx,y~X

Now, if z ( p , q) < 6, then

and so n o f ( q ) E U,(g o n ( p ) ) ,which implies

Since 5 5 bO,we have that f(U6(p))is connected. Notice that U,(f ( p ) )is the connected component of f ( p ) in n-' (u,(~ o n ( ~ ) )Then ) . f (Ua(p))C U, ( f ( p ) ) , and thus f is uniformly continuous. (1) : Suppose g is an expansive homeomorphism with expansive constant e, and let 7 = min{e, 60). For p, q E if z(f i ( p ) , f " q ) ) 5 7 for i E Z , then we have d(gi((.~)),gi(49>> =) '(PI, f i ( 9 ) ) l 7, i E Z ,

x,

4f

and so ~ ( p=) n(q). Since

z(p,q) 5 50, it follows that p = q.

52.2 Expansivity

63

Conversely, suppose f is expansive. Choose 0 < 6 < 60 such that if 2(p, q) < 6 then 2( f (p), f (9)) < 60 and f (p), f -l (q)) < 60, and put 7 = min{e, 5 ) where e is an expansive constant. For x, y E X with d(x, y) 5 7 we take p, q E such that n(p) = x, n(q) = y and Z(p, q) = d(x, y). If d(gi(x), gi(y)) I: 7 for i E Z, then

z(

since Z(f (p), f ( 9 ) ) < 60. Inductively, we have Z(fi(p), fi(q)) 5 7 for all i 1 0. In the same way, 2(fi(p), fi(q)) 5 7 for all i I: 0. By expansivity we have p = q, and therefore x = y. (2) : This is proved in the same manner as (1). (3) : Suppose g is c-expansive (with expansive constant e > 0). Let 7 = min{e, 60) and (pi), (qi) E Since s o f = g o n, it is clear that pi)), ( ~ ( q i ) )E Xg.If a(pi, qi) I: 7 for i E Z, then d(n(pi), ~ ( q i ) )_< 7 for i E Z, and SO pi)) = (n(qi)), which implies (pi) = (qi). Conversely, suppose f is c-expansive and satisfies the condition (C), and suppose d ( ~ iyi) , -< 7 (i E Z)for (xi), (yi) E X, where 7 = min{e, 6) and O < 6 5 60 is chosen such that 2( f (p), f (9)) I: 60 whenever 2(p, q) < 6. For each q?, E T-'(~-,) such that 2(p!nq,!n) 5 7, n 2 0 we take pl, E n-'(2-,), and define (pl), (ql) E by

xf.

x

py = f i + n ( p ~ n ) ,qp = f i+n(qln) for i 2 -n, P?-I E f -'(pf),

Ef

for i

I: -n.

Then ( ~ ( p r ) )= (xi) and (~(9;)) = (yi), and z(pl, q?) 5 7 for all i 2 -n. By the condition (C) it follows that d(p;, q;) + 0 as n + oo. Thus xo = yo. Similarly, we have xi = yi for i E Z.

Remark 2.2.35. -

The condition (C) in Lemma 2.2.34 (3) is always satisfied if X is compact. However, it remains a question of whether it is true for general case. Let X be a compact metric space. Topological dimension of the space X is said to be less than n if for all 7 > 0 there exists a cover a of X by open sets with diameter < 7 such that each point belongs to at most n 1 sets of a. (If the topological dimension is of interest, the reader should see Hurewicz and Wallman [H-W]).

+

Theorem 2.2.36. Let X be a compact metric space. X is 0-dimensional if and only if it is totally disconnected (i.e. the connected component of each point is a single point). Before starting with the proof we prepare the following

CHAPTER 2

64

Lemma 2.2.37. Let 0 be the collection of open closed subsets of X . Then 0 is a base of X if and only if X is totally disconnected. Proof. Take x , y E X with x # y. Then we can find an open set U, such that x E U, and y # U,. Suppose 0 is a base of X . Then x E B c U, for some B E 0.Since X \ B is open closed and y E X \ B , the connected component of x , c(x), is a subset of B . However, since y is arbitrary, we have

and therefore X is totally disconnected. For fixed x E X let G be an open neighborhood of x. Then it suffices to find an open closed subset B satisfying x E B C G. If X is totally disconnected, then each x E X is expressed as { x ) = U ( x ) where { U ( x ) ) is the collection of open closed subsets containing x. Indeed, let L = U ( x ) . To see { x ) = L it suffices to prove that L is the connected component of x. If this is false, then L is expressed as L = A U B where A and B are closed and A n B = 0. Thus we have that x E A or x E B . We suppose x E A. Choose open subsets A1 and B1 such that A c A1, B c B1 and Al n B1 = 0. Then L c Al U B1. Since L = U ( x ) ,there is U ( x ) such that L c U ( x ) c Al U B1. Then it is easily checked that A: = U ( x )n Al and B i = U ( x )n B1 are open closed in X . Since x E A, A: is an open closed subset containing z and L n A: = 0. This contradicts the definition of L. Therefore, for y E X we can find an open closed subset H, satisfying y E H, and x # H,. Since X \ G is compact, a finite cover {H,, ,-. ,H,, ) of X \ G exists. Notice that each H,,satisfies x @ H,,. H = Uy HVi is open closed and x # H. Therefore x E B C G where B = X \ H .

n

n

n

Proof of Theorem 2.2.36. Suppose X is totally disconnected. By Lemma 2.2.37 the collection of all open closed subsets is a base of X . Let I' be an open cover of X and let x E X . Then there is U, E I' such that x E U, and then there is an open closed V, satisfying z E V, C U,. Since {V, : x E X ) covers X , choose a finite cover { W l , ,W k ) and define

..

.. . .-

Then IFl,. ,Fk) is a refinement of r, and a point of X belongs to only one set of { F l , ,Fk). Therefore X is zero dimensional. Suppose X is zero dimensional. Take arbitrarily a, b E X with a # b. We set U = X \ { a ) and V = X \ {b). Then { U , V ) covers X. Let a = { F l , . . - ,F k ) be a closed cover of X such that Fi r l Fj = 0 for i # j and a is a refinement of {U,V ) . Without loss of generality we suppose a E Fl. Since X = Fl U F and Fl n F = 0 where F = F2 U ... U Fk, The connected component of a, c(a), is a subset of Fl and so c(a) c X \ {b). Since b is arbitrary, we have that c(a) c n { X \ { b ) : b E X , a # b ) = { a ) , i.e. X is totally disconnected.

52.2 Expansivity

65

We here construct the Cantor set which is of particular importance. To do so we proceed the argument as follows. First, denote the closed interval [O,1] by C l . Next, delete from Cl the open interval (113,213) which is its middle third, and denote the remaining closed set by C 2 . Clearly, c 2 = [0,1/31 UP/3,11. Next, delete from Cz the open intervals (119,219) and (719,819) which are the middle thirds of its two pieces, and denote the remaining closed set by C3. It is easy to see that

Figure 5 If we continue this process, at each stage deleting the open middle third of each closed interval remaining from the previous stage, we obtain a sequence of closed sets C,,, each of which contains all its successors. The Cantor set C is defined by m

and it is closed. C consists of those points in the closed interval [0, 11 which ultimately remain after the removal of all the open intervals

Clearly C contains the end points of the closed intervals which make up each set Cn ; . 0, 1, 113, 213, 119, 219, 719, 819, The set of these end points is clearly countable. However, the cardinal number of C itself is the cardinal number of the continuum.

CHAPTER 2

66

To prove this it suffices to exhibit an injection f of [0,1) into C. We construct such a map as follows. Let x E [O,1) and let x = .blb2 be its binary expansion. Each bn is either 0 or 1. Let a, = 2bn and regard

---

f (x) = 3-'al

+ 3-2a2 + - - -

as the ternary expansion of a real number f (x) in [O,l). The reader will easily convince himself that f(x) is in the Cantor set C. In fact, since a1 is 0 or 2, f(x) is not in [1/3,2/3). Since a2 is 0 or 2, f(x) is not in [1/9,2/9) or [7/9,8/9) ; etc. Also, it is easy to see that f: [O,1) -t C is surjective. This shows that C is the set of the numbers x E [O,1] with

For T 2 1 we call the set C n [3-'43-'(i subinterval with mnk T if

+ I)] (0 5 i 5 3'

We denote as I ( i , r ) the i-th Cantor subinterval with rank Obviously

T

- 1) a Cantor

from the left.

Remark 2.2.38. Let X be a compact totally disconnected metric space having no isolated points. Then X is homeomorphic to the Cantor set C. This is checked as follows. Let d and d' be metrics for X and C respectively. Since X is totally disconnected, we can choose open closed subsets X1,X2,.-• , X 2 ~ I ( k> 1 0) such that

Since each of Xi has no isolated points, as above we can choose open closed subsets Xi,l,. ,Xi,2~,such that

..

52.2 Expansivity

67

Continue inductively this manner. Then there is a sequence { k j ) g l of positive integers and open closed subsets Xi,, ... ,im(1 5 it 5 2", 1 5 15 m) satisfying the conditions 6)

diam(Xi,,... ,im) < diam(X)/am (1 5 it 5 2" ,1 5 15 m).

(ii)

Xil,... ,im-1,j fXil,... l ,im-l,jl = 0 ( j # jl), 2 hm

(iii)

Xil, ,im-l =

U Xi,, ...,im-l,j-

j= 1

-..

,2"}, then we have {x) = If (il, ,in,. - - ) E njm,,{l,2, for some x E X. Define k(m) = CEl kj for m 2 1 and

nGl Xi, ,...,ij

n:=,

Then there is y E C such that {y) = I(p(m), k(m)), and so we define h(x) = y. Then h: X + C is a one-to-one continuous surjection ~ choose 6 Indeed, for E > 0 take mo 1 1 such that 3kl+"'+k-.0 > 1 / and such that

If d(x, y) < 6(2, y E X), then h(x) and h(y) belong to the Cantor subinterval with the rank CT1k,. Thus we have dl(h(x), h(y)) < E and therefore h is continuous. That h is bijective is easily checked.

Theorem 2.2.39. Let f : X + X be an expanding map of a compact metric space X. Then the topological dimension of X is finite. Proof. To see dim(X) < oo we consider the inverse limit system ( Xf , a) of (X, f). Since f is expanding, by definition f is a local homeomorphism and positively expansive. Then, by using Theorems 2.1.1 and 2.2.10, we see that there is a base of neighborhoods for X f each of whose members is the direct product of a neighborhood of X with Cantor set. Since f is positively expansive, a : X f + X f is expansive. If X f is finite dimensional, obviously the dimension of X is finite. That dim(Xf) < ca follows from the following result. Theorem 2.2.40 (Maiib [Mall). If f : X + X is an expansive homeomorphism of a compact metric space X , then the topological dimension of X is finite . For the proof we need the following lemmas. Let e > 0 be an expansive constant and fix 0 < E < e/2.

CHAPTER 2

68

Lemma 2.2.41. There is a 6 > 0 such that if d(x, y)

for some n

< 6 (x, y E X )

and

2 0, then d(fn(x), fn(y)) > 6.

Proof. If this is false, then there are sequences x,, y, E X , m, that

> 1, > 0 such

Since X is compact, we suppose that fn' (x, ) + x and fn' (y,) + y as n + 00. Then d(x, y) > E and d(f ,(x), f ,(y)) 28 for all n E Z.Therefore Lemma is proved.

<

Lemma 2.2.42. For all p

whenever d(x, y)

> 0 there exists N

= N(p)

> 0 such that

> p.

Proof. If this is false, then there are sequences s,, y, E X with d(x,, y,) 2 p such that s ~ ~ { d ( f ' ( x n )f,j ( ~ n ) ): Ijl I n) I E If x, + z and y, + y as n + oo, then d(x,y) p and d(fn(x),fn(y)) I E for all n E Z.

>

Proof of Theorem 2.2.40. By using Lemmas 2.2.41 and 2.2.42 we derive the conclusion. Let 6 be as in Lemma 2.2.41 and choose a cover {Ui : 1 I i 5 1) of X by open sets with diameter < 6. We show that dim(X) I L2. For each n 0, choose 6, > 0 such that d(x, y) I 6, implies d(fj(z), fj(y)) < E for all Ijl 5 n. We define

>

-

.

and x y for x, y E U$ if there exists a finite sequence x = 2 0 , 21, . , xp = y such that d(x,,x,+1) < 6, for all 0 5 r 5 p - 1 and x, E Ucj for all 0 I r I p. Denote as nk Ui,J , 1 I 12 5 k(i, j, n) the 6,-components of Ucj (i.e. the equevalence class of Ucj under the relation x y). It is easily checked that each u,$ is open and they cover X. We claim that N

lim (sup diam(~,yt))= 0 , 93

n+oo

52.2 Expansivity

69

Indeed, if this is false, then we could find p > 0 and a large number n, say n > 2 N ( p ) , N ( p ) given in Lemma 2.2.42, such that in some [I$ there are z, y with d ( x , y) > p. Let z = xo, X I , . .. ,z p = y be a sequence in Ur, such that ~ ( X , , X , + ~ ) 5 6, for all 0 5 r 5 p - 1. Let us put

Then a,

> E by Lemma 2.2.42.

Choose r such that a,, I

since f -n(xo), f -"(x,) fn(x,)) < 6 and so

E

E

if r'

From the choice of 6, it follows that


and a,

> E.

Ui- Since f n(xo),f

Then s, 5 2~ and therefore

n(2r) E

For simplicity put xb = f-"(xo) and x', = f-"(x,).

Since d(x;,x',)

< 6, by Lemma 2.2.41

U j , we have d ( f"(xo),

Then

we have

thus contradicting d( f n(zo),f n ( ~ , ) )< 6. Therefore ( 1 ) is proved. It only remains to show that for each n, any point of X belongs to at most L2 sets of the cover

Suppose that

~{uC:L :15m<s}#0.

~2%

= since they are 6,If (i,, j,) = (i=,j=), then we have components of U C j m and have nonempty intersection. This implies that to different values of m correspond different values of the couple (i,, j,). Therefore, s 5 L2.

Remark 2.2.43. Theorem 2.2.40 is true for a c-expansive continuous surjection. This is checked in the way as follows. Let 3 = {Fi : 1 5 i L) be a finite closed cover of X such that each Fi has sufficiently small diameter. For n > 0 we define Fcj = fn(Fi) n f-"(Fj) for 1 5 i,j I 1. Then 3" = {Fc,) is a finite closed cover of X. Use this cover F n in the proof of Theorem 2.2.40. Then we can show that the dimension of X is finite.

<

70

CHAPTER 2

Theorem 2.2.44 (Maii6 [Mall). If f: X + X is an expansive homeomorphism of a compact metric space, then every minimal set for f is zerodimensional. For the proof we must investigate the structure of orbits for expansive homeomorphisms. We first prepare the following

Lemma 2.2.45. Let X be a compact connected metric space and A a proper closed subset of X . If x E A and C is the connected component of x in A, then C n BA # 0 where BA denotes the boundary of A. Proof, Let C be the collection of all open and closed subsets K under the relative topology of A such that C c K c A. Since X is compact, we have C = n { K : K 6 C). To obtain the conclusion suppose C n 8A = 0. Then U{X\K : K E C) > BA. Since OA is compact and X \ K is open and closed, BA is covered by the finite union of X\Ki. Since Ki = K is open and closed, we have K E C and K r l a A = 0. Notice that K is expressed as K = U n A for some open subset U of X because K is open in A. Since A = BA U int(A), we have K = U n A = U n (BA U int(A)) = U n int(A). Thus K is open in X . Since K is closed in X , we have X = K and so 8A = 8. Therefore, A = X which is a contradiction.

ni

Let 7 > 0. For x E X let C,(x) be the connected component of x in B,(x) = {y E X : d(x, y) 5 71, and write BB,(x) = {y E X : d(x, y) = 7). We prepare some lemmas that are used in proving Theorem 2.2.44.

Lemma 2.2.46. If, for some x E X and some m WS(x,d) # 0, then X contains a periodic point.

>

0, fm(Ws(x,d)) n

Proof, Take y E W8(x, d) r l f m(Ws(x, d)) and put z = f -m(y). Then f "(2) E Ws(x,d) = W8(z,d). Thus, limn,,d(fn o fm(z), fn(z)) = 0. Since X is compact, a subsequence { f n j (z)} converges to w . Therefore, d(w, f "(w)) = limn,, d(fjn 0 fm(z), fjn(z)) = 0. Define Ci(x) and C:(x) as the connected components of x in W,"(x,d) n B6(x) and W,"(x, d) n B6(x) respectively. Let e > 0 be an expansive constant for f . We fix 0 < E < e / 2 .

Lemma 2.2.47. If dim(X) > 0, then there exists 0 < 7 < 0 < 6 < y there is a E X such that

E

such that for

Proof. Let C,(x) denote the connected component of x in B,(x). Since dim(X) > 0 , X is not totally disconnected. Thus there exist x E X and 7 > 0 such that C,(x) n ~ B , ( x )# Q by Lemma 2.2.45, and so C ~ ( Xn) B B ~ ( x )# 0 for 0 < 6 < 7.

$2.2 Expansivity

Suppose there is 0 < 6

< -y

such that

X!(Y) n a B 6 ( ~ = ) 0 for all y E X. Then we prove that for some a E X X:(a) n aBa(a)

# 0.

Figure 6 To find such an a E X we construct a collection of compact connected sets A,, n 2 0, and a sequence of points xn E A, such that for a certain subsequence {m,) the following conditions hold : (i)

(ii) (iii) (i.1

f -** (xn) = x n + ~ , f -mn(An) 2 An+l, An n a B e ( ~ n # ) 0, fj(~nc ) B 6 ( f j ( ~ n ) ) , 0 I j < mn-1.

If we establish the above conditions, then the conclusion is obtained as follows. Take a subsequence {x,,) of {x,) such that x,, -,a as i -t oo. We fix {xni) and define

It is clear that the point a is contained in A. We show that A is connected. Indeed, suppose A is not connected. Then there exist nonempty closed sets Fl, Fz such that Fl fl Fz= 0 and A = Fl LJF2. Since {x,,,,) is a subsequence of the sequence {x,,), we have x,,,, -, a as j -, oo. The point a belongs to Fl or F2. Without loss of generality we

CHAPTER 2

72

suppose the former, i.e. a E Fl. For b E F2 let {b,,,,) be a sequence such that b,,,, E A,,,, and bniSj-t b as j -+ oo. Since Fl n F2 = 0, there exist neighborhoods U ( F i )of Fi such that U ( F l )n U ( F 2 ) = 0. But each A,,, is connected. Thus we can find yniSj E A,,,, satisfying YniVj6 ~ ( J ' UI u(F2). ) If y,,,, 4 y as j + oo, then we have y E A, but y @ FlUF2, thus contradicting. By (iv) we have

Since x, = f

(xn+l) by (i), from (iv)

from which

f j ( ~ n + lc ) B e ( f j ( ~ n + l ) ) ,mn

<j

< mn-1 + m,.

Figure 7 By induction on n we have

f j ( A n ) C B e ( f j ( ~ n ) ) ,0 5 j < m o + m l + . - - + m n - 1 . If n

-t

oo, then f j ( A ) C B c ( f j ( a ) )for 0

A

Since X n i S j

+ a,

<j

< oo, and thus

c W,d(a,d).

there is y,,,, E A,,,,,

fl

aB6(xni,,)satisfying

5 = !im d(yni,j,X n i q j) = d(y,a). 3 --"=a

52.2 Expansivity

73

Therefore, A n BB6(a) # 8. Since A C Wi(a, d) and A is connected, we have E;(a) n 6B6(a) # 0. The remainder of the proof is only to construct the sequence {A,) of closed connected subsets satisfying (i), (ii), (iii) and (iv). Let x and 0 < 6 < E be as above. Put A. = E6(x), then A. n 6B6(x) # 0. Notice that E:(y) n BBa(y) = 8 for all y E X. To obtain the conclusion suppose Ao, 111,. ,An-1 are constructed. Then we have

-.

Indeed, if A,-1 C W;(x,-l,d), connected component of x,-1

then E;(X,-~) > An-l since E;(X,-~) is the in B6(xn-i) n W,U(X,-~,~).Thus

thus contradicting. Therefore, A,-~\W,U(X,-~,~)# some m > 0 we have

8, from which choose a point y. Then, for

For this m we write m,-1 = m and x, = f - m ~ - ~ ( x n - l ) .Denote as A, the connected component of xn in B6(xn)n f-mn-l(An-l). Then (ii) holds, and A, n BB6(xn) # 8 implies (iii). By (v) we have (iv).

L e m m a 2.2.48. For 0 < 6 < E there exists N > 0 such that for all x E X and y E W,d(x, d) with d(y, x) = 6, there is 0 < n 5 N such that

Proof. If this is false, then there exist sequences x, E X, y, E W,d(x,, d) such that d(x,, y,) = 6 and d(f-j(xn), f-j(y,)) 5 E for 0 5 j 5 n. If xn -, x and y, + y as n -, oo, then x # y and d(fn(x),fn(y)) 5 E for all n E Z, thus contradicting.

Lemma 2.2.49.

There exists 60 > 0 such that

for allx E X and0 < 6 < 60. Proof. If this is false, for n > 0 then there exist x, E X and 0 such that w,d(~n,d)n B6,(~n)# W&(xn,d) n B 6 , ( ~ n ) -

< 6, < lln

74

CHAPTER 2

Thus we can find yn E Wic(xn,d) and mn

Since yn E Wad,(%,, d), for all m with m

> 0 satisfying

> -mn

from which d(f m(x), f m ( ~ )5) 2&, m E Z where fmn(xn) + x and fmn(yn) + y as n -t oo. Therefore, x = y, but d(x, y) 2 E. This is a contradiction.

Wl(x, dl

Figure 8

L e m m a 2.2.50. Let Z = inf{d(x, y) : d(f-'(x), f - ' ( ~ ) ) > E,X, y E X) and > 0 be as in Lemma 2.2.49. For 0 < 6 < min{b0,Z/2) there exists let N = N ( 6 ) > 0 such that if x E X and A c W,S(x,d) is a compact connected set containing x and if A n Ba(x) # 0, then there exist 0 < m < N, a, p E f -m(A) and compact connected sets Aa,Ap such that

Proof. Let N = N(6) > 0 be as in Lemma 2.2.48. Since A n dB6(x) # 0 and x E A c W,"(x,d) by assumption, for y E W,"(x, d) with d(x, y) = 6 we can find 0 5 m 5 N - 1 such that

52.2 Expansivity

By (f) and the fact that A C W:(X, d)

f -m(A) c Wcd(f -m(x), 4.

(4 Since f -"(x)

E f -m(A), we have

@I

f-m(x)~Wi(~,d) forw~f-~(A).

Figure 9 F'rom (g) and (h) it follows that for all w E f-m(A)

By (e), d(f-l o f-m(z), f-I o fWm(z))> E for some z E A, and so d(f-m(z), f -" (x)) 2 Z.

Figure 10 Since z , x E A, we have diam(f -"(A)) 2 Z. Since 36 < Z, we can find such that d(a,p) > 36. Obviously a $! B6(P) and P # Ba(a). Therefore, (c) holds. Let A,, A, be the connected components of a and p in f -m(A) n B ~ ( ( Y ) , f-m(A) n Bs(@) respectively. From (g) it follows that (a) holds. Since

a,p E f-m(A)

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76

A, c f -"(A) and Ap c f-m(A), by (i) we have A, c Wzd,(a,d) and Ap c W.&(P, d). Thus (d) holds. It remains to check (b). This is followed by replacing X, A and C with f-m(A), f-m(A) n &(a) and A, respectively (in fact, since fqm(A) n Ba(a) is a closed ball with radius 6 of a, we have 8(f -"(A) n B6(a)) = {x E f -"(A) : d(x,a) = a) ). Proof of Theorem 2.2.44. It is enough to verify that if f: X -+ X is minimal and expansive then dim(X) = 0. Suppose dim(X) > 0 and 7 > 0 are as in Lemma 2.2.47. For 0 < 6 < E choose N = N(6) > 0 as in Lemma 2.2.48 and let 60 > 0 be as in Lemma 2.2.49. Take Z > 0 as in Lemma 2.2.50, and for 0 < 6 < min{60,Z/2, 7) define

If 71 = 0, then X is a set consisting of finite points. Indeed, for all n there exist x,, y, E X and 0 5 in,j, 5 N(6) such that

We can suppose that x, n 2 0. Thus

-+

x, y,

4

y as n

-+

>0

ca and j, = j,i, = i for all

Notice that i # j. Suppose i - j > 0, then we have fj-;(x) = y E W&(x, d) C W" (x, d) and f j - i ( w a ( ~ d)) , n ~ " ( 3d), # 0 since fj-i(x) E f j - i ( ~ " ( x , d ) ) . By Lemma 2.2.46, X contains a periodic point xo and thus the orbit Of(xo) equals X (since f : X + X is minimal). It remains to check the case 71 > 0. For this case we shall derive a contradiction by showing the existence of a nonempty open set U and a point p such that fn(p) # U for all n 2 0. If we have U and p as above, then B n U = 0 and f (B) c B where B is the closure of the semi-orbit {p, f (p), . -1. Since A= f n ( B ) is nonempty and f(A) = A, we have X = A, thus contradicting. We first construct a collection of compact connected sets A,, a sequence of points x, E A, and a nonempty openset U c X that satisfy the following

.

nz=o

(4 (b)

An n aBa(xn)

# 0,

(c)

An c W&(xn,d), f-""(A,) > A,+,

for some 0 5 m, <_ N(6),

(d)

fj(nn)nU=O

forall O < j < m , .

$2.2 Expansivity

Since 6 define

< 7, there is a

77

E X with X$(a) n 6B6(a) # x0 = a,

0 by Lemma 2.2.47.

We

A0 = E:(xo)

and show the existence of U # 0 with diam(U) < y1/2 such that U n A. # 0. Suppose U n Ao # 0 for all open sets U. Then A. is dense in X and so A. = X (since A. is closed). Since diam(Ao) 5 26 < e and f is expansive, X consists of one point set, thus contradicting dim(X) > 0.

/

Figure 11

If mo = 0, then (a), (b) and (d) hold for Ao. To construct Al satisfying (c) we use Lemma 2.2.50. In fact, since don 6B6(xo) # 0 and A0 C W,"(xo,d), by Lemma 2.2.50 we can find 0 < ml < N(6), a, /3 E f (Ao) and compact connected sets A,, Ap satisfying (a), (b), (c) and (d) of Lemma 2.2.50. Then

<

, y E f j ( ~ p for ) some 0 5 j ml. Thus Indeed, if y E U n {Ujm0 f j ( ~ p ) ) then there is z E Ap such that fj(z) = y. By assumption of (e), for x E U fl f j ( ~ , ) there is w E A, such that fj(w) = x. Since x, y E U,

and since w E A, and d(w, z )

tE

>6

Ap (by (c) and (d) of Lemma 2.2.50).

By Lemma 2.2.50 (d), w E W,d(a,d) and by Lemma 2.2.50 (a), a E W&(p,d). Thus, w E W&(z, d). However d(f i(w), fj(z)) = d(x, y) 1 71 by definition, thus contradicting (f).

CHAPTER 2

78

Since p E f -ml(Ao) n Ba(P), we denote as Al the connected component of (AO)f l B6(P) and put xl = 0. Then we have /3 in f (a') (b')

A1 n 8B6(21) # 0 A1 c W2dC(~1 9 d )

(C'1

A1 C f-"'(Ao),

(d')

{Uf ' ( ~ 1 ) ) n u = 0

(by Lemma 2.2.50) , (by Lemma 2.2.50 (d))

,

ml

(by (el)

j=O

Replace A. by Al and repeat the above technique. Then there exist a compact connected set A2 and m2 > 0 satisfying (a'), (b'), (c') and (d'). Continuing inductively this technique, we have {A,), {x,) and U satisfying (a), (b), (c) and (d). By (c) we have A, 3 f ml (Al) 3 . .. 3 f ml+...+mm (An) 3

n,,,

and thus f-"l(p) E

-..

f mi+"'+m- (A,) # 0, from which take a point p. Obviously By (d) we have f-ml+j(p) 4 U for 0 5 j 5 m l and thus

and consequently f -'(p) $ U for all i

> 0.

Let f : X -t X be a continuous surjection of a compact metric space. The map f is said to be minimal if, for all x E X, the orbit of(2) = {fn(x) : n 2 0) is dense in X. A closed subset E is said to be a minimal set if f ( E ) = E and flE: E -t E is minimal.

Remark 2.2.51. It remains a question of whether Theorem 2.2.44 is true for a c-expansive continuous surjection. We remark that the question can not be shown by the technique of the proof of Theorem 2.2.44, nor by the way through the inverse limit as in Theorem 2.2.39 (because the map is not a local homeomorphism). 82.3 Pseudo orbit tracing property

A sequence of points {xi : a < i < b) of a metric space X is called a Spseudo orbit of f if d(f (xi), xi+l) < 6 for i E (a, b- 1). Given E > O a S-pseudo orbit {xi) is said to be &-traced by a point x E X if d(fi(x), xi) < E for every i E (a, b). Here the symbols a and b are taken as -oo 5 a < b oo if f is bijective and as 0 5 a < b 5 oo if f is not bijective. We say that f has the pseudo orbit tracing property (abbrev. POTP) if for every E > 0 there is S > 0 such that every 6-pseudo orbit of f can be &-traced by some point of X. For compact spaces this property is independent of the compatible metrics used.

<

$2.3 Pseudo orbit tracing property

79

Remark 2.3.1. In the case where X is compact, a homeomorphism f : X -+ X has POTP if for every E > 0 there is 6 > 0 such that every (one sided) 6-pseudo orbit {xi : i 0) of f can be €-traced by some point of X. Indeed, let {xi : i E Z) be a &pseudo orbit of f . For each n 2 0 define a (one sided) &pseudo orbit {zr : i 0) by z r = xi-,, for all i 2 0, and let 2;" be a tracing point for {zl : i 2 0). Then it is easily checked that an accumulation point of {fn(zn)) is a tracing point for {xi : i E Z).

>

>

Theorem 2.3.2. Let X be a compact metric space and denote as id the identity map of X . Then id : X -+ X has P O T P if and only if X is totally disconnected. Proof. If X is totally disconnected, for E > 0 there exists a finite open cover {Uo,. ,Un) of X such that U;n Uj = 0 for i # j and diam(Ui) < E for 2. Choose 6 with 0 < 6 < min{d(Ui,Uj) : i # j) and fix a &pseudo orbit {xk : k E Z) for id. Then there is Ui such that {xk) c Ui and so we can find in Ui an €-tracing point for {xk). Therefore id : X -t X has POTP. Conversely, suppose dim(X) # 0. Then there is a closed connected subset F such that diam(F) > 0. Since F is compact, diam(F) = d(xo, yo) = €0 for some xo, yo E F. Let ~1 = ~ 0 1 3 .Since F is connected, for any 6 > 0 we can construct a &pseudo orbit from xo to yo in F which is not €1-traced in X. This is a contradiction.

..

Theorem 2.3.3. Let f : X + X be a continuous map of a compact metric space and let 12 > 0 be an integer. Then f has POTP if and only if so does

fk. Proof. We first notice the following (I), (2) and (3). (1) Let E > 0. Then there is E > ~1 > 0 such that each €1-finite pseudo orbit {xi : 0 5 i 5 12) satisfies

and d(x, y)

< €1

implies

(2) Let €1 be as in (1). Then there is 61 > 0 such that each C1-pseudo orbit f k is ~1-tracedby some point. (3) Let €1 and S1 be as in (1) and (2). Then there is 6 > 0 such that each 6-finite pseudo orbit { z i : 0 5 i 5 12) is 61-traced by zo E X. With these properties we show that each 6-pseudo orbit {yi : i 2 0) for f is €-traced by some point. Write

for

For fixed i, {yki+j : 0 5 j 5 12) is a &finite pseudo orbit for f . By (3) we have

80

CHAPTER 2

If j = k then d ( f k(yki),Yki+k) = d ( f k ( x i ) xi+') , < 6'. Thus { x i ) is a 6'pseudo orbit for f k . By ( 2 ) there is y E X such that d ( f k i ( y ) , x i ) < for i 2 0. Thus, d ( f k i ( y ) Y, k i ) < &I for i 2 0. On the other hand, since {yki+i : 0 5 j 5 k) is an el-pseudo orbit for f , by (1) we have

and d ( f "+'(Y), fi(yki)) < &/2,

0

5j 5k

since d( f " ( y ) , yki) < E X . Therefore,

, Since i is arbitrary, we have d ( f n ( y ) y,) 6-pseudo orbit {yi).

< E for

n >_ 0 and y &-traces the

Theorem 2.3.4. Let X be a compact metric space. If f : X -,X is a homeomorphism with POTP, then so is f-'. Proof. For every E > 0 let 6 > 0 be a number such that each 6-pseudo orbit { x i ) is &-tracedby a point y E X. Choose 6' > 0 such that d(x,y) < 6' implies d ( f ( x ) ,f ( y ) ) < 6 and put g = f-'. I t is enough to see that each 6'-pseudo orbit { y i ) for g is &-tracedby some point. Since d(g(yi),yi+l) < 6' for i E Z , we have d(yi, f(yi+l)) < 6 for i E Z. Letting xi = y-i for i E Z, { x i ) is a 6-pseudo orbit for f and thus there is Y E X with d ( f i ( Y ) , x i ) < E for i E Z. This implies d ( f - " y ) , ~ _ ~<) E for i E Z and therefore d(gi(y),yi) < E for i E Z.

Theorem 2.3.5. Let X and Y be metric spaces and X x Y the product topological space with metric

where dl and dz are metrics for X and Y respectively. Let f : X g: Y + Y be continuous maps and let f x g be the map defined by

+

X and

Then f x g has POTP i f and only if both f and g have POTP. Proof. Suppose f x g has POTP. For every E > 0 let { x i ) and { y i ) be 6-pseudo orbit for f and g respectively. Then { ( x i ,yi)) is a 6-pseudo orbit for f x g. Thus there is ( x ,y) E X x Y with d ( ( f x g)i(x,y), ( x i ,yi)) < E for i 2 0. Then, d l ( f i ( x ) ,x i ) < E and d ~ ( ~ ' ( yi) y ) ,< E for i 2 0. Also the converse is proved.

$2.3 Pseudo orbit tracing property

81

Theorem 2.3.6. Let f : X + X be a continuous map of a compact metric space X and let h: X -+ Y be a homeomorphism. Then g = h o f o h-' has POTP i f and only i f so does f . Proof. For every E > 0 there is 61 > 0 such that d(x,y) < ~1 implies dl(h(x), h ( y ) )< E where d' is a metric for Y. If f has POTP, then there is b1 > 0 such that each 61-pseudo orbit { x i ) for f is el-traced by some point. Let 6 > 0 be a number such that d'(x, y) < 6 implies d(h-l(x), h W 1 ( y ) < ) 61. Then it is enough to see that each 6-pseudo orbit {yi) for g is &-traced by some point. To do this put xi = h-'(yi) for i 2 0. Since dl(g(yi),yi+l) < 6 for i 0, we have d ( f ( ~ i ) , ~ i +=ld(h-' ) 0 g(Yi),h-'(Yi+l)) < 61 for i 0. Thus { x i ) is a 61-pseudo orbit for f and SO d ( f i ( ( y 9 x i )< e l ( i 0 ) for some y E X . Therefore, d'(h o f ' ( y ) )h, ( z i ) )= d'(gi o h ( y ) ,yi) < E for i 2. 0.

>

>

>

Theorem 2.3.7. Let X be a compact metric space. A continuous surjection f : X + X has POTP if and only if for every E > 0 there is 6 > 0 such that for ( x i ) E X' with d(xi,xi+l) < 6 (i E Z ) there exists a point (yi) E X f so that d(yi,xi) < E for i E Z. Proof. This is easily checked as in Remark 2.3.1. By Theorem 2.3.7 we can give the definitions of &pseudo orbit and &-tracing point for continuous surjections as follows. If ( x i ) E X' has the property that d ( f ( x i ) xi+l) , < 6 for i E Z , then ( x i ) is called a 6-pseudo orbit of f . If (yi) E Xf satisfies d(yi,x i ) < (i E Z ) , then ( y i ) is called an s-tracing point for ( x i ) .

Theorem 2.3.8. Let X be a compact metric space. If a continuous sujection f : X + X has POTP then u : Xf + Xf obeys POTP. Proof. Let a = diam(X) and 7 > 0 be a number such that

E

> 0. Choose N > 0 with

c ~ / 2 ~<-s,~ and let

By POTP of f there is 6' > 0 such that any 6'-pseudo orbit of f is 7-traced. Choose 6 > 0 with 0 < 2N6 < 6'. Let k > 0 and suppose { ( x p ): 0 5 n 5 k ) is a finite 6-pseudo orbit of u in Xf.Then we have

where i ( ( x i ) ,( y i ) ) = CZ-, d(zi,yi)/2Iil, and hence { x l N : 0 5 n 6'-pseudo orbit of f , from which we can find y E X such that

d ( f n ( ~ ) , x ! ! N5) 7

(0 I n I k).

I k ) is a

CHAPTER 2

82

We put yi-N = fi(y) for i 2: 0 and take y i - ~E f - l ( ~ ; + ~ - N )for i (yi) E Xfand

< 0.

Then

<

because of d(f "(yi), 29) 5 ~ / 8 ( l i l N ) by the fact that d(fn(y-N), xEN) 5 7 .

Theorem 2.3.9. If u : Xf + Xf has P O T P and f : X homeomorphism, then f has POTP.

-+

X is a local

Proof. For 8 > 0 let 6 > 0 be a number such that every &pseudo orbit of u is €-traced by some point of Xf.Let a = diam(X) and choose N such that 0 < ~ r / 2 ~< -6.~Then we can find 7 > 0 such that if d(x, y) 5 7 then there are {xi : -N i 5 N ) and {y; : - N 5 i 5 N} such that xo = x, f (xi) = xi+l and yo = y, f (pi) = Yi+l and d(xi, yi) < 618 for lil 5 N. Let { t i : 0 5 i < oo} be a 7-pseudo orbit of f . Then for i 2 0 we can find (zi) E Xf such that zi = zi and {(zi,)} is a Cpseudo orbit of u. Hence there is (x.) E Xf such that i(ui(xn), (zk)) 5 E for 0 < i < 00, and then

<

Theorem 2.3.10. Let X be a compact metric space. A positively expansive map f: X + X has P O T P if and only if f is an open map (i.e. expanding). Proof. If f is expanding, then f is a local homeomorphism and X has a hyperbolic metric (Theorem 2.2.10). Thus, by using the technique of Theorem 1.2.1 (2) we see that the map has POTP. Conversely, suppose f has POTP. Let U be an open set of X and for x E U choose E > 0 such that the €-neighborhood, Uc(x), is contained in U. Then there is 0 < 6 e such that every 6-pseudo orbit of f is €-traced, where e is an expansive constant for f . If z E Ua(f (x)), then a sequence {x, z, f (z), f 2(z), ,f i(z), ) is a &pseudo orbit of f , and so it is E-traced by some point y E X. Then d(f'(f(y)), fi(z)) < 6 e for all i 2 0, which implies f(y) = z. Since y E Uc(x), we have f(U,(x)) > Ua(f(x)). Therefore f (U) is open in X .

<

...

<

$2.3 Pseudo orbit tracing property

83

Theorem 2.3.11. Let X be a compact metric space. If X is connected and f : X + X is an expanding map, then the set of periodic points off i s dense in X . Proof. Let D be a hyperbolic metric for f and let (60, 8) be Eilenberg's constants for (X, f). Then there are el > 0 and X > 1 such that D(x,y) el (x, y E X ) implies D( f (x), f (y)) 2 XD(x, y). Notice that 8 is chosen such that 8 < el. Then D(f(x), y) < 8 implies Ua,/x(x) n f-'(3) # 0 where U,(x) = { z E X : D(e,x) < a). It is enough to see that for each v > 0, U,,(x) contains a periodic point of f . Fix 0 < v < e1/2. Since f has POTP by Theorem 2.3.10, there is 6 > 0 such that every 6-pseudo orbit o f f is v-traced by some point of X. Let {Ul,. .,U,) be a finite open cover of X such that the diameter of each Ui is less than 8. Then we can find r > 0 such that (n 1)8/Xr-' < 6. Take and fix xo E X. Since X is connected, there is a finite sequence

<

.

+

< <

such that yj, yj+l E Uij for 0 j k and k 5 n. Find y: E X with properties that f(y:) = yk and D(y:, fr-'(xo)) < $/A. Since D(yk-I, f(yk)) < 8, there is y:-l E X such that f (yk-,) = yk-l and D(Y:-~, y:) < $/A. Continuing this fashion, we obtain a finite sequence

< <

such that f (y;) = yj-1 and D(Y;-~,y,!) < 8/X for 0 j k. Since d(yk, fP-'(xo)) < $/A, we have yi E X such that f(y;) = yk and D ( Y ~fP-2(xo)) , < 8/X2. Since D(Y:-~, f(yi)) < 8/X, we have also yi-lE X with f (I/;-,) = and D(y:-, ,y:) < 8/X2. In this manner, a finite sequence

is constructed and by induction on 1 a finite sequence

exists. Then we have

and hence {xo, %,-I,. .. ,X I , xo, x,.-1,. ..) is a periodic 6-pseudo orbit of f . Since f has POTP, there is a v-tracing point p E U,(xo) for this Cpseudo orbit. Clearly f '(p) is also a v-tracing point. Thus d(fa(p),f "'(p)) < 2v < el for all i 2 0. Since f is positively expansive (with expansive constant el), we have p = f '(p).

CHAPTER 2

84

Theorem 2.3.12. Let X be a compact metric space and denote the product space by xZ= n r a X i where each Xi is a copy of X . A metric for X' is defined by

Then the shift map u : xZ-, xZhas POTP. Proof. For every E > 0 choose 6 with 0 &pseudo orbit for o. Then, for i E Z

for all k E Z, and so d(xi+,,

< 26 <

E.

< 2146 (i, k E Z).

Let {xi : i E Z) be a

Define a point x by

Then x E X' and (ui(x))&= x;+& for i, k E Z. When k 2 0, we have

and when k < 0 the inequality ~(x:,x;+~) < 2lk1+'6 is also calculated. Therefore, d(xi7ui(x)) 26 < E for i E Z, i.e. the point x is an &-tracing point.

<

Let X be a compact metric space with metric d. A product space XZ = {(xi) : xi E X, i E Z) has metrics

cW i€Z

214

and

, : i E Z) sup{--d ( ~ iyi) 2111

for x = (xi),y = (yi) E x'. The reader can check that these metric generate the same topology of xZ. Theorem 2.3.13. Under the notations of Theorem 2.3.1.2, if topological dimension of X is non-zero, then u : xZ+ xZis not expansive. Proof. This follows from facts that dim(xz) < oo whenever u is expansive (In fact, since dim(X) > 0, we have dim(xz) = Eidim(Xi) = oo, [H-W]).

52.3 Pseudo orbit tracing property

85

Theorem 2.3.14. Under the notations and the assumptions of Lemma 2.2.94, g: X + X has POTP if and only iff + has POTP.

:x

x

is uniformly continuous follows from the proof of Proof.That f:X -+ Lemma 2.2.34. We first show that if f:X -+ W has POTP then so does g:X -bX. Let 60 > 0 be as in Lemma 2.2.34. For e > 0 there is 0 < 6 < 60 such that Let {y;) be any each 6-pseudo orbit {pi) for f is e-traced by some point of &pseudo orbit of g. F i x k > 0 and choose p-k E such that ~ ( p - ~=)y-k and fix p-k. Inductively we choose pi E X such that ?r(pi) = yi for i 2 -12. Since f(p;) E X,we have n o f(p;) = go?r(p;) = g(y;). Since y;+l E Ua(g(yi)), there is a unique pi+l E Ua(f (pi)) such that ~ ( p ; )+ = ~ yi+l. By the choice of {pi : i 2 -k), {pi) is a &pseudo orbit for f and thus there is an etracing point yk of {pi), i.e. a(fi(yk),pi) 5 E for i -k. Let n(yk) = zk. Then, for

x.

>

i20

If k -r oo, then {yi : i E Z) is a 6-pseudo orbit and there is a subsequence {zkj) such that aka + z as j + m. Thus d(gn(zkj),y,) 5 e for n 2 -kj and so d(gn(z), y,) 5 E for all n E Z. Therefore g has POTP. To show the converse, let 60 > 0 be as above. Then, for e > 0 we can find 0 < 6 < 6012 such that each 6-pseudo orbit for g is e-traced by some point of X. Since f : -+ X is uniformly continuous, we may suppose that q) < E implies j(f (p), f (9)) < 6012. Let {pi} be a 6-pseudo orbit for f and let y; = for i. Obviously {y,) is a &pseudo orbit for g. Indeed, since a(f (pi),pi+l) < 6 for all i, d(g(~i),yi+l) = d(n 0 f (pi), n ( ~ i + i ) = )

a(f (pi),pi+l) < 6.

Thus there is z E X such that d(gi(z), yi) < e for all i. Since d(z, yo) there is q E Ua(po) such that r(q) = z. Note that

< E,

- all i. Since n is locally isometric, we have z(q,po) < e when i = 0. If for -d(f-'(q),pi-1) < E for i 2 1, then we have a(fi(q),pi) < E. Indeed, since d( f 0 f (q), f (pi-I)) < 6012 and z(f (~i-l),pi)< 6, we have a ( fi(q),pi) < 60 and d(fi(q),pi) = d ( r 0 fi(q),~(pi))= d(gi(z), yi) < E Thus the point q &-tracesthe one sided 6-pseudo orbit {pi : i L 0). Similarly we can show that ;i(fi(q),pi) < E for i 5 0. Therefore the conclusion is obtained.

'-'

CHAPTER 2

86

Theorem 2.3.15. Let f be a linear map of the euclidean space Rn and let d be the euclidean metric for Wn. Then f has POTP under d if and only i f f is hyperbolic, i.e. it has no eigenvalues of modulus 1. Proof. If f is hyperbolic, then Rn splits into the direct sum of f-invariant subspaces, Wn = Es@EU, such that f : Es + Edis contracting and f : EU-+ Eu is expanding. See the proof of Lemma 2.2.33. Thus, in the same manner as the proof of Theorem 1.2.1 (2) we have that both f l E . and flEu have POTP. Since f is linearly conjugate to f l E . x f l E Y , by Theorem 2.3.5 it follows that f has POTP. Conversely, suppose f has an eigenvalue with absolute value one. Then there is an f -invariant subspace F such that f : F + F is an isometry under d. Let E > 0. Since f is a linear map, by using the Jordan form of f (in the real field) we can choose K = K(E) > 0 such that 11 f"(x)11 -+ m as i -+ m whenever llxll < E and 11 fi(x)ll > K for some i 2 0. Since f l F is an isometry, for any 6 > 0 we can construct in F an &pseudo orbit which comes from the origin and goes to a set of points x with llxll = 2K, which implies that f does not have POTP. Theorem 2.3.16. A toral endomorphism has POTP if and only if it is hyperbolic. Proof. This is obtained from Theorems 2.3.14 and 2.3.15. Let k be a natural number and Yk = { l , . . . ,k). A metric d for :Y is defined by d(x, y) = 2-"' if m is the largest natural number with xn = y, for all In\ < m, and d(x, y ) = 1 if x0 # yo. Such a metric is uniformly equivalent to the metric d defined as above. If S is a closed subset of Yf with a(S)= S, then 01s : S -+ S is called a subshift. We sometimes write u p as a : S -+ S. A subshift a : S -+ S is said to Yk be of finite type if there exist some natural number N and a subset C C with the property that x = (xi) E S if and only if each block (xi,-.- ,x;+N) in x of length N 1is one of the prescribed blocks. The smallest such natural number N is called the order of the subshift of finite type.

nr

+

Theorem 2.3.17. Every subshift of finite type is topologically conjugate to one of order 1. Proof. We just take a new symbolic space yZ consisting of the allowable blocks of length N, and then the conjugacy cp ::Y + yZ is given by

The shift 5 :yZ -+ yZ is of order 1 and cp o o = 5 o cp holds.

$2.3 Pseudo orbit tracing property

87

Let A be a k x 12 matrix of 0's and l's, called a transition matrix. We define the compact set Then XA is invariant under the shift map (a(XA) = EA). Such a shift map olzr is called a Markou subshift. Every subshift of finite type with order 1 is a Markov subshift.

Theorem 2.3.18 (Walters [W3]). Let S be a closed subset of :Y and let a : S -, S be the subshift. Then a : S + S has POTP if and only if a is a subshift of finite type. Proof. If a : S + S is a Markov subshift, obviously o is of order 1. Let E > 0 and take m > 0 such that 2-" < E . Then xi = yi for lil 5 m when d(x, y) < 2-(m+1). Let {xi : i E Z) be a 2-(m+1)-pseudo orbit for u p , i.e. {xi} c S

and

d(a(zi), xi++')< 2-(m++')

for all i E Z. Then xf = z:+' for all i E Z. Now put xn = xg for n E Z and define x = (xn) E.:Y Then we have and thus x E S. Since (uix)j = xi+j = xf for Ijl 5 m and i, we have

for i E Z. Therefore x is the &-tracingpoint of {xi). If a : S + S has POTP, then we show that a is a Markov subshift. Since o ::Y +:Y is expansive ( under d ), so is 01s. Let E > 0 be an expansive constant for u p and let 0 < 6 < ~ / 2 .Then there is N > 0 such that Denote by B the set of all blocks (xi-N, 2N 1 for all x E S and write

+

- . ,xi,. .- ,xi+N) in x with length

Obviously, S(B) > B and u(S(B)) = S(B). It is easy to see that S(B) is closed and a l s ( ~is) of finite. Since ( Y ( - ~ , ,y i + ~ E) B for each y E S(B) and i E Z, for i E Z there is xi E S such that yi+j = xf for Ijl 5 N. Thus, ~ ( U ~ ( ~ < ) ,6Zfor ~ )i E Z, i.e. {xi} is a 6-pseudo orbit for 01s. Since u p is POTP, ther is z E S such that 2 all i E Z. Then we have d(ai(z), xi) < ~ / for for all i E Z. Since z, y E S and 01s is expansive, we have z = y and so y E S. Therefore, S(B) = B.

CHAPTER 2

88

Remark 2.3.19. The family { f , : s E [O,1]) of tent maps f , : [O, 21 + [O, 2) defined by

has the following properties : ( 1 ) f , has POTP for almost all parameters (with the exception of Lebesgue measure zero of [O, 2]), ( 2 ) the set of parameters for which f , does not have POTP is locally uncountable. This is a result due to Coven-Kan-York [C-K-Y]. A set is said to be locally uncountable if its intersection with any open set is uncountable. We introduce the family of functions cp,, n 2 1, on 21 by cpn(s) = f f ( 1 ) . A point sE 21 is said to be a periodic point with period n if f r ( 1 ) = 1 (cpn(s)= 1) but f ! ( l ) # 1 for 1 5 k < n. Show that periodic parameters are dense in 21 by using the mean value theorem. Then ( 1 ) will be obtained. ( 2 ) will be shown by applying the kneading theory and Theorem 2 of [D-G-PI. See [C-K-Y] for the details.

[a,

[a,

[a,

52.4 Topological Anosov maps (TA-maps)

E

Let f : X + X be a continuous surjection of a compact metric space. Let We set for x E X

> 0.

as before (henceforth, as the metric d is fixed, the local stable set W,d(x,d ) is written by W,d(x))and for x = ( x i ) E X f

Lemma 2.4.1. Let f: X + X be a c-expansive continuous suljection with expansive constant e. For 7 > 0 there exists n, > 0 such that for x = ( x i ) E Xf a n d x ~ X (1) fn(W,"(x))C W , d ( f n ( x ) )for n 2 n,, ( 2 ) if y = ( y i ) E X f and d(yi,x i ) 5 e for i 5 0 ( i.e. yo E W,U(x) ), then d(y-n, 2 - n ) 5 7 for n 2 n, ( i.e. W,U(x)c f n ( W y ( a - n x ) ) for n 2 n, ).

Proof. If ( 1 ) is false, then there are xn, yn E X and mn 2 n such that yn E W,"(xn) and d ( f m n ( x n )f, " " ( y n ) ) 2 7. Fix i > 0. If fmn-i(xn) + xLi and f mn-i(yn) -+ yLi as n + oo, then d(x'_,, yLi) 2 e since d( f j ( x n ) , f j ( y n ) ) 5 e for 0 5 j < m,. We have d(xb,yh) 1 7 when i = 0. Thus yb E W,"(xf)

$2.4 Topological Anosov maps (TA-maps)

89

where x' = (x:) because fi(x'_,) = xb and fi(yl_i) = yb for i 2 0. Since yn E W,d(xn), we have

and hence y; E W,d(xb). Since W,U(x1) n W,d(x&)3 yk, we have xb = yb by c-expansivity, thus contradicting. If (2) is false, then we can find (x:), (yl) E Xf and mn 2 n such that e for i 5 0. From this we obtain a d(x", Y-, 1> - 7 and d(x?, y l ) contradiction.

<

For a continuous surjection f : X + X we define the stable and unstable sets WS(x)={y E X : n-mo lim d ( f n ( x ) , f n ( y ) ) = ~ ) WU((xi)) = {yo E X : 3(yi) E

lim d ( ~ - ~ , y - ; )= 0) Xf ~ . t .i-00

for x E X and (xi) E Xf.

Remark 2.4.2.

Let f : X -t X and 9: Y t Y be continuous surjections of compact metric spaces. Suppose h o f = g o h holds for some homeomorphism h: X 4 Y. Then the following are easily checked : (1) if x E X and WS(s) is the stable set of f , then h(Ws(x)) is the stable set of g at h(x), (2) if (xi) is a point in Xf and WU((xi)) is the unstable set of f , then (h(xi)) belongs to Yg and h(Wu((xi))) is the unstable set of g at (h(~i)). t X be a c-ezpansive continuous suvjection with expansive constant e and let e > E > 0 and x E X. Then

Lemma 2.4.3. Let f: X

and for x = (xi) E Xf

Proof. This follows from Lemma 2.4.1. I0 Now we shall give Anosov property for continuous surjections of metric spaces, in a general setting. We say that a continuous surjection f : X -, X is a topological Anosov map (abbreviated TA-map ) if f is c-expansive and

90

CHAPTER 2

has POTP. If, in particular, f is a homeomorphism, then f is said to be a topological Anosov homeomorphism if it is expansive and has POTP. A TAmap f : X --+ X is special if f satisfies the property that Wu((x;)) = Wu((yi)) for every (xi), (yi) E Xf with xo = yo. The notion of special TA-maps is a generalization of that of special Anosov differentiable maps. Let X and Y be metric spaces. We say that two continuous maps f : X -,X and g: Y --+ Y are topologically conjugate if there exists a homeomorphism h: Y --+ X such that f o h = h o g. In this case any orbit of g is mapped by h to a homeomorphic orbit of f . If h is a continuous surjection, then f is said to be topologically semi-conjugate to g, in other words, f is called a factor of 9.

Remark 2.4.4. Let f : X -+ X and g:Y --+ Y be continuous surjection of compact metric spaces. If f and g are topologically conjugate, then (1) f is a TA-homeomorphism if and only if so is g, (2) f is a TA-map if and only if so is g, (3) f is a special TA-map if and only if so is g. Indeed, (1)and (2) follow from Theorem 2.3.6 together with Theorems 2.2.5 (3) and 2.2.30 (3). By Remark 2.4.2 (2) we obtain (3). In the remainder of this chapter we discuss the relation between TA-maps and the topological stability of homeomorphisms (or self-covering maps) on closed topological manifolds. Let X be a compact metric space with metric d. A homeomorphism f : X --+ X is said to be topologically stable in the class of homeomorphims if for E > 0 there is 6 > 0 such that for a homeomorphism g with d(f (x), g(x)) < 6 for all x there is a continuous map h so that h o g = f o h and d(h(x),x) < E for all x. A self-covering map f is said to be topologically stable in the class of selfcovering maps if f satisfies the conditions as above in the class of self-covering maps.

Theorem 2.4.5 (Walters [W3]). If a homeomorphism f: X -+ X of a compact metric space is topological Anosov, then f is topologically stable in the class of homeomorphisms. Proof. Let e > 0 be an expansive constant for f and fix 0 < E < e/3. Let 0 < 6 < e/3 be a number with the property of POTP. By expansivity it is checked that there is a unique x E X which &-traces a given 6-pseudo orbit {xi). Indeed, let y E X be another €-tracing point of {xi}. Then we have

for all i E Z and thus x = y. Let g: X --+ X be a homeomorphism with d(g(x), f (x)) < 6 for all x E X. Let x E X. Since d(f o gn(x),gn+'(x)) < 6 for n, {gn(x)} is a 6-pseudo orbit of f . Thus there is a unique point h(x) E X whose f-orbit &-traces {gn(x)}.

$2.4 Topological Anosov maps (TA-maps)

91

This defines a map h: X -+ X with d ( f n o h ( x ) ,gn(x)) < E for n and x E X . Letting n = 0, we have d ( h ( x ) ,x ) < E for x E X . Since d(f o h o g ( x ) ,gn+l ( 3 ) )< E for n E Z and

for n E Z, we have h o g ( x ) = f o h ( x ) for x E X. Finally, we show that h is continuous. Let X > 0. Then we can choose N > 0 such that d( f " ( x ) ,f " ( y ) ) < e for In1 5 N implies d(x, y) < A. This is checked as follows. Suppose this is false. Let a be an open cover of X with diameter < e. Then there is E > 0 such that for each j > 0 there are a,, y, E X with d ( x j , y j ) > E and Aj,i E a ( - j 5 i 5 j ) with xj,yj E f - j ( ~ ~ ,Since ~). X is compact, we can suppose that xj -+ x and yj -+ y. Thus x # y. Consider the sets AjVofor j. Then infinitely many sets of them coincide since a is finite. Thus x j , yj E A. for infinitely many j and so x , y E 20.Similarly, for infinitely many Aj,, they coincide and we have A, E a with x , y E Therefore x , y E 00 f--(a,,), thus contracting the fact that f is expansive. Choose q > 0 such that d(x,y) < q implies d(gn(x),gn(y)) < e/3 for In1 5 N. If d ( x ,y) < q then

ni=-j

nn=-,

Therefore, d ( x ,y)

f-"(x,).

< q implies d(h(x),h ( y ) )< X and continuity of h is proved.

Figure 12

Remark 2.4.6. Let h: X -+ X be as above and suppose that X is a closed topological manifold. Then h : X + X is surjective. For the proof the readers may refer to Remarks 6.7.10 and 10.6.4 which will be mentioned in Chapters 6 and 10 respectively. However, in general h is not surjective. We give an example due to Walters.

92

CHAPTER 2

-t Y : where Y2 = ( 0 , l ) . Then a is expansive Consider the shift a : -t and has POTP. Let m > 0 and define g: by

given as before. If Then d ( g ( x ) , o ( x ) )5 112" where d is the metric for m is sufficiently large then we have h o a = g o h for a continuous map with d ( h ( x ) , x ) < 1/2,(x E YZZ). Since g2m+1(x)= x for all x E ,Y : we have h ( x ) = h o gam+l(x) = a2,+l o h ( x ) for all x E @. Therefore h is not surjective.

Theorem 2.4.7. Let M be a closed topological mandfold and f: M + M a self-covering map, but not injective. If f is a TA-map and has topological stability in the class of self-covering maps, then f is positively expansive. For the proof we need some properties of TA-maps on closed toplogical manifolds. Thus we shall postpone the proof until Theorem 6.7.11 in $6.7 of Chapter 6. Let f: X + X be a homeomorphism of a compact metric space. A point p E X is nonwandaring for f if, for any neighborhood U of p and any integer no > 0 , there exists n with In1 > no such that f " ( U )n U # 8. The set R( f ) of nonwandering points is closed and f -invariant (f ( R (f ) = R ( f ) ) . The limit sets w(q) and a ( q ) , for q E X, are contained in R ( f ) . Such a set R ( f ) is called the nonwandering set. Let g: Y + Y be a homeomorphism of a compact metric space and h: X + Y a continuous surjection. If g o h = h o f , then we can easily prove that h ( R ( f )) c R(g). If, in particular, h is bijective, then h ( R ( f )) = R(g).

Theorem 2.4.8. Let f : M -t M be a homeomorphism of a closed topological manifold. I f f is topologically stable in the class of homeomorphisms, then the set of all periodic points of f , Per( f ), is dense in R( f ) . Proof. Since f is topologically stable, for E > 0 there exists 6 > 0 satisfying the definition of topological stability. We may suppose that 6 < E . Take and fix x E R ( f ) , and choose a coordinate neighborhood U of x contained in U 6 / 2 ( x ) (where U 6 / 2 ( = ~ ){ y E M : d ( x ,y ) < 6/21). Notice that U is chosen such that it is connected. Since x E R( f ), there is 12 > 0 such that f " ( U ) n U # 0 and f ' ( ~ )n U = 0 for 0 < i < k. Thus f q x ' ) E U for some x' E U. Since U is a connected coordinate neighborhood, we can construct a homeomorphism g: M + M satisfying g 0 f k ( x ' ) = x', g(U) = U, ~ ( y=) y for all y $2 U.

52.4 Topological Anosov maps (TA-maps)

93

Clearly, since d(g(x),x ) < 6 for x E M , we define cp = go f . Then, d(cp(x),f ( x ) ) < 6 for x E M . Therefore we can find a continuous surjection h: M 4 M such that h o p = f o h , d(h(z),x)<& ( X E M).

I t is easy t o see that cpk(xt) = xt. Since h o cpk = f k o h, we have h(xt) =

<

f k ( h ( x ' ) ) and thus h ( x l ) E P e r ( f ) . On the other hand, since d ( x , h ( x l ) ) d ( x , x l ) d(xt,h ( x t ) ) < 2&, we have h ( x t ) E U2e(x). Since E is arbitrary, it follows that P e r ( f ) is dense in GI( f ).

+

Theorem 2.4.9(Walters [W3]). Let M be a closed topological manifold with dimension 2 2. lj f: M -t M is topologically stable in the class of homeomorphisms, then f has POTP. For the proof we need the following two lemmas.

L e m m a 2.4.10. Let f : M -t M be a homeomorphism of a closed topological q positive numbers. Then for manifold. Let k 2 0 be an integer and let ~ , be ) d ( f( x i ) , < r ( 0 5 i < k - 1 ) there any set of points 1x0,X I , . . , ~ k with exists a set of points {xh,xi, ,xL) such that

-

(i)

~ ( x ~ , x : ) < TO, s i s k , d ( f ( ~ : ) , 2 : + ~ ) < 2O~ < , i
(ii) (iii)

Proof. This is proved by induction on k. For k = 0 this is true. Suppose the lemma is true for k - 1 and we prove it for k. Let T > 0 and q > 0 be given. Choose X > 0 such that d(x,y) < X implies d( f ( x ) ,f (y)) < T . W e may , Let { x o ,x l , . -.,x k ) be a r-pseudo orbit o f f . By suppose q < m i n { ~A). assumption we can choose {xb,x i , . . ,xL-,) such that d(xi,x i ) < X(0 i k - l ) , d ( f ( ~ ~ ) , x ~ + ~( O )
< <

we can choose xk E M so that xk # x i i f j 5 k d ( f ( x ' , - ~ )x',) , < 27.

- l,d(xL,xk) <

q and

Lemma 2.4.11. Let M be a closed topological manifold with dimension 2 2. Then for E > 0 there is 6 > 0 such that if a finite collection 3 = {(pi,qi) E M x M : 1 i 5 T ) satisfies the following conditions :

<

(i) (ii)

d(~i,qi)<6, l < i < r , if i # j then pi # pj and qi # qj,

CHAPTER 2

94

then there exists a homeomorphism f : M -+ M such that

Proof. Let n = dim M . From the dimension theory we can take a finite open cover V of M which can be represented as the union of n 1 families V l , ,Vn+1, such that each element of V is a connected coordinate neighborhood and for 1 L n 1 all elements of Ve are disjoint. Moreover, given E > 0 we can choose the open cover V such that the diameters of all elements of V are less than E. Let 6 > 0 be a Lebesgue number of V and assume that a finite collection 3 = {(pi,qi) E M x M : 1 5 i r ) satisfies the conditions as in the lemma. Let 3 e = {(pi,qi) E 3 : 3 5 E V e s.t. pi, qi E Vj). Then 3 1 U . . U Fn+l = F. Since dim M = n 2 2, for each 1 5 L 5 n 1 there is a homeomorphism fe : M + M such that

+

...

< < +

<

-

+

( 1 ) f t ( ~ i=) qi for (pi,%) E 3 e , ( 2 ) fe(pi) = Pi for (pi,qi) 9 3 e , and ( 3 ) fe is the identity outside elements of Ve. Let f = f i o . . . o fn+l. Then f ( p i ) = q i f o r a l l ( p i , q i ) E 3 , a n d d ( f ( x ) , x ) ( n 1)e for x E M .

+

<

Proof of Theorem 2.4.9. Let e > 0 be given and let 6 correspond to e as in the definition of topological stability. We may assume that S has the property as in Lemma 2.4.10 (by taking 6 sufficiently small if necessary). Suppose { x o ,X I , ,x k ) is any pseudo orbit such that d(f ( x i ) ,xi+l) < 6/2 for 0 i 5 k - 1. By Lemma 2.4.10 there exists { x h , x i , , x i ) such that d(xi,x:) < E(O 5 i 5 k ) , d ( f ( x : ) , ~ i + < ~ )6(0 i 5 k - 1),xi # x i if i # j ( i , j < k ) and f ( x i ) # f ( x i ) if i # j ( i , j k - 1). By Lemma 2.4.11 there is a homeomorphism cp : M + M with d(cp(x),x ) < 6 ( x E M ) and cp o f ( x i ) = x:+,(0 i k - 1). Let g = cp o f . Then d(g(z),f ( 3 ) )< 6 ( x E M ) and g(x:) = xi+,(O 5 i k - 1). By topological stability there is a continuous surjection h: M -t M such that d ( h ( x ) ,x ) < E and h o g ( x ) = f o h ( x ) for all x E M . Then

- -.

<

< <

< <

<

Thus, for each pseudo orbit { x o ,X I , ,x k ) with d(f ( x i ) ,xi+l) < 6/2(0 5 i 5 k - 1 ) there exists y E M such that d( f i ( y ) , x i ) < 2 4 0 5 i k - 1). From the proof of Theorem 2.3.3 it follows that f has POTP.

<

$2.4 Topological Anosov maps (TA-maps)

95

Remark 2.4.12. It is known that Theorem 2.4.9 is also true for the case where dim(M) = 1. In fact, we can prove that if a homeomorphism f of the unit circle S1 is topologically stable in the class of homeomorphisms, then Per(f) is a finite set and each periodic orbit, Of(p), is sink or source, i.e. there is a neighborhood U of Of (p) in S1such that f '(z) + Of (p) (i + oo) for x E U or f'(x) -* Of(p) (i + -00) for x E U. From this fact we obtain that the map has POTP. It remains a problem of whether or not if a homeomorphism f : M + M of a closed topological manifold is topologically stable in the class of homeomorphisms then the restriction fin(f) : R(f) + R(f) is expansive. This problem is a topological version of the stability conjecture. The conjecture was solved by MaiiC [Ma3], in C1differentiable dynamics.

CHAPTER 3 Nonwandering Sets

For a continuous surjection of a compact metric space, we shall introduce the important subsets in dynamics, recurrent set, Birkhoff center, nonwandering set and chain recurrent set each of which includes its predecessor in this sequence. These sets serve as the foundations for developments of dynamics of the maps. Moreover we shall investigate the relationship between the continuous surjection and the shift map of the inverse limit space.

$3.1 Chain recurrent sets

-

Let f: X -t X be a homeomorphism of a compact metric space. For x, y E a X and a > 0, x is a-related to y (written x y) if there are a-pseudo orbits of f such that xo = x, X I , . . .,xk = y and yo = y, yl,.. ,ye = x. y for any a > 0, then x is related to y (written x y). The set If x CR(f) = {x E X : x x} is said to be the chain recurrent set of f. It is clear that CR(f) = f (CR(f)) and O(f) C CR( f). The relation is an equivalence relation in CR( f ) satisfying (1) u u for every u (reflexivity), (2) u v + v u (symmetry), (3) u v and v w + u w (transitivity). Let be an equivalence relation in X. If x E X , the subset by [XI = {y E X : y x) is called the equivalence set of x. Thus the equivalence set of x is the set of all points which are equivalent to x. We show that the class of all distinct equivalence sets forms a partition of X. By reflexivity, x E [XI for each point x in X , so each equivalence set is nonempty and their union is X. Any two equivalence sets [xl] and [xz] are either disjoint or identical. To see this suppose [xl] and [x2] are not disjoint, i.e. they have a commom point z. Since a x1 and z X Z , by symmetry and transitivity we have xl xz. Thus x1 E [x2]. Let y be any point of [xl], so that y XI. Since x1 XZ, transitivity shows that y xz SO that y E [x2]. Thus we have [xl] c [xz] and the same reasoning shows that [x2]c [xl]. Lemma 3.1.1. CR(f) is closed.

-

N

--- -

N

-

-

-

N

-

.

-

-

--

Proof. Let a > 0. Since X is compact, f : X -+ X is uniformly continuous. Thus there exists 0 < 6 < a12 such that d(x, y) < 6 implies d(f (x), f(y)) < a/2. Let {x(')} be a sequence in CR(f) such that x(') + x as k X: t. Then d(x('), x) < 6 for some k > 0. Since x!') E CR(f), there is a 612-pseudo ,x,, x ( ~ ) }Therefore . {x, X I , . ,x,, x} is an a-pseudo orbit. orbit {x('), 21, Since a is arbitrary, we have x E CR(f).

.

.

$3.1 Chain recurrent sets

Theorem 3.1.2. I f f has POTP, then R(f) = CR(f). Proof. It is enough to see that CR(f ) c R( f ). If x E CR(f ), then for every a > 0 there is a pseudo orbit {xi) such that xo = x,X I , . ,xk = x and d(f i(y), xi) < a(0 5 i 5 k) for some y E X. Therefore, fk(u,(z))n U,(x) # 0 where U,(x) = {y E X : d(x, y) < a). Since a is arbitrary, we have x E R(f).

.

Let X be a topological space and 2X the family of all nonempty closed subsets of X. The exponential topology of 2X is defined by assuming that the family of all sets B(G) = { F E 2X : F C G), c(H)={FE~~:F~H#~) is an open subbase of 2X provided G and H are open subsets of X. It is easily checked that the family of all sets

where each of Gi is an open set of X, is a base of 2X.

Remark 3.1.3. If X is compact, then so is 2X. It is enough to prove that every cover of 2X whose sets belong to an open subbase of 2X contains a finite subcover. Suppose that

where Gt and H, are open in X. Put Fo = X \ U, Ha.Then, for each s we have FOn Ha = 0, i.e. FO! I$ C(Ha) and thus Fo E B(Gt,) for some to, i.e. Fo C Gto and so X\Gt, C X\Fo = H..

U 8

Since X\Gto is compact, there are si(l 5 i 5 n) such that

Let F E 2X. Then there exist the following two cases : (i) (ii)

F c Gto, F C Gt,,i.e. F n (X\Gt,) # 0.

For (i) we have F E B(Gt,), and for (ii) we can find j such that F n Haj # 0, i.e. F E C(Ha,). In any case, F E B(Gto)U C(H.,) U U C(H.,). Therefore 2X is compact.

---

CHAPTER 3

98

Remark 3.1.4. When X is BausdorfF, X is compact if so is 2X. Indeed, let X = Ut Gt where each Gt is open. Obviously, 2X = Ut C(Gt). Since 2X is compact, we have

for some n > 0. Let xo E X. Then {xo} E C(Gtj) for some j 5 n. This implies that xo E Gtj and therefore

Lek X be a metric space with bounded metric d. A metric p defined by p(A, B) = max{sup d ( A ,b), sup d(a, B)}, bE B

where d(A, b) = inf{d(a, b) : a

A, B E 2'

aEA

A}, is called the Hausdorffmetric for 2X.

Remark 3.1.5. If a metric space (X, d) is compact, then so is (2X,p). To obtain this it is enough to see that 0 ( 2 X ) = ~ ( 2 where ~ ) ~ ~( 2 ~ ) denotes the exponential topology of 2X and 0 ( 2 X ) p denotes the topology induced by p. We first show that 0(2X), c 0 ( 2 ~ ) ,i.e. each open ball with center A, R = { F E 2X : p(A,F) < E},

...

is open in 2X. Since A is compact, for each k 2 1 there are a:, ,a:, that for each x E A we have d(at,x) < 1/k for some i. Let us put

Then it is enough to prove that

If F 6 R, then p(A, F) < E - l/k for some k. Thus

such

53.1 Chain recurrent sets

99

(2) and (5) imply that F C G. On the other hand, letting x = a t , by (4) there i s y e Fsuchthatd(y,af)<&-1/k. ThusFnGf#Oforl
On the other hand, let x E A and let i be an integer such that d(x, a t ) < l/k. Since F n G: # 0, there is y E F such that d(y, a:) < E - Ilk. It follows that d(y, x) < E and so d(x, F ) < E. Thus, if x E A then d(x, F ) < E. This fact and (6) yield F E R. Next we show that 0 ( 2 X ) c 0(2X),. We can, of course, restrict ourselves to the case when a subbase of 2X is contained in 0(2X),. Let A = {F E 2X : F C G} where G is open in X and G # X. Then A is a member of the subbase of 2X. If A is a compact proper subset of G, then d(A,X\G) = E > 0. Thus we have A > R where

and so FER*FCG. Indeed, suppose p E F\G. Then d(p, A) 2 E and thus p(F, A) 2 E, so F $ R. Therefore, R C A. Let A = {F E 2X : F n G # 0) where G is open in X. Let A be a compact subset such that A n G # 0. For a E A n G put d(a, X\G) = E. Since p(A,F) < E for F E R, we have d(a, F ) < E and so F n G # 0. Therefore, R c A. Making use of Hausdorff metric, we have the following result.

Theorem 3.1.6. Let f : X 4 X be a homeomorphism of a compact metric space. Then the chain recurrent set of f lcR(f) coincides with CR( f ) . Proof. We give here a proof due to Robinson [Ro]. Let x E CR(f) and Cn = {x!")) be a periodic lln-pseudo orbit through x. Then Cn is a finite set. In the Hausdorff metric, there is a subsequence Cn, that converges to some f-invariant compact set C c X. If we establish that for every y E C and E > 0 there is a periodic E-pseudo orbit {zi} through y with t i E C, then it follows that y E CR(f1c) c CR(f). Since y E C,we have C c CR(f) and x E C C CR(flc) C CR(flcR(f)). This is true for all E CR(f) and thus C R ( f ) C CR(f IcR(~)). It remains only to show that there is a periodic €-pseudo orbit through y with zi E C. Since f : X + X is uniformly continuous, there is ~ / 3> 6 = 6 ( ~ / 3 )> 0 such that d(a, b) < 6 implies d(f (a), f (b)) < €13. Since Cn,

loo

CHAPTER 3

converges to C, we can find n = nk such that l l n < €13 and the distance from Cn to C in the Hausdorff metric is less than 6. Suppose that {xin)) has a period j and xi;! = xin) for all i. For each zin) take t i E C with d(xin),ti) < 6, zi+j = t i for all i, and d(f (r), r+l)

ti

= y for some i. Then

< d(f (ti), f (xin))) + d(f (xin)), xi;),) + d(xi;),,

ziti) < E .

Therefore {ti) is a periodic €-pseudo orbit in C through y.

Theorem 3.1.7. Let f : X + X be a TA-homeomorphism of a compact metric space. If flcR(f ) is topologically tmnsitiue, then X = CR(f ). Proof. Suppose X # CR(f). Then we shall derive a contradiction by showing (X\CR( f )) n Per( f ) # 0 where Per( f ) denotes the set of all periodic points of f . Since w(xo) and a(xo) are contained in CR(f) for xo E X\CR(f), we have cl({xo,f(zo),...)) n CR(f) # 0 and cl({x~,f-'(x~),--.))ncR(f) # 0. Let 0 < E < d(xO,CR(f)) and let 6 > 0 be a number with the property of POTP. Then d(CR(f ), f "(zo)) < 6 and d(CR(f), f -"(xo)) < 6 for some large n . Thus d ( ~ ~f n(xo)) + ~ , < 6 and d(x-,-I, f -n(xo)) < 6 for some xn+l, x-,-l E CR(f 1. Now construct a 6-pseudo orbit

from x-,-1 to %,+I. Since flCR( f ) is topologically transitive, we can find in CR(f) a Cpseudo orbit

from xn+l to x-,-l. By combining the above two 6-pseudo orbits, a periodic 6-pseudo orbit is constructed. Since f has POTP, the periodic 6-pseudo orbit is &-traced by a point y E X. Since f is expansive, y is periodic and since d(xo, CR(f)) > E we have y $ CR(f). This is a contradiction.

Theorem 3.1.8 (Aoki [Ao~]).Let f: X + X be a homeomorphism of a compact metric space. I f f has POTP, then so does If, in addition, f is expansive, then the set of all periodic points, Per(f ), is dense in R( f). This is given in Theorem 3.4.2 for continuous surjections and so we here omit the proof of Theorem 3.1.8 since the technique of the proof is similar.

Remark 3.1.9. There exists an example of a TA-homeomorphiim f of a compact metric space such that X # CR(f). This is easily constructed as follows. As before let a : YF + Y? be the shift map, and S = {(xi) E Y? : (xi, E C, i E Z) where C = ((0, O), (0, I), (1,l)). Then a : S + S is

$3.1 Chain recurrent sets

101

a Markov subshift. Since u p is expansive and has POTP, by Theorem 3.1.8 the set Per(uls) of periodic points is dense in C R ( u l s ) . However S contains only two periodic points x = (. ,0,0, ) and y = ,1,1,. ). The point z = (. ,0,0,1,1, ) is in S and not periodic. Therefore, z 6 Per(uls) = C R ( q s ) # S. Let X be a compact metric space. A closed subset E of X is isolated for the homeomorphism f : X + X if f ( E ) = E and if there is a compact neighborhood U of E such that fn(U) = E.

.-

..

- -.

(a

.-

..

nrm

T h e o r e m 3.1.10. Let X be a compact metric space. I f f : X + X is expansive and f: C R ( f ) + C R (f ) has POTP, then C R ( f ) is isolated. Proof. Let e > 0 be an expansive constant for f . For 0 < /3 < e / 2 let a > 0 be a number with the property of POTP. By uniform continuity of f we can take 0 < 7 < min{a/2,e/2} such that

f n ( U ) . Since Now let U = { y E X : d(y, C R ( f ) )< 7 ) and take y E f i ( y ) E U for i E Z, there is xi E C R ( f )with d ( f i ( y ) , x i )< 7 for i E Z. Thus

for i E Z. Since f l C R ( f ) has POTP, there is a +-tracing point x E C R ( f ) and for i E Z

Since f: X

-, X

is expansive, we have x = y and therefore y E C R ( f ) .

A homeomorphism f : X + X of a metric space is topologically mixing if for nonempty open sets U, V there is N > 0 such that U r l f " ( V )# 0 for all n 2 N. If f is topologically mixing, then it is topologically transitive (see Remark 2.2.1). T h e o r e m 3.1.11 (Topological decomposition theorem). Let f : X + X be a homeomorphism of a compact metric space and let C R ( f ) be the chain recurrent set. If flCR( f ) : C R ( f ) C R ( f ) is a TA-homeomorphism, then the following properties hold : (1) (Spectral decomposition theorem due to Smale) C R (f ) contains a finite sequence Bi (1 I i 5 L ) o f f -invariant closed subsets such that (i) C R ( f ) = B; (disjoint union), (ii) flBi : Bi + B; is topologically transitive. ( 2 ) (Decomposition theorem due to Bowen) For each Bk there exist ak > 0 and a finite sequence Ci (0 5 i 5 ak - 1 ) of closed subsets such that (i) Ci n C j = 0 ( i # j ) , f ( C ; )= Ci+l and f a k ( C j )= Ci,

u:=,

CHAPTER 3

(ii) Bk = lJ;2i1 Ci, (iii) f p i : Ci -+ Ci is topologically mixing. The sets Bi and C j are called basic sets and elementary sets respectively. For more general case (i.e. continuous surjections case) we shall give the proof of this result in Theorem 3.4.4 of 53.4.

Theorem 3.1.12. Suppose f: CR(f) -+ CR(f) is a TA-homeomorphism. If, in addition, f: CR( f ) + CR( f ) is topologically transitive and CR( f ) contains a fied point of f , then f : CR( f ) + CR(f ) is topologically mixing. Proof. This follows from Theorem 3.1.11 (2).

53.2 Stable and unstable sets For A a subset of a compact metric space X define w8(A) = U{w8(x, d) : x E A), and for E

Wu(A) = U { w U ( x ,d) : x E A)

>0

w:(A)

=u{~:(x,d):xEA),

W z ( A ) = U { ~ z ( x , d ) : xA). ~

Here the subsets W8(x,d) and Wu(x,d) are defined as in 52.2 of Chapter 2.

Theorem 3.2.1. Let CR(f) be the chain recurrent set for f . Iff: CR(f) -, CR(f) is a TA-homeomorphism and if CR(f) splits into the finite union CR(f) = R1 U...U RL of basic sets Rj, then for 1 j t!

< <

W8(Rj) = {X E X : lim d(fn(x), Rj) = O), n-r oo

Wu(Rj) = {X E X : lim d(f-"(x), Rj) = 0) n--too

and for each E

> 0 there exists an open set Uj > R j such that

Proof. Since each R j is open and closed in CR(f), f l R jhas POTP. Thus, for > 0 there exists a > 0 such that every a-pseudo orbit of Rj is €12-traced by some point in Rj. Let 0 < y < min{a/2,~/2) be a number such that E

$3.2 Stable and unstable sets

103

Suppose limn,, d ( f n ( y ) , R,) = 0 for y E X. Then we can find N > 0 with d ( f n ( y ) , R j ) < 7 for n N , and so there is xn E Rj such that d(xn,f n ( y ) ) < 7 for all n 2 N. Let xn = fn-N(xN)for n < N. Then { x , : n E Z) is an a-pseudo orbit in R j . This follows from the facts that if n N then

>

>

and if n < N then

Let x E R, be an a/2-tracing point of {x,). Then

Thus, f N ( y ) E W , ' ( f N ( x )c ) w 8 ( f N ( x )and ) so y E f - N ( ~ d ( f N ( x=) ) ) W 8 ( x ) .Therefore, y E W 8 ( R j and )

To show the second statement let Uj = { y E X : d ( y , R j ) < 7 ) and take f-"uj). Since d ( f k ( z ) , ~ , )< 7 for k 2 0, there is yk E Rj with d(fk(z)),yk) < 7 for k 0. Put y-i = f b i ( y o ) for i > 0. Then { y k : k E Z) is an a-pseudo orbit of Rj. If x E R, is an a/2-tracing point of { y k ) , then for 1220 %

E

nk>,

and thus

E

>

E W,'(x)c W,'(Rj).

Theorem 3.2.2. Under the assumptions of Theorem 3.2.1, given y exists Ri ( R j )such that lim d(fn(y),Ri)=O (lim d(f-"(y),Rj)=O)

n-w

n+w

and furthermore X = If i # j , then

e

e

i=l

j=1

UW 8 ( R i=) U W u ( R j ) .

E

X there

CHAPTER 3

104

Proof. Since f : C R ( f ) + C R ( f ) has POTP, we have that C R ( f ) = R ( f ) . Thus w ( z ) C C R ( f ) = R1 U U Re for every x E X . For fixed x E X , to show the existence of Ri with w ( x ) c Ri, it is enough to see that Ri with Ri n w ( x ) # 0 is only one in { R i : 1 5 i 5 e). To do so suppose

...

for j # j'. Let U, be an open neighborhood of R, and suppose they satisfy U, n Ub = 0 for a # b. Choose an open neighborhood V , of R, such that

Since w ( x ) n R j # 0 and w ( x ) n R j , and { fmi(x)) such that lim f n ' ( x ) € R j ,

i-m

# 0, there exist

subsequences { f n i ( x ) )

,lim f m i ( z ) E R p .

a-m

Without loss of generality we suppose that { n i l and { m i ) are chosen such that nl < m l < n2 < m2 < - . - . Then, for I

> 0 sufficiently large i 2I

f n i ( x ) E V j , f m i ( x ) E Vjl.

since f(V,) C Ua(l 5 a 5 e) and UanUb= 0 for a # b, we have fni+'(x) E U j and fni+'(z) $ Ua for all a with a # j. Thus, for i I there exists ni < ei < m i such that f ( 2 )$ 4 n Vj,.

>

Since X \ (4U 41) is closed and { f e * ( x ): i 2 I ) c X \ ( V j U Vy),there is a subsequence of { f ' i ( x ) )for which the limit point z is contained in X \ ( & ~ & I ) . Thus, z $ R j U Rjl. However, since z E w ( x ) , there exists Rk such that z E Rk for 12 # j, j'. This is a contradiction. Therefore, w ( x ) c R, for some j. Finally we show that d ( f " ( x ) ,R,) = 0 whenever w ( x ) c R j . If this is false, then there exist €0 > 0 and an infinite sequence { n i l such that d(f "'(x), R j ) €0 for all i 2 0. If f n i ( x ) + z, then d(z, R,) 2 EO. But we have z E R j since z E w(x), thus contradicting. The proof of the rest of the statement is easy and therefore omitted.

>

Let f : X -, X be a homeomorphism of a compact metric space. If f is expansive and f : C R ( f ) -, C R (f ) has POTP, then C R ( f )is an isolated set (by Theorem 3.1.10). Moreover, by Theorem 3.1.11 C R ( f ) splits into the disjoint union C R (f ) = R1 U s U Rc of basic sets. Then each R, is an isolated set.

.

$3.2 Stable and unstable sets

105

A cycle for the family {R; : 1 5 i 5 L ) is a sequence R;, Ril = R,, and

for 1 1 j < k. The sequence Ril , such that

... ,Rib = R;,

xj @

, -.,Rib such that

... ,zk-l

is a cycle if there are points xl ,

k

U Rim,

a(xj) C Rij,

w ( y ) C Ri,+,

m=l

for 1 5j

< 12. For this case we write Ril < R;, <

< R;,

= R;,.

Theorem 3.2.3. If f : X -r X is eqansive and f : CR(f) POTP, then f has no cycles.

4

CR(f) has

Proof. I t only shows no existence of families {R;,) such that Ril < Ria < < R;, = R,,.If, in particular, R; > Ri for some i, then there are x, y E R; such that E WU(x)n wd(y), z $ cR(f). Take and fix E > 0 sufficiently small such that d(z,CR(f)) 2 2 ~ .Since f: CR(f) + CR(f) has POTP, we let 6 > 0 such that every 6-pseudo orbit off in CR(f) is &-traced. Since 612 > d(f -"(E), f -'(x)) and 612 > d(f '(z), f '(y)) for k large and f : Ri + Ri is topologically transitive, there exist w E Ri and m > 0 such that

Thus a (212

+ m)-periodic &pseudo orbit

is constructed. If p E CR(f) is an &-tracingpoint of the pseudo orbit, then it is easily checked that p is a (2k + m)-periodic point and d(z,p) < E . However, p $! CR(f), thus contradicting. In the same way we can derive a contradiction for the more general case Ril < Ria < < R;, = Ria.

..

Let Ri be a basic set. We say that Ri is a sink if Wd(Ri) is a neighborhood of Ri in X and Ri is a source if Wu(Ri) is a neighborhood of Ri in X.

Theorem 3.2.4. Let f: X -r X be an expansive homeomorphism having POTP and let R, be a basic set. Then Ri is a sink if and only if WU(Ri)= Riy and Ri is a source if and only if Wd(Ri) = R;. Proof. If Ri is a sink, then we have Wd(Ri) > B,(Ri) = {x E X : d(x, Ri) 5 E ) for E > 0 sufficiently small.

CHAPTER 3

106

Suppose z E W u ( R i \) R;. Then f -"(z) E W U ( R in) Be(Ri)for some n > 0. Since Be(R;)C W d ( R ; )we , have f -"(z) E W d ( y for ) some y E Ri. Let z E R; be a point such that f-"(z) E W U ( z ) Since . f-"(2) is a wandering point, we have Ri > R;, thus contradicting. Therefore, W u ( R i )= R;. For E > 0 small enough let 6 > 0 be a number such that every &pseudo orbit of f is &-traced. If W U ( R i = ) R;, then we show that Ri is a sink, i.e. W a ( R iis ) a neighborhood of R;. If this is false, then there is y E B611(Ri) \ R; such that f n ( y ) -* Rj for some j # i. Take z E Ri with d ( y , z ) < 6. Then

is a 6-pseudo orbit. Let w E X be an &-tracing point of the 6-pseudo orbit. Since E is small, we have

However, since w is a wandering point, we have Wu(R,) # R;, thus contradicting. Therefore, Ri is a sink. The proof of the rest is done in the same argument and therefore we omit it.

Theorem 3.2.5. Under the assumptions of Theorem 3.2.4, f has sinks and sources. Proof. Let R; be a minimal basic set with respect to the ordering relation >. Then Ri is a sink. Indeed, if R; is not, then we can take z E W U ( R i\) R;. Since X = Ui W d ( R i )we , have z E W a ( R j for ) some j. Thus there are x E R; and y E Rj such that z E W U ( xn ) W u ( y ) .Since z is a wandering point, we have R; > Rj. However, since R; is minimal, we have a contradiction. Theorem 3.2.6. Under the assumption of Theorem 3.2.3, let C be an elementary set containing in a basic set R;. Then for each z E C,W u ( x )n C is dense in C for a = s, u. Proof. Notice that fnO(C) = C for some no > 0. Since C is closed, for c > 0 there are wl , ,wq E C such that U: Bc(wi)> C. To show that W a ( z n )C is dense in C, let z E C and take E > 0 small enough. Since f;cO is topologically mixing, we can take N > 0 such that

..

Let 6 > 0 be a number such that every 6-pseudo orbit of flCR( f) is &-traced by some point. Since c > 0 is arbitrary, let c = 614. Then f " ~ ~ E ( BS14(wj) z ) for some j and thus B 6 1 4 ( ~Cj B ) ~f nON(x)). ~ ~ ( Take w E f " o N ( ~ , ( z )n ) B ~fnoN(z)) ~ ~ nC ( R ( f ) . Then a 6-pseudo orbit

$3.3 Recurrent sets and Birkhoff centers

107

is constructed in CR(f). If y E CR(f) is a &-tracing point of the 6-pseudo orbit, then y E Wi(f n ~ N ( x )and ) so f - n ~ N ( y )E W,d(x). Thus f - n ~ N ( y )E B,(x). Since f - " ~ ~ ( wE) BE(%)and d(f -n~N(y),f - n o N ( ~ ) )< E, we have d(f-nON(y),z) < 28 and thus Ws(x) f l Bz,(z) # 8. Since z is arbitrary in C, Ws(x) n C is dense in C.

53.3 Recurrent sets and Birkhoff centers Let X be a compact metric space. A point x E X is said to be recurrent for a homeomorphism f if, for any neighborhood U of x, there exist infinitely many n with fn(x) E U (in other words, x E a(x) n w(x)). Every recurrent point belongs to R(f). The closure of all recurrent points, c(f), is called the Birkhoff center. The notion of chain recurrent point is a generalization of that of recurrent point. Let F : X + R be a function. Given a neighborhood U6(x) with radius 6 centered at x we define

Then M(x) = lim6,0 M(x, 6) and m(x) = lim6,0 m(x, 6) satisfy -00 5 m(x) 5 M(z) 5 oo. The function F : X -t R is said to be upper semicontinuous if M(x) = F ( x ) for x € X , and to be lower semi-continuous if m(x) = F ( x ) for x E X.

Remark 3.3.1. If the set of all recurrent points is dense in R(f), then the set is a Baire set in R(f) (a Baire set means the intersection Aj of sets Aj, j 2 0, that are open dense). Indeed, let F(x) = inf{d(f n(x), x) :n > 0). Then the set is contained in {x E R(f) : F(x) = 0). Since F(x) is upper semi-continuous, for k 2 1, {x E R(f) : F(x) < 1/k) is open dense in Q(f) and therefore the set of all recurrent points is a Baire set in R(f).

nj

Let f : X + X be a homeomorphism of a compact metric space. A closed f invariant set, A, is called a basic set for f if A is an isolated set and f l A: A + A is topologically transitive. Let A be a basic set for f and let U be a compact neighborhood of A such that f n(U) = A. Define

n m :

Then we have

CHAPTER 3

108

where

W a ( A ) ' ={ x E X : d ( f n ( x ) , A )4 0 as n + oo), WU(A)'= { x E X : d(f-"(x),A) + 0 as n + oo). If f : CR(f ) + C R ( f ) is a TA-homeomorphism and A is a basic set as in Theorem 3.2.1, it follows that Wo(A)' = W o ( A )for a = s, u. Let {A1,. ,At) be a finite family of basic sets for a homeomorphiim f of a compact metric space. We say that the family has no cycles if there is no sequence Ail,. ,Ai, of distinct members such that Ail = Ah and

.

..

for 1 5 j

< k.

Theorem 3.3.2 (Malta [Mall). Let f : X 4 X be a homeomorphism of a compact metric space. If the Birkhof center c ( f ) is an isolated set for f and admits a decomposition c( f ) = Al U .. U At into basic sets having no cycles, then c ( f )= R ( f ) . From this theorem we have the following

Theorem 3.3.3. Let f : X + X be a homeomorphism of a compact metric space. If f : X + X is expansive and f:c( f ) 4 c( f ) has POTP, then c( f ) =

N f 1. Proof. Replace c ( f ) by R ( f ) in Theorem 3.1.8. Then we have that the set of points is dense in c ( f ) and by Theorem 3.1.11, c ( f ) splits into the disjoint union c( f ) = d l U ... U At of basic sets Aj . By applying Theorem 3.1.10 we see that c ( f ) is isolated. Replacing {&) by { A i ) in the proof of Theorem 3.2.3, we have that cycles for the family { A i ) do not exist. Therefore the conclusion is obtained by Theorem 3.3.2. all periodic

We first prepare some lemmas which need to prove Theorem 3.3.2. Let A be an isolated closed f-invariant set and let U be a compact neighborhood of A such that flyrnfn(U) = A. Define W:",,(A, U ) as above. For a compact neighborhood V of A in W i c ( A ,U ) with f ( V ) C V , the set

D

= cl(V - f ( V ) )

is called a fundamental domain for WP,(A,U ) . In a similar way we define a fundamental domain for W&(A, U). If x E Wa(A)'\ A, then there exists k E Z such that f k ( x )E D. If D n A = 0, we call D a proper fundamental domain.

$3.3 Recurrent sets and Birkhoff centers

Lemma 3.3.4. If A is isolated, then

is a proper fundamental domain for WLc(A, U). Proof. Since f (WL,(A, U)) = f (U) n Wtoc(A,U), we have

and f (int(U))n Wt",(A, U) is an open neighborhood of A in Wic(A, U). Therefore, D" n A = 0.

Lemma 3.3.6. Let V be a neighborhood of D" in X. Then U' = WZ,(A, U) U O+(V) is a neighborhood of A in X. Here O+(V) = U,>, f ,(V). -

< <

Proof. If this is false, for W a neighborhood of A in X we have W U'. Thus there is a sequence x, E U \ U' which converges to x in A. Since Zn # Wlu,,(A,U) = fn(U), we have 0-(2,) = {f-j(x,) : j 2 0) U. For each n let m, > 0 be an integer such that f -'(x,) E U for 0 5 k 5 m,, but f-(mm+l)(xn) # U. Thus y, = f -ma(xn) E U \ f (U). Notice that m, -+ oo as n + oo. We may suppose that y, converges to y E U (notice that U is compact). Since f (Wk(A, U)) C int( f (U)), we have y # f (Wtoc(A,U)). To show that y E WL,(A,U) suppose y # WL,(A,U), then there exists k > 0 such that f k ( ~) ) U. Thus f k ( ~ , ) = f-"n+'(x,) # U for n sufficiently large. Since m, -r oo as n -+ oo, we have 0 < m, - k 5 m, for n sufficiently large, which is a contradiction. Therefore, y E W,"b,(A, U) \ f (WLc(A, U)) c D" and thus y, = f -mn (x,) E V when n is large. We have x, E U', but contradicting our assumption.

n,,

Lemma 3.3.6. Let Al and A2 be isolated sets of X. If Al n A2 = 0, then

(ii)

~ [ w ~ ( A-~a21 ) '# 0

Proof. Let U be a compact neighborhood of A2 such that 0:- f "(U) = A2. Denote as D" a proper fundamental domain for WLc(A2, U). To show (i) let x, E Wu(Al)' be a sequence such that x, -r x E cl(WU(A1)') n A2. Define F = cl(U,>, O(x,)). Then F C cl(Wu(A~)').Thus it sufficies to prove that F n Dd # 6 Let V be a small neighborhood of D" such that V n A2 = 0, and

CHAPTER 3

110

define U' = Wcc(A2,U )U O+(V). Let k > 0 be an integer such that xk E U'. Since xk E Wu(Al)', we have xk $! Wu(Az)' and there is nk 0 such that xk E f nr (V). Thus U,,, Of(xn) n V # 0 and therefore F n D8 # 0. To show (ii) if x E cl(WU(Al)') r l [WU(A2)'- A2], then 0 # a(%) C cl(Wu(Al)') n A,. Thus cl(Wu(Al)') f l Wa(A2)' # 0. From (i) we have cl(WU(Al)') n [w8(A2)'- A,] # 0.

>

Lemma 3.3.7. k t {A1,. ,At) be a family of disjoint isolated sets having no cycles and satisfy X = WU(Ai)'. If cl(Wu(Ai)') n W8(A,)' # 0 or cl(WU(Ai)') f l [W8(Aj)' - A,] # 0 if i = j , then there exists a sequence Aj = A,,,-. - ,A,, = Ai such that

~6,

for 1

< a 5 k.

Proof. By Lemma 3.3.6 we have cl(WU(Ai)') n [Wa(Ajo)' - A,,] # 0, from which we choose a point x. Since X = U: WU(Ai)', there is 1 jl 5 I such that x E Wu(Ajl)'. Then we have

<

but x E cl(WU(Ai)')n [WU(Aj,)' - A,,] and so by Lemma 3.3.6 (ii) we have c1(Wu(Ai)') n wa(nj,)I # 0. NOWsuppose Ajl ,.. . ,Ajp are defined for p < I such that Aja # Ai for lIa
If j, # i, then we can obtain A,,+, by the same process used to get A,, . Since 1 j , 5 t , by the assumption of no cycles we have j , = i for some a which proves the lemma.

<

Lemma 3.3.8. Let A be an isolated set for f and let x E X . If a(x) nA # 0 and x $! WU(A)', then (i) (ii)

nrm

Proof. (i) : Let U be a compact neighborhood of A such that f "(U) = A and let D8 = D8(A, U) be a proper fundamental domain for Wtoc(A,U). To

$3.3 Recurrent sets and Birkhoff centers

111

obtain (i) it suffices to prove that 0-(x) n V # 0 for a neighborhood V of D d such that V n A = 0. By Lemma 3.3.5, U' = W;,(A,U) U O+V is a neighborhood of A. Since a(%) n A # 0, there exists k > 0 such that f-&(x) E U'. By the assumption x # WU(A)', thus f-&(x) # W:,(A,U) and so f -&(x) E f "(V) for some n 3 0, which implies 0-(x) n V # 0. (ii) : Let U be as above such that x # U. Since f (WL,(A, U)) c Wic(A, U), we have 0 - ( x ) n Wic(A, U) = 0. Since a(x) n A # 0, there is a sequence nk with nk -, oo as k + oo such that xk = f-nh(x) E U. Since xk # Wic(A, U), we have O+(xk) $Z U. For each k > 0 let mk 3 0 be such that fm(xk) E U for 0 5 m 5 mk and f mh+l(xk) # U. Then we have

and mk - nk < 0. Thus yk E 0-(x). Indeed, if not, we have 0 5 nk 5 mk and so f nh (xk) = x E U, which is a contradiction. Since xk converges to A, given N > 0 there is ko > 0 such that xk E f-"(U) for k ko. Then fn(xk) E U for 0 n 5 N and so mk 3 N for k ko. Thus mk + oo as k + oo. Without loss of generality we suppose that yk converges to y E cl(U - f -'(U)). To see y E WU(A)' -A it suffices to prove that y E WU(A)' since A c int(f-'(U)). If y # WU(A)', then there exists j > 0 such that f -j(y) # U. Therefore, there exists ko > 0 such that for 12 2 we have mk - j > 0 and fmk-j(xk) = f -j(yk) # U, which is a contradiction since 0 < mk - j 5 mk.

:n

>

<

>

..

Lemma 3.3.9. If {Al, ,At) is a family of disjoint isolated sets for f such that a(x) c ~ f Ai =for ~all x E X, then a(x) C Ai for some i.

This lemma can be derived immediately from lemma 3.3.8. Lemma 3.3.10. Let F be a compact f -invariant subset of X and let Q be a compact neighborhood of F such that - fn(Q) = F. Then there is a compact neighborhood V of F such that

n,,,,

V

c Q, f (V) c int(V).

n;=,

Proof. Define A, = f "(Q) for r 2 0. Then {A,) is a decreasing sequence of compact sets converging to F. Since Q is a compact neighborhood of F, we have A, c int(Q) for large r . For U = Q n f -'(Q) we have

and so fix an T large enough such that A,

c int(U). Then

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112

Since A, is a neighborhood of F, there is n so large that f "(A,) C int(A,). If n = 1, then we obtain the conclusion. When n 2 2, we can take a small compact neighborhood W of A, such that

Define E = f "-'(w)

U A,. Then f "-'(A,)

C int(f "-'(W))

c int(E) and

Thus we have fn-'(E) C int(E). As above choose a small compact neighborhood G of E satisfying E C G c U and fn-'(G) C int(E), and repeate inductively the above argument. Then our required set is produced.

Lemma 3.3.11 (Filtration lemma). If {Al,. . ,At) is a family of disjoint isolated sets for f having no cycles such that a(x) C Ai for all x E X , then (1) there exist compact sets

u:=,

such that f (Xi) C int(Xi) and (2)

Wf)c u:=, Ai.

nrafn(Xi - Xi-l)

= Ai,

Figure Proof. (1) : We define a relation Ai 5 Aj on {Ai : 1 5 i 5 L) if there is a ,Ajh = Ai such that [Wu(Aj,-l)'-Aj,-l]~[W*(Aj,)'sequence Aj = A,,, A,.] # 0 for 1 5 a 5 k. By the assumption of no cycles this relation is a partial ordering. I t induces a total ordering Al < A' < -.. < At of the family where A; < Aj if Ai 5 Aj and Ai # Aj.

53.3 Recurrent sets and Birkhoff centers

113

We define X j by induction. Let Xo = 0 and suppose that we have defined compact sets 0 = X o c X 1 c...cxj for j

< L such that f (Xi) c int(Xi),

6)

n 00

(ii)

fn(Xi-Xi-l) =Ai

( O s i 5 j).

-00

+

Let F = U{Wu(Ai)' : i 5 j 1). Then F is closed and F n Ak = 0 for k > j 1. Indeed, let {x,) be a sequence in F converging to x E X. Without loss of generality we suppose that for all n > 0 we have xn E Wu(Ai)' for some fixed i 5 j 1. By Lemma 3.3.9 there is 1 5 k 5 L such that a(x) c Ak, i.e. x E Wu(Ak)'. Thus x E cl(Wu(Ai)') f l Wu(Ak)'. By Lemma 3.3.7 we have A k 5 A i ~ d s o k 5 i 5 j + 1 , i . e .x E F . T h u s F n A k = O f o r k > j + l . Now let Q C X be a compact neighborhood of F such that

+

+

Then

n:==, fn(Q) = F.

Indeed, if x E

n:=o

fn(Q) then f-n(x) E Q for Since a(x) c Ai for some

> 0. Thus 0-(x) c Q and therefore a(x) C Q. i < j + 1, we have x E Ui<j+l WU(Ai)' = F. n

emm ma

Since f ( F ) = F, by 3.3.10 we have a compact neighborhood Vj C Q of F such that f (Vj) C int(Vj). Define Xj+1 = X j U Vj. Then f (Xj+l) C int(Xj+l). To conclude (1) it remains to see that nFwfn(Xj+1-Xj) = First we observe that

is closed. Indeed, if xk E S is a sequence converging to x E X , then fn(x) E Xj+1 for all n since Xj+l is compact. This follows from the fact that if fn(x) E X j for some n then fn+l(x) E int(Xj), and thus fn+l(xk) E int(Xj) for large k (which contradicts xk E S). Thus a(x) C S for all x E S and so a(x) c Xj+1 for all x E S, from which we have S c Wu(Aj+l)'. Similarly, w(x) c S for all x E S. By using the dual of Lemma 3.3.8 we have S C W8(Aj+l)'. Therefore,

Lemma 3.3.9 ensures that X =

u:=, Wu(Ai)' and so we observe Xe = X.

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114

( 2 ) : To obtain the conclusion it suffices to show that if y E X-U;, Ai then there exist y' E 0 - ( y ) and a neighborhood V of y' such that f n ( V ) n V = 0 for all n > 0. Choose A j such that a ( y ) c A j c i n t ( X j )- Xj-1. Take yl E 0 - ( y ) such that 0 - ( y l ) C X j - Xj-1. Since yl 4 A j , there exists > 0 such that

However, O+(yl) C X j and so f n O ( y l )E Xj-1. Let mo be an integer such that f F r n ( y 2 )E Xj-1 for 0 5 m L mo and f-(rn0+1)(y2) 6 Xj-1. Then

Take a neighborhood W C i n t ( X j - l ) of f ( X j - 1 ) and a neighborhood V of y' such that W n V = 0 and f ( V ) C W . Then we have f n ( V ) C f n - ' ( W ) C W for n > 0 and therefore V n fn(V) = 0 for n > 0.

L e m m a 3.3.12. If { A 1 ,-.. , A t ) is a family of disjoint isolated sets for f having no cycles and if x E X satisfies a ( x ) c W U ( A i ) ' , then x E Wu(Ai)'.

u:=~

The reader will understand the relationship between this lemma and Theorem 3.3.2.

= ~ and a ( x ) C ~ f W= u (~A ; ) ' ,then Proof of Lemma 3.3.12. If x 6 ~ f Wu(Ai)' a cycle exists. Indeed, choose A;, ,1 < il < t! such that a ( x ) n A;, # 0. Since a ( x ) (Z Ail, by Lemma 3.3.8 (i) we have a ( x ) n [W8(A;,)'- A;,] # 0. Since a(") c Wu(Ai)', there exists 1 5 i25 t! such that

u:,

[Wu(Ai,)' - A*]

n[w8(~i,)' - Ail] # 0,

a(%) n A;, # 0.

If i2 = i l , then we have a cycle. If i2 # i l , then we repeat the process as above. Then we have 1 5 i3 t! such that

<

Continuing this, we have a cycle.

Lemma 3.3.13. If y E a ( x ) , for E > 0 thew is an €-pseudo orbit y = yo, Y i , . ,Yk = y in a ( x ) (i.e. d ( f (yi-I), yi) < E for 1 5 i < k ) . Proof. For E > 0 let 0 < 6 < €12 such that if d ( p , q) < 6 then d(f ( p ) ,f ( 9 ) ) < €12. Since f d n ( x ) converges to a ( x ) as n + oo, there is no > 0 such that d( f - n ( x ) , a ( % )<) 6 for n 2 no. Since y E a ( x ) , there are nl,k 2 0 such that nl - k 2 no, d( f ( x ) ,y) < 6, d( f ( 2 1 , ~< ) 6-

$3.3 Recurrent sets and Birkhoff centers

Let sf = f-"l(x). Since n l - k 2 no, when 1 5 j 5 k we have

Thus there is yj E a(%)such that

and then

d ( f j + ' ( x f ) ,f (

<~12-

~ j ) )

Letting yo = yk = y we have that for 1 5 j 5 k

>

L e m m a 3.3.14. Let { C , : n 1) be a family of f-invariant compact sets such that if n 1 m then C , c C,. If x , E C , and y, E a(%,) converges to En, for E > 0 there is an &-pseudo orbit y = %,%, ,% = y yEC= in C (i.e. d ( f ( y i _ l ) , ~<)c for 1 5 i 5 k ) .

nz=l

..-

Proof. Let 0 < 6 < ~ / be 3 a number such that d(p,q) < 6 implies d(f ( p ) , f ( 9 ) ) < ~ 1 3 Take . no > 0 such that if x E C,, then there is y' E C with d ( x , y') < 6. Take n > 0 such that y, E a(%,)c C,, and d(y,y,) < 6. By Lemma 3.3.13 there is an &/3-pseudo orbit y, = 20, z l , ,zk = yn in a(x,). Thus we have

- -

for 1 5 j 5 k.

L e m m a 3.3.15. If A is f -invariant and compact and if R( f i A ) = A, then A cdf). Proof. For n > 0 define

< l l n for some k > O ) , A , = { y E A : d ( f - k ( y ) , y ) < l l n for some k > 0). A: = { y E A :d ( f k ( ( y y, )

Clearly A; and A$ are open dense subsets of A, and so is A, = A; n A:. Thus A = An is a Baire set of A, i.e. cl(A) = A. Since x E a ( x ) n w ( x ) for x E A , we have A = c l ( A ) c c ( f ) .

n;=,

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116

Lemma 3.3.16. If x E a ( x ) , then x E c ( f ) .

Proof. a ( x ) is compact and R(fl,(,)) = a(%). By Lemma 3.3.15 we have c c ( f). Proof of Theowrn 3.3.2. If we establish that a ( x ) C c( f ) for all x E X, then we have Q ( f )C c ( f )by Lemma 3.3.11, and therefore the conclusion of Theorem 3.3.2. Suppose there is x E X such that a ( x ) q! c ( f ) . Let 3 be the family of compact f-invariant subsets C of X such that there is x E C with a ( x ) @ c(f ). Introduce to 3 an ordered relation by inclusion. If 3 possesses a minimal set E, then a ( x ) @ c ( f ) for some x E X. By Lemma 3.3.16 we have x # a ( z ) . Thus a ( x ) is a proper subset of E. Since f ( a ( x ) ) = a ( % )and a ( x ) is compact, a ( x ) E 3, which is a contradiction. Therefore, t o obtain Theorem 3.3.2 it suffices to show the following Lemma.

41 .

Lemma 3.3.17. 3 possesses a minimal set C.

Proof. Let 3' = {Cp : P E B} be any totally ordered subfamily of 3. Let C = n { C p : P E B},then we need to prove I: E 3. If E $! 3,then a ( % )C c ( f )for all x E C. B y Lemma 3.3.11 the family admits a filtration and Q ( f i c )c ~ f Ai = nC. ~ Take xp E Cp with a ( z p )@ c(f ) for 3/ E B. Since a ( z p )n Ai) # 0, without loss of generality we suppose that a ( x P )n A1 # 0 for all p E B. Let U be a compact neighborhood of A1 such that nZrnf n ( U ) = A1 and Un Ai) = 0. Since a ( x P )n Al # 0 and a ( x p )q! Al, by Lemma 3.3.8 we conclude that a ( x p )n Da # 0 where D" = Da(Al,U) C U is a proper fundamental domain for W&,(Al). Choose yp E a ( x p )n Da for P E B and take a subsequence ypn such that ypn converges to y E C n Da. Recall that Da n C( f ) = 0. Since y $! c( f ) and C admits a filtration, there is a compact set K c C and y' E O f ( y )such that

(u:

(uL2

f ( K ) c int(K),

y' E K - f ( K ) .

Let W C K be an open set in C such that y' # W and f ( K ) c C. Since f ( K ) is compact, there is E > 0 such that if z E E and d(z,z') < E for some z' E f ( K ) then a E W . By Lemma 3.3.14 there exists an E-pseudo orbit y'=yo,yl,...,y~=y'inC.Thenwehaveyj~Wforl~j~k. Indeed, since f(y0) = f (y') E f ( K ) and d ( f ( y o ) yl) , < E, we have y1 E W C K . Suppose we have proved that yj E W for j 2 1. Then f ( y j )E f ( K ) and d ( f ( y j ) ,yj+l) < E. Thus yj+l E W . Consequently, y' E W, which is a contradiction. Therefore, E E 3.

53.4 Nonwandering sets of TA-maps

117

R e m a r k 3.3.18. For a continuous surjection f : X + X of a compact metric space, the Birkhoff center c(f) is defined by c(f) = cl{x E X : x E w(x)). As Theorem 3.3.3 we can show that iff: X + X is c-expansive and f : c(f) + c(f) has POTP, then c(f) = R(f ). The proof will be given in Remark 3.5.4 of 53.5.

53.4 Nonwandering sets of TA-maps We shall discuss the decomposition theorem of the nonwandering set for TA-maps. As before, let X be a compact metric space with metric d and let f: X -+ X be a continuous surjection. Recall that x E X is a nonwandering point if for any open neighborhood U of x there is n > 0 such that fn(U) n U # 0. Denote as R(f) the set of all nonwandering points of f ( R(f) is called the nonwandering set of f). Then R(f) is a non-empty closed subset and f (O(f 1) c R(f 1. T h e o r e m 3.4.1. Let f : X + X be a continuous sujection of a compact metric space. If f has POTP, then f (R( f )) = R( f ). Proof. Suppose R(f) - f (Q(f )) # 0. Then there are a: E R(f) - f (R( f )) and > 0 such that

E

Let 6 > 0 be a number with the property as in the definition of POTP. Since x E fl(f), there is n > 0 and an n-periodic &pseudo orbit (xi) E xZwith xo = x. Since f has POTP, there is (yi) E X f such that d(yi,zi) < E for all i E Z (Xf = {(xi) E XZ : f(xi) = xi+i,i E Z)). Hence, {fni(yo) : 0 5 i < oo) C B,(xo). Since X is compact, a subsequence of {fni(yo)) converges to some y' E X and, hence, y' belongs to the w-limit set of yo. Since f n(y') E Be(x) and f "(y') E f (R(f)), this can not happen. Let R(f) be the nonwandering set of a continuous surjection f . As before let

nf = {(xi) E ~ ( f ) ' : f(xi) = xi+l,i E Z).

The chain recurrent set of the continuous surjection f , CR(f), is defined in the same way as given for a homeomorphism. Then f (CR( f )) c CR( f ) and C R ( f ) is closed (cf. see Lemma 3.1.1). I f f has POTP, then R(f) = CR(f) (cf. see Theorem 3.1.2). T h e o r e m 3.4.2. If a continuous surjection f has P O T P then so does fln(f). If, in addition, fin( f ) is c-expansive, then the set of all periodic points, Per( f ), is dense in R( f ) .

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118

Proof. For E > 0 let 6 > 0 be as in the definition of POTP. Since R(f) = CR(f), the 6-relation A is defined on R( f ) as in 53.1. By this relation, a(f ) is decomposed as a union R(f) = Ux Ax of distinct equivalent sets Ax. We first prove that each Ax is open in R(f). Take and fix x E Ax. For any y E Ax let xo = x, X I , . ,xp = y be a 6-pseudo orbit in R( f). Choose 0 < 7 < 613 such that f (U,(xo)) c Ua(xl). To get the conclusion it is enough to see that U,(xo) n R(f) c Ax. For any x; E Uy(xo) n R( f ) , {xk, xl, ,xp) is a 6-pseudo orbit in R(f ). Since x, y E R( f ), a 6-pseudo orbit 0 = {yo = y, yl , ,ye = x) exists. If f(ye-1) E cl(U,(zo)) n n ( f ) , then ( 0 \ {ye)) U {xk) = {yo,yi,... ,ye-i,xb) is 6 a 6-pseudo orbit since d(f (yc-l),xb) 27 < 6. Thus we have y xb. When f (ye-1) g! cl(U,(x~)) n R(f), then there is z E cl(U,(xo)) n R(f) such that

..

...

-

<

and so d(xk, z) 1 27. Since z E R(f) = CR(f), we have z a, i.e. there is a periodic 7-pseudo orbit {zo = z, zl , ,zb, z) in R( f ). From

---

N

---

the sequence (0\ {ye)) u (20, ,zb, xb) = {YO,. ,ye-I, 20, ,zb, 2;) is a 6-pseudo orbit from yo to xb. Therefore, xk E Ax. Since R(f) is compact, {Ax) is finite and R(f) is covered by finite open sets Ax. Since each Ai is open and closed in R(f), we have d(Ai,Aj) = inf{d(a,b) : a E Ai,b E Aj) > 0 if i # j. Put 61 = min{d(Ai,Aj) : i # j). For 0 < a < min{6,61) let (xi) be an a-pseudo orbit in R(f)'. Then we must prove that an €-tracing orbit of (xi) is chosen in Rf. If xo E Ai for some i then each point of (xi) belongs to Ai (since x is always a-related to f(x) for x E R(f)). Take xa,xb E (xi) (a < b). Then x, 2, xb, from which we can find a (L1 It2)-periodic 6-pseudo orbit (y)E R(f)' such that for i 0

>

Put It = kl

+

+ Ftz for simplicity. Since f

has POTP, there is yavb E X such that

for i 2 0, and so

) is discrete, then there is t > 0 such that fe(yavb) = If D = ~ l { f ~ ~ ( y:"i *2~0) yaqb, and then yayb E R(f ). If D is not discrete, then a subsequence of

53.4 Nonwandering sets of TA-maps

converges to some aalb E X and d(fi(zagb), z,)

119

< E for 0 < j < k.

Thus we

have for any a' > 0. This follows from the fact that for any a' > 0 there is io > 0 so large that

>

for in io. Therefore, E CR(f) = R(f ). We put z: = f ' ( ~ ~ for 3 ~i )2 a and choose 2:-, E f -'(zapb) n R(f) and z:-i-l E f -'(z:-~) r l R(f) for i 2 1. Then za,a = (z:) E R(flz and d(z:,xi) E for a i b. Since R is compact, a subsequence of {z,,~) converges to r = (zi) E ~ ( f as) a~+ -00 and b + oo. Hence d(zi,xi) 5 E for i E Z since d(z:,xi) 5 E (a i 5 b) and a, b are arbitrary. Clearly z = (zi) E Rf. To show density of Per(f), take x E R(f). Then there are I > 0 and an I-periodic &pseudo orbit (xi) E R(f)' with xo = x. Since fin(f) has POTP, there is (zi) E Rf such that d(zi, xi) < e for i E Z. Then d(ze+i, zi) 2 E for i E Z. Choose e > 0 less than c-expansive constant, then we have (ze+') = (z;) and therefore fe(zo) = ze = zo E UU,(x).

< <

<

<

<

A continuous surjection f : X -, X of a metric space is topologically transitive if there is xo E X such that the orbit O+(xo) = {xo, f (xo), . - . )is dense in X . Let X be a compact metric space. A continuous surjection f : X -, X is topologically transitive if and only if for U, V nonempty open sets there is n > 0 such that fn(U) n V # 0. For the proof suppose O+(xo) is dense in X and U, V are nonempty open sets. Then there are n > m > 0 such that fn(xo) E U and fm(xo) E V. Thus fk(u)nV#Ofor k=n-m. Let {Ui : j 2 1) be a countable base for X. For x E X the following are equivalent :

Remark 3.4.3.

@ 0 + ( x ) n Un

fm(x) E X

=0

\ Un

for some n, for every m 2 0 and some n,

00

~

X

n f -E" ( X \ U n )

forsomen,

m=O

U n f-"(x\Un). 00

~

X

w

E n = l m=O

Since Uz=of-"(Un) is dense in X by the assumptions, X = r)z=o(X \ f -m(Un)) is nowhere dense. Thus Urz1

\ Uz=o f-"(Un) f -"(X \ Un)

nz==,

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120

= U=:l n:=,(X \ fem(Un)) is a set of first category. Since X is a compact metric space, {x E X : cl(O+(x)) # X ) is a set of first category.

A continuous surjection f: X -+ X of a metric space is topologically mixing if for nonempty open sets U,V there exists N > 0 such that U n f n(V) # 0 for all n 2 N. Topological mixing implies topological transitivity.

Theorem 3.4.4 (Topological decomposition theorem). Let f: X -+ X be a continuous sujection of a compact metric space. If f: X + X is a TAmap, then the following properties hold: (1) (Spectml decompsition theorem due to Smale) R(f) contains a finite sequence Bi (1 5 i 5 L) off -invariant closed subsets such that Bi (disjoint union), (i) R(f) = (ii) flsi : Bi + Bi is topologically transitive, (such the subsets Bi are called basic sets.) (2) (Decomposition theorem due to Bowen) For B a basic set there exist a > 0 and a finite sequence Ci (0 5 i 5 a - 1) of closed subsets such that (i) Ci n C j = 0 (i # j ) , f (Ci) = Ci+l and fa(Ci) = Ci, (ii) B = : : U: Ci, (iii) fbi : Ci + Ci is topologically mixing (such the subsets Ci are called elementary sets.)

u:=,

Proof. Since f: X + X has POTP, we can prove that R(f) = CR( f ) as in Theorem 3.1.2. Thus R(f) splits into the union R(f) = Ux Bx of the equivalence classes Bx under the relation which is defined in CR(f). Each Bx is closed and f(Bx) = Bx. If BAis open in R(f), then R(f) = ~ f Bi= for some k > 0 (since O(f) is compact). In this case flBi is topologically transitive. For, let U and V be non-empty open sets in Bi. Since x y for x E U and y E V, we can find in Bi a tracing point for a pseudo orbit from x to y (since flsi has POTP). This shows that U n f ' ( ~ )# 0 for some I > 0. To obtain (I), it is enough to show openess of Bx. For E > 0 there is 6 > 0 such that any 6-pseudo orbit of fln(f) is €-traced by some point of R(f). Then, for p E Ua(Bx) n Per(f) there is y E Bx such that d(y,p) < 6 (here Ua(Bx) denotes an open neighborhood of Bx in R(f)). By Lemma 2.4.3 we have

-

-

and

for x E X f . Since has POTP, we have Wa(y) n Wu((pi)) # 0 for a periodic orbit (pi) E Of with po = p, and Wa(p) n Wu((yi)) # 0 for an orbit (yi) E Rf with yo = y. Since we can choose such that each yi of the orbit (yi)

~

53.4 Nonwandering sets of TA-maps

is in B x , in our case suppose that so does (yi). Then we have y which p E B x . And so

121

-

p, from

To prove ( 2 ) ,take p E P e r ( f )n B with f m ( p ) = p and put C p = c l ( W d ( p )n B ) . Then C p is open in B . For, take q E U6(Cp)r l P e r ( f ) n B ( f n(q) = q for some n > 0 ) , then we can find x E W d ( p )r l B such that d(x,q) < 6.

Figure Let (qi) E Bf be an n-periodic orbit with qo = q. Since f l B has POTP, there is x' such that x' E W U ( ( q i )f )l W 8 ( x ) nB , and SO d(x:, qi) -+ 0 (i -+ -00) for some ( x i ) E Bf with xb = 2'. Since xb = x' E W 8 ( x )= W d ( p ) ,we have d ( f i ( x b ) f, i ( p ) ) -+ 0 as i -+ oo. Note that x'_,,~ + q a s k -+ oo. Fix 12 > 0 and let j = mnk i for i 0. Then

+

>

>

since fmn"x'_mnk) = xb and fmnk(p) = p. Thus xLmnk E W 8 ( p )for k 0 and so q E Cp. This shows that C p is open in B . We next show C p = Cq for q E C p n Per( f ). Let f m(p) = p and f n(q) = q. For -y > 0 let n, > 0 be as in Lemma 2.4.1. Fix E > 0 and choose 6 > 0 as in the definition of POTP. Let x E W d ( q )n B. Then there is J, > 0 such that mnJ, 2 n, implies d( f mnJ* ( x ) ,q) < 612. Since q E Cp, we can choose y E Ualz(q)n W d ( p )n B , and so d ( f m n J ~ ( xy)), < 6. Thus

for some orbit ( x i ) E Bf with x = xo (since is an orbit (2:) E Bf such that

flB

has POTP). Therefore there

CHAPTER 3

122

and so

d ( f ' ( t 2 ) , f i ( y ) -+ ) O

d(tlmnj, ,f """

as i + oo,

(x-mnJ, )) = d(zTmnj,

,Z)< 7.

Since a; E W 8 ( y )= W 8(p),we have

and so tImn

J,

c, c cp.

E W s ( p ) n B .Since 7 is arbitrary, we have x E Cp and therefore

On the other hand, suppose p 4 C,. Then we have 0 < d = d ( K ,C,) where K = Cp - C,. Since q E Cp, there is z E W 8 ( p )n B such that d(z,q) < d. Clearly t E C,. But

that f mnj(z)4 C,, which is a contradiction. It is easily checked that if C, nCqr # 0 for q, q' E Per( f ) n B then C, = C,, . Since Cfm(p)= Cp,there is the smallest integer a > 0 such that a 5 m and Cfa(,]= C,. Thus B = C, U Cf(,) U U Cfa-l(,) since f l B is topologically transitive. We finally show that is topologically mixing. Let U and V be nonempty open sets in Cp. Take q E V n Per(f) with fn(q) = q. Choose E > 0 with U,(q) c V. Then, for 1 j n - 1 there are zj E U n w S ( f a j ( q )and ) N j > 0 such that t 2 N j implies

SO

--.

fh

< <

<

j 5 n - l . Put Therefore, f a ( n t + n - j ) ( ~ ) n V # 0 for t 2 N j and 0 N = max{Nj : 0 5 j 5 n - 1). Then 8 2 nN implies fa8(U)n V # 0.

53.5 Inverse limit systems

Let f : X + X be a continuous surjection of a compact metric space. In the previous chapter, for expansivity and POTP we have discussed the relationship between the continuous map and the shift map of the inverse limit space. This section will investigate those relationships for w-limit sets, recurrent sets, nonwandering sets, chain recurrent sets and Birkhoff centers. Denote as c ( f ) the closure of { x E X : z 6 w(x)). Then c ( f ) is called the Birkhoff center for the continuous surjection.

53.5 Inverse limit systems

123

Theorem 3.5.1. Let f : X + X be a continuous sueection of a compact metric space and let X ; = l&(X, f ) be the inverse limit space with the shift map a : X ; + X;. Then ( 1 ) if n > 0 is an integer and Fn is the set of periodic points of f with period n (i.e. Fn = { x E X : f n ( x ) = x)), then l i i ( F n ,f ) is the set of periodic points of a with period n and the natural projection po : lim(Fn, f ) + Fn defined by ( x i ) r I+ xo is bijective, t ( 2 ) if x E X and x = ( 2 ,x l , x 2 , ) E X ) , then w ( x ,a ) = l&(w(x), f ) is the w-limit set for a , ( 3 ) i f R(f ) = { x : x E w ( x ) ) is the recurrent set, then R ( u ) = lii(R(f ), f ) is the recurrent set for a , ( 4 ) c(o) = l i i ( c (f ), f ) is the Birkhof center for a , ( 5 ) a ( u ) = l i i ( R ( f ) ,f ) is the nonwandering set for a , ( 6 ) C R ( o ) = l&(CR(f), f ) is the chain recurrent set for a .

<

Proof. For i 0 let pi : X ; + X be the natural projection to the i-th coordinate. W e first claim that i f a ( A ) = A is a closed set o f X ; then A = l@(A, f ) where A = p 0 ( A ) Indeed, since pi(A) = p i ( a i ( A ) ) = po(A) = A for all i 2 0, we have A c l i i ( A , f ) . I f y E l&(A, f ) , then pi(y) E A for all i 2 0. Since A = p i ( A ) for i 2 0, there exists zi E A such that pi(zi) = pi(y). Then pj(zi) = p,(y) for j 5 i. Since A is compact, { s i ) converges t o z E A. Therefore, z = y E A. ( 1 ) : I f x E Fn, then ( 2 ,fn-'(x), fn-2(x),... , x , E l&(Fn, f ) is a periodic point of a with period n. Coversely, i f x E X> is a periodic point of o with period n, then x is expressed as x = ( X O , 2 1 , , xn-~,xo, ), which implies xo E Fn, and so x E lim(Fn, f ) . Therefore l h ( F n ,f ) = { x E X ) : t a n ( x ) = x ) . I t is clear that po : l i i ( F n ,f ) -, Fn is bijective. ( 2 ) : Clearly po(w(x,a ) ) c w(x). For y E w ( x ) there is a sequence f ni ( x ) such that fn'(x) 4 y as i --t oo. Then { a n i ( x ) ) has an accumulation point y E w ( x ,a ) . Thus po(y) = y and w ( x ) = po(w(x,a ) ) . Therefore we have 4x9 0 ) = 9 ( w ( x ,f ), f 1. ( 3 ) : I f x E R ( a ) , then x E w ( x ) c l i i ( w ( x ) ,f ) and so pi(x) E w(xi) = ~ ( x )Therefore, . x E l&(R(f), f ) . Let x E l i i ( R ( f ) ,f ) and put x = po(x). Then pi(x) = xi 6 R ( f ) for all i 2 0. Thus we have pi(x) E w(xi) = w ( x ) and thus x E l&(w(x), f ) = w ( x ) by (2). Therefore, x E R ( a ) . ( 4 ) : By ( 3 ) we have . a - )

and so c ( a ) = cl(R(u)) c l&(c( f ), f ). To obtain the conclusion take x E l$(c(f), f ) . Since p;(x) = xi 6 c ( f ) for i 2 0, for E > 0 there is N > 0

CHAPTER 3

124

such that for any m 2 N there is ym E { y E X : y E w(y)) (so close to x,) satisfying d(yi,xi) < € 9

---

where yrn-i = f'(ym), for 0 5 i 5 m. Here we choose points ym+l, grn+2, such that ym = (YO,YI,... ,~rn,~rn+l,...)E l@(X,f)Then we have

J(Y-,R(.))

4

" M M C 3 < 2, i=m+l

where M is the diameter of X . Therefore, x E c ( a ) since ym + x as m + oo. ( 5 ) : Since pi o o = f o pi for i 2 0, we have pi(n(o)) c n(f)and thus c li@(Q(f >,f 1. To show n ( a ) > l i i ( O ( f ) ,f ) let M be the diameter of X. Let x = ( x OX, I , - - - , x m , - - . ) E l i i ( n ( f )f ,) and take m > 0 such that M/2m < €14. Since xm E Q(f ) and f '(x,) = xm-i for 0 5 i 5 m , by uniform continuity we can choose ym E X satisfying

n(.)

for some It that

> 0.

Here we take points ym+l, ym+2,

are in l i i ( X ,f ). Then

and similarly d ( x , y l ) < €12. Since

and

y;+,

,

such

$3.5 Inverse Inverse limit systems

125 125

we have have

and thus

I E Therefore, E fl(a). (6) : The : chain chain recurrent set E E Indeed, let = = pseudo orbit ==

I

++

8.

in X) is a subset of l@(CR(f),f). 2 0.2 e e> 0>0take an ~ / 2 ~ = = belonging belonging to Then we

{xk

and SO < E<Efor 0for05 i5< i n. < Thus == ,zk-',xk = = x,} is is an &-pseudoorbit from from x, to x,. Since ESince is is arbitrary, we have have x, E E and therefore E l&(CR(f),f) E since since is is chosen arbitrarily. chosen To prove prove that l&(CR(f),f) c c for e e> 0>0choose choose m >m 0>0such that e ewhere where M is is the diameter of X. X. Since Since f:X X + X +X is is uniformly uniformly M/2" < < continuous, there is 0 0< 6<< < such such that d(z, y)

< 6<(3, y Ey E

= (=x o , x 1 , ~ ~ ~ &pseudo orbit == Thus which which

xk

max{d(f(x),

: :< i<5 i m) 5 < ~
E l&(CR(f),a), E since since x, E E there is is a , x k = x=, such such that each each x i is in << €14 €14 for for 0 05 5 j j5 5 n -n 1-and 1 0 0< i
.- -.- -

CR(f)) = CR( = f), for 0for0< j<j 5 n5 1we 1 can find points find Since f( Since such that

- - .)

==

* ** ** * ,f(x&),2&,x~+1, * *

is a member of lii(CR(f), f) and = = = =holds. holds. Then is an &-pseudoorbit. This follows from from the fact fact that

E 2

M 2,

<-+-<€

for 0 05 j5j5 n 5n 1.-Since ESince Eis arbitrary, we have have E E

0 0

- --., ,

126

CHAPTER 3

Theorem 3.5.2. Let f : X -t X be a continuous surjection of a compact metric space. If A is a minimal subset of X , then the shift map a : Af + Af is minimal. If A is a minimal subset of X f , then po(A) is a minimal subset of X . Here Af = l i i ( A , f) and po : Af -, A is the natuml projection to the 0-th coordinate. The proof is straightforward and so we leave it to readers.

Theorem 3.5.3. ( 1 ) f : X -, X Xf 4Xf ( 2 ) f: X -+ X X f is so.

Let f : X -, X be a continuow surjection. Then is topologically tmnsitive if and only if the shift map a : is so. is topolqgically mixing if and only if the shift map a : X f +

The proof leaves to the readers.

Remark 3.5.4. We now give the proof of the statement described in Remark 3.3.18, i.e. if a continuous surjection f: X -, X is c-expansive and f : c ( f ) -t c ( f ) has POTP, then c ( f ) = R ( f ) . B y Theorem 3.5.1 (4) and Theorem 2.3.8, the shift map a : c ( u ) 4 c ( a ) of the inverse limit space of ( c ( f ) ,f ) has POTP, and by Theorem 2.2.29 the shift map a : X f + X f is expansive. Since X is compact and a : X + X is a homeomorphism and c ( a ) is the Birkhoff center of a on X f , we have C ( U ) = R ( a ) by Theorem 3.3.3. Therefore, c ( f ) = R( f ) by Theorem 3.5.1 (4) and (5).

CHAPTER 4 Markov Partitions

We have studied dynamics of TA-maps and symbolic dynamics in the previous chapters. Our main task in this chapter is to show that every TAhomeomorphism has Markov partitions, and by them the TA-homeomorphism is realized on a symbolic dynamics. This symbolic dynamics is applied often to the ergodic theory of TA-homeomorphisms.

84.1 Markov partitions a n d subshifts Markov partitions for Anosov diffeomorphisms were first constructed by Sinai [Sill. After that, Bowen [Bo2] found Markov partitions for basic sets of Axiom A diffeomorphisms. Ruelle [Ru] constructed Markov partitions for homeomorphisms satisfying some conditions that means local product structures. Let f : X + X be a homeomorphism of a compact metric space. Let A(€) = {(x, y) : d(x, y) 5 E ) for E > 0. Then we say that f has local product structu~-e if the following conditions (A) and (B) are satisfied : (A) There are 60 > 0 and a continuous map [ , ] : A(60) -+ X such that for x, y, z E X

when the two sides of these relations are defined. (B) There exist 0 < b1 < such that for each x E X, and 0 < p < letting

the conditions hold : (a) Nx is an open set of X and diam(Nx) < 60, (b) [ , ] : V{(x) x Vt (x)+ N, is a homeomorphism, (c) N, > BP(x) where Bp(x) = {y E X : d(x, y) 5 p). If f : X + X has POTP, for 6 > 0 small we have Wi(x) n W,U(y) # 0 when x is very near to y. If, in addition, f is expansive, then we see that Wi(x) n W,U(y) is a set consisting of single point and it is denoted as [x, y]. Throughout this section let f : X + X be a TA-homeomorphism of a compact metric space.

CHAPTER 4

128

Theorem 4.1.1. The homeomorphism f has a local product structure. Proof. Let e > 0 be an expansive constant for f and ~IX EO = e / 4 . Then there is 0 < So < EO such that W a x ) n Wz(y) = [x,y] for x,y E X with ~ ( x , Y ) 60. First we show that [ , ] : A(60) -+ X is continuous. Suppose a sequence {(x,, y,)) of A(&) converges to (3, y) E A(60). Put z, = [x,, y,]. Since X is compact, there is a subsequence {z,,) of {z,) that converges to z E X. Since znj E W,d,(x,,), we have d(fi(xnj), fyz,,)) 5 EO for i 0 and nj, and so d(fi(x), f'(z)) I EO for i 2 0. Thus, z E W:o(x). Similarly, z E Wz(y) and z = [x, y]. This shows that {z,) converges to [x, y]. It is clear that [x, x] = x for all x E X. Since [z, y] E W:o(x), we have [[x,y], z] E W&,(x) n W t ( z ) and then [[x,y], z] = [x, z ] by expansivity. Similarly, [x, [y, z]] = [x, a]. It is easily checked that f [x, y] = [f(x), f (y)] by uniform continuity. To conclude the theorem we must prove (B). To do so define gl : X x A(6o) -t a by 91(x, (Y,4 ) = 4x9 [Y, 21)

<

>

for x E X and (y, z) E A(60). Then gl is continuous and gl(x, (x, x)) = 0. By uniform continuity of gl we can find 0 < < b0/2 such that diam({x, y, z)) < 261 implies d(x, [y, z]) < 6013. If (y, z) E V{(x) x (x), then d(x, [y, z]) < 6013 and thus diam(N,) < 60. To show openness of N, let w E N,. Then there are y E Vt(x) and z E V[ (x) with w = [y, el. Since d(x, w) < 6013, we can find maps

by p,(v) = [v, x] and p,(v) = [x,v] for v E B6013(w). They are clearly continuous. Since w = [y, z], we have p,(w) = [y, x] = y and p,(w) = z. Thus there is a neighborhood U C BSo13(w)of w in X such that p,(U) C V{(x) and p,(U) c V&(x). If v E U, then v E N, since v = [[v,XI,[x, v]] by expansivity. This implies that N, is open in X. Therefore, (a) was proved. (b) is easily checked as follows. Define a map h: N, -+ V{(x) x V[ (2) by

Obviously, h is continuous and h is the inverse map of [ homeomorphism. To see (c) put 9 2 ( ~Y) , = diam{~,[Y,x],[x,Y])

,

1; i.e. h is a

for (x, y) E A(&). Then g2 is a continuous map and there is 0 < p < b1 such that d(x, y) < p implies g a ( ~ , y )< 61. This shows that [y,x] E Vt(x) and [x,y] E V6d,(x) and therefore y = [[y,XI, [x, y]] E N,.

$4.1 Markov partitions and subshifts

129

Remark 4.1.2. Let x E X. Then [y,z] E N, for y, z E N,. For y, z E N, there exist u1, uz E V z ( x ) and vl ,vz E Va",( x ) such that Y = [ ~ vi] i , and 2 = [ U Z ,V Z ] . Thus [Y,z] = [[ui, v i ] ,[ U Z ,vz]]= [ui,vz] E N,. For convention we write

for x , y E X.

Lemma 4.1.3. For x, y E X with d(x,y) of x in V z ( x ) ( a = s, u ) , and the maps

< p, D:,,

is an open neighborhood

are homeomorphisms. Proof. Since N, is open in X , D k , is open in V z ( x ) . If d(x,y) 5 p, then x E B,(y) C N, and x E Dg,. Thus D:, is an open neighborhood of x in V c ( x ) . Let z E D,",,. Since z E N,, we have [z,y] E V{(y). Since z E V{ (x) c N, and y E B,(x) C N,, we have [ r ,y] E N,. Thus [z,y] E Dlf,,. Similarly we have [z,x] E D,",, for z E D,",,. That [D,",,, y] = D,",, follows from the properties of [ , 1. By (b) the map [ ,y] : D;,, + D,",, is a homeomorphism and [ ,x] : D,",, -t D,",, is its inverse map. The same result is true for a = s.

Lemma 4.1.4. For x E X (a) f Va",(.) n V,dl(f(4)is open in V,dl(f( x ) ) , (b) f -'Vz ( 2 )n V z (f ( 2 ) )is open in V; ( f

-'

-'

(3)).

Proof. Take w E f (N,) n Va",(f( x ) ) . Then f -'(w) E N, and f-'(w) = [y,z] for some y E V$ ( x ) and z E Va",( x ) . Since V; ( z ) C W E ( z ) ,we have f ( w ) E W,.O(z). On the other hand, since w E V;(f ( x ) ) ,we have w E V;(f ( x ) )c W.&,(f ( 2 ) ) . By expansivity we have f -'(w) = z and so w = f ( z ) E f Va",( x ) . Therefore

-'

Since N, is open in X, we have (a) and similarly (b).

A subset R of X is called a rectangle if diam(R) 5 p and [x,y] E R for x , y E R. Throughout this section let R denote a rectangle of X.

CHAPTER 4

130

Remark 4.1.5. Since p is chosen such that p < 61,we have [x, y] = Vad,(x) r l V<(y) for x, y E cl(R). Thus cl(R) is a rectangle. For convention we write

for x E X. Denote as int Ve(x, R) the interior of VU(x,R) in V<(x) (u = S, U) and write dVd(x,R) = V8(x, R) \ int V8(x, R),

BVu(x, R) = Vu(x, R) \ int Vu(x, R).

Figure 15

L e m m a 4.1.6. Let x, y E R. Then (a) R = [VU(x,R), V8(x, R)1, (b) [dvU(x,R), V8(x, R)] = [dVU(y,R), V8(y, R)1, (c) [vU(x,R),aV8(x,R)] = [VU(y,R), dV8(y, R)]. Proof. Since R is a rectangle, clearly [VU(x,R), V8(x,R)] C R. Let z E R, then [t,X] E R n V ~ ( X= ) vU(x,R) and [x, z] E V8(x,R). Thus

(a) was proved. Since diam(R) p and y E R, we have R C Bp(y) c N, and so VU(x,R) C D,",,. Similarly, VU(y,R) C D;,, . Since [VU(y,R), y] = Vu(y, R), by Lemma 4.1.3 we have [dVU(y,R), x] = dVU(x,R)

<

$4.1 Markov partitions and subshifts

131

and thus

(b) was proved. (c) is shown in the same way. We write BdR = [BVU(x,R), Vd(x,R)],

a U R= [Vu(x,R), BVd(x,R)].

By Lemma 4.1.6,BdR and BURdo not depend on x E R. Since R is a rectangle, BdR c R and BURc R. Denote as int(R) the interior of R in X and write

Lemma 4.1.7. (a) int(R) = [intVu(x,R), intVd(x,R)] for x E R, (b) BR = 8"R U BUR.

Proof. Since R C N, and N, is open in X , the interior of R in N, coincides with int(R). By Lemma 4.1.3, (a) follows from (B) (b). Thus

OR = R \ int(R) = [Vu(x,R), Vd(x,R)] \ [intVu(x,R), intVd(x,R)] = BdR U BuR.O

Remark 4.1.8. int(R) is a rectangle. This follows from Lemma 4.1.7 (a). Lemma 4.1.9. Let x E R. Then intVu(x, R) = V0(x,int(R)) for a = s,u.

Proof. Since int(R) is open in X , Vd(x,int(R)) is open in G ( x ) . Thus Vd(x,int(R)) c intVd(x,R). Let z E intVd(x,R), then z = [x,z] and by Lemma 4.1.7 (a), z E int(R). Therefore, z E Vd(x,int(R)). The same result is true for a = u. Lemma 4.1.10. Let x E R. Then cl(Vu(x, R)) = Vu(x, cl(R)) for a = s, U.

Proof. Since cl(R) is a rectangle, we have

Let z E Vd(x,cl(R)), then z E V;(z) and z = [x, z] E [x,cl(R)]. Thus Vd(x,cl(R)) = [x, cl(R)]. Since [x,cl(R)] is closed in X , we have cl(Vd(x,R)) C Vd(x,cl(R)). Similarly, cl(VU(x,R)) C Vu(x, cl(R)). By Lemma 4.1.6 (a), R = [Vu(x,R), Vd(x,R)]

CHAPTER 4

132

and thus R C [cl(VU(x,R)), cl(V8(x, R))l which is closed in X. Thus

Therefore, V8(z,cl(R)) = [x, cl(R)] = [x, [cl(VU(x,R)), cl(V8(x, R))] = [x,c1(V8(x,R))] = c1(V8(x,R)).O

/ "'(f

Rj)

Figure 16 A rectangle R is said to be proper if R = cl(int(R)). A Markov partition for a homeomorphism f: X + X of a compact metric space is a finite cover {R1,. .. ,R,) of X such that (a) each Ri is a proper rectangle, (b) int(Ri) fl int(Rj) = 0 for i # j, (c) let x E int(Ri) n f -'int(Rj), then f v 8 ( x , R ) c v8(f (x), Rj)

and

f VU(x,Ri) > Vu(f (x), Rj).

54.2 Construction of Markov partitions

With the preparation of 54.1 we construct a Markov partition for a TAhomeomorphism f : X + X of a compact metric space by using the method in Bowen [Boll.

$4.2 Construction of Markov partitions

133

We say that a subset A of X is ?-dense if for every x E X there exists y E A such that d(x,y) < 7. Choose 0 < P < min(pl2, e ) such that d(x,y) < P implies max{d(f ( x ) , f ( y ) ) , d ( f - ' ( x ) , f - ' ( y ) ) } < 61. Since f has POTP, let 0 < a < PI2 be a number such that any a-pseudo orbit is P/2 traced by some point of X. We choose 0 < 7 < a12 such that

-.. ,P,) be a 7-dense finite set of X and define

Let P = { P I ,

Given q E E ( P ) there is a unique 8(q) E X which pl2-traces q. This implies that for any x E X there is q E E ( P ) with x = 8(q). Thus 8 : E ( P ) X is surjective and the diagram

01x

le f

commutes

.

x

Here a denotes the shift map defined as usual. Now define

To = {B(q): E E(P),qo = P,},

1 5 s 5 r.

.-

Then diarn(T,) 5 P and T = {TI,. ,T,)is a cover of X. Fix 1 x, y E T,, then there exist q, q' E E ( P ) such that

< s < r . If

x = 8(q), y = 8(q1) and qo = q&= P,. Put q; = q, for j have

2 0, q; = qi for j 5 0 and write q* = (9;) E E(P). Then we

Since /3 < e and

[ x ,YI = P(q),fl(ql)I E W , ( @ ( q n ) )Wpu(9(q1>>, we have [ x ,y] = 8(q*),which is contained in T,. Therefore, T, is a rectangle.

CHAPTER 4

134

We next prove that B : C ( P ) + X is continuous. If this is false, then there = j y , Ijl 5 N , is X > 0 so that for every N we can find q N , j N E C ( P )with such that d ( e ( q N ) , @ ( g N>) )X.

qy

Let xN = e ( q N )and y, = @ ( j N )Then .

By taking a subsequence we have x , -+ x and y, + y as N + oo. Then d ( f j ( x ) ,f j ( y ) ) 5 2P for all j E Z and so x = y. On the other hand, since d(x,y) 2: A, we have a contradiction. Continuity of 0 ensures that Ts is closed in X. Let x E X and define

T ( x ) = {Tj E T : x E T j ) , T * ( x )= {Tk E T : Tk n T j #

0 for some Tj E T ( x ) ) .

n5=laTj, where each aTj is the boundary in X, is open in

The set Z = X \ X. Here we define

Lemma 4.2.1.

Proof. For x

E

Z* i s dense in X .

X we define

Since diam(Tk) 5 P , we have

and thus

where yk E T k . Since V U ( y kT, k ) C Dik,,, by Lemma 4.1.3 we have [dVu(yk, Th), X ] C D,U,yk. But [dVU(yk, Tk),XI is nowhere dense in D&,, and so is V t ( x ) . Thus [a:, x] is nowhere dense in V; ( 2 ) . Let x E Z , then there exists an open neighborhood U, c N , of x in X such that T ( x ) = T ( y ) for any y E U,,from which 8; = d,Q for o = s,u. Let us define u: = ux~ I V ; ( X ) \ [~:,x~,v;,(x) \

I~,~:II.

54.2 Construction of Markov partitions

135

By (B)(b) we have that UL is dense in U,. For y E UL there exist yl E V{(x) \ [a:, x] and y2 E V4 (x) \ [z,a:] such that y = [y, ,yz]. Thus V; (y) n 4 = 0. Indeed, if z E V; (y) n a:, then

thus contradicting. Similarly, 5;(y) n 8: = 0. Therefore, y E Z* and so Ui c Z* # 0 for x E Z. Since U, is an open neighborhood of x, Z* is dense in Z. Since Z is dense in X, Z* is dense in X. For T,, Tk E T with Ti n Tk #

0 we define

Figure 17

Remark 4.2.2.

4 If Tj n Tk # 0 then T;,& = Tj n T k and Tj = U,= ,

Tlk.

Remark 4.2.3. Each TG is a rectangle. Indeed, it is clear that TjVk is a rectangle. To see that TjVk is a rectangle, let x, y E Tik. Then z E V; (y)nTk # 0 and V<(x)nTk = 0. Since z = [z,y] = [z, [x,y]] and d(z, [x, y]) 5 2P < 6i, we have z E V<([x,y])nTk # 0. If z' E Vg:([x, y])nTk # 0, then z' = [[z,y],z'] = [x, z']. Since d(zl, x) < 2P < 6i, we have z' E V; (x) n Tk # 0, a contradiction. Thus V i ([x, y]) f l Tk = 0 and [x, y] E Tik. This implies that Tik is a rectangle. It is checked by the similar way that Tik and Tik are rectangles.

136

CHAPTER 4

Lemma 4.2.4. For T j ,Tk E T with T j n Tk # 0

for 1 5 n 5 4. Proof. Take x E int(Tlk)and z E V; ( x )naaTj# 0. By the definition of aaTj there exist zl E 8VU(x, T j )and 2 2 E V a ( xT, j )such that z = [ z l ,zz]. Since

we have

z = [[a, zz],[a,z l ] ]= 21 E aaTj. This contradicts x E int(Tj) by Lemma 4.1.7 (b). I f z E V; ( x )n aaTk,then z = [zl,z2] for some 21 E aVU(z,Tk) and some Z z E Va(Z,Tk). Thus 2 = [ [ % I , ~ 2 1 [z2, , ~ l ]=]z1 and so z E EVU(z,Tk).Since d(x,z) 5 p, by Lemma 4.1.3 there is a homeomorphism [ I., : Di.2 D;,, +

and then [x,z] = z since z E V;(x). Note that D;,, n T l k is a neighborhood of x in D:,, and that VU(z,Tk)has no interior points in Tk. Then we can find the following

v E D;,, 17T l k

with [v,z] $ V U ( zTk). ,

For this v we have V;(v) n Tk = 0. Indeed, if w E V;(v) n Tk # 0, then [v,z ] = [[w,v],z ] E V U ( zTk). , But [v,z] $ V U ( zTk), , thus contradicting. Therefore, V;(x) n (aaTjf l &Tk) = 0 when x E int(T5). By the similar way we can prove that

Suppose that

Then x E: aVU(x, Tzk).If x E aVU(x, T j ) ,then

and so V t ( x )n aaTj# 0 , thus contradicting. Therefore, x

E

int(VU(x, Tj)).

54.2 Construction of Markov partitions

137

Since d(x, yk) 5 p for yk E Tk, by Lemma 4.1.3 there is a homeomorphism [ , ~ k ]: Dz,,, -+ Dih , , . Since ~ ; ( x ) n aaTk = 0 and B V ~ ( ~ ~c, T PT ~~ ), we have V&(x) r l aVu(yk, Tk) = 0 and thus [z, yk] $i! 8VU(yk,Tk). Since Tk is closed, VU(yk,Tk) is closed in X by Lemma 4.1.10. Since Vu(Yk,Tk)c Dih,=,BVU(yk,Tk)is the boundary of Vu(yk,Tk) in D:h,z. Thus there is a neighborhood U,"c VU(x,Tj) of x in D,",, such that

namely V;(v)nTk # 0 for d l v E U,"or V&(v)nTk = 0 for all v E U;. On the other hand, since U,"c Vu(x,Tj), we have that either V;(v)nTk # 0 for all v E Uz, or V{(v) n Tk = 8 for all v E U,U. Therefore, U,"c TZk which contradicts x E aVU(x,Tzk). It was proved that for x E TZk

Similarly we can prove that V;(x) n (aUTjU aUTk)= 8 implies x $i! aUTZk. Therefore the conclusion is obtained by Lemma 4.1.7 (b)

Remark 4.2.5. Let x E Z*. If Tj n Tk# 0 for Tj E T(x) and Lemma 4.2.4 we have x E int(Tlk) for some n.

Tk

E T, by

For x E Z* define R(x) = n { i n t ( ~ s :) Tj n T k # 0 for Tj E T(x), Tk E T and x E int(Ttk)}.

Remark 4.2.6. R(x) is an open rectangle and for y E R(x), R(x) = R(y) and T(x) = T(y). The former is clear. If Tk E T(y) and Tj E T(z), then y E Tj n T k # 0 and so y E R(x) C TjPkand Tk E T(x). Therefore, T(y) C T(x). It is clear that T(x) c T(y). Thus T(x) = T(y). By definition, R(x) = R(y) for y E R(x). Since T is finite, so is {R(x) : x E Z*).Thus there exist such that Z* = R(xl) U U R(zm)

XI,...

,xm E Z*

- ..

is a disjoint union.

Lemma 4.2.7. Let x E R(xi) n f -'R(x

j)

for i

# j. Then

Proof. Fix x E R(si) n f-'R(xi) and let v E X. Then v = B(q) for some q E Z(P). Suppose qo = Pa and ql = Pt for Pa,Pt E P = {PI,. ,P,).

..

CHAPTER 4

138

If w E V8(v,T,),then there is q' E C(P)such that w = B(qt) and q; = P,. Thus w = [v,wl = [B(q),B(9')l = B(99') so that

f (w)= f

--

0 B(qq1) =

B o a(qql)E Tt.

- ..

Here qq' = (- ,q Y z , q i l , 90,q1, qz, ). Since w = [v,w] E V; ( v )and d(v,w) < P, we have d ( f ( v ) ,f ( w ) ) < and so

Since w is arbitrary, we have

and in the similar way

Let y E V d ( xR, ( x ~ ) Then ). y E V { ( z ) and R(x)= R(y)= R(xi). We first prove that T ( f( x ) )= T ( f( y ) ) . I f f ( 3 )E T j and f ( 2 )= B o a(q)where ql = Pj and q, = P,, then x = B(q) E T,. By (1)we have

and thus f ( y ) E Tj. Similarly, f ( x )E T j when f ( y ) E Tj. Next we prove that if T j E T ( f( x ) )= T ( f ( y ) ) and T j n Tk # 0 for Tk E T then f ( ~ ) , f ( ~E ) Tzk. Since f ( y ) E V d ( f ( x ) , T j f) ,( ~ and ) f ( y ) belong to T:,k u T i k or T l k u Ti!. Suppose V; ( f ( y ) ) n Tk = 0 and V { ( f( x ) )n TI # 0. Note that this assumption is equivalent to

Take f ( z ) E V U ((fx ) , T j )nTk. Let f ( x )= B o ~ ( qfor ) q E C(P)with and qo = Pa. By ( 2 ) we have

ql

= Pj

Let f ( z ) = 8 o a(q1)for q' E C(P)with q{ = Pk and q; = Pt. Then a E Tt and z E T,nTt # 0. Since x E T,, we have T, E T ( x )= T ( y ) .Since z E Vu(x,T,)n Tt and x, y belong to the same set T:,, it follows that Vu(y,T,)nTt# 0. Thus there is z' E VU(y,T,)n Tt and then t"

= [a,y] = [a,z'] E V8(z,Tt)

$4.2 Construction of Markov partitions

and by (1)

f (z") E VS(f(k),Tk)Since f (z),f (y) E Ti, we have which contradicts (3). Now we denote R(f (%))I= n { ~ : k

:Tj n T k

# 0 for Tj E T(f (z)),Tk E T and f (x) E T};.

Then, int(R(f (x))') = R(f (x)) = R(xj) since f (x) E R(xj). Since diarn(R( xj)) 5 P, we have f VS(x, R(x,)) C Vb)t(f (2)) and thus

5

(x) r l Vz (f (x)) is open in Vb)t(f (3)) and Lemma 4.1.4 (a) ensures that f R(xj) is also open in X. Thus fVa(x, R(xi)) is open in Vb)t(f(x)). Therefore, by Lemma 4.1.9 fVS(x,R(xi)) C int(VS(f (x), R(zj))) = VS(f (x), R(zj)). Similarly we can prove that Vu(f (x), R(xj)) C f Vu(x, R(xi)). With the above preparations we have the following theorem.

Theorem 4.2.8. Let f : X -r X be a homeomorphism of a compact metric space. I f f is a TA-homeomorphism, then there ezists in X Markov partitions with arbitrarily small diameter.

---

Proof. Let R j = cl(R(xj)) for 1 5 j 5 m. Then 7Z = {R1, ,k }is ' is a cover of X , and each Rj is a a Markov partition of X. Indeed, R proper rectangle with diam(Rj) 5 P and int(Ri) n int(Rj) = 0 for i # j. The remainder of the proof is to check that if x E int(Ri) n f-'int(Rj) then f VS(x,Ri) C Va(f (x), Rj) and f VU(x,Ri) 3 Vu(f (x), Rj). By Lemma 4.2.7 and by Lemma 4.1.10 and then for y E Ri n f-'(Rj)

The proof of one half has been given. The proof of the other half is similar and is therefore omitted.

CHAPTER 4

140

Remark 4.2.9. Let f : X -+ X be a homeomorphism of a compact metric space and let R(f) be the nonwandering set of f in X. If f is a TAhomeomorphism, then has POTP by Theorem 3.1.8 and it is expansive. Thus R(f) has Markov partitions by Theorem 4.2.8. R(f) is expressed as the finite disjoint union R(f) = URj of basic sets R j which are open and closed in R(f) (see Theorems 3.1.2 and 3.1.11). Thus f l n j : R j + Rj is a TA-homeomorphism. Therefore, each Rj has Markov partitions.

54.3 Symbolic dynamics

Let f : X -+ X be a TA-homeomorphiim of a compact metric space. Let R = {R1, ,R,) be a Markov partition of a basic set R, and define the transition matrix A = A('R) by Aij =

1 { ,,

if int(Ri) n f otherwise

.

-'(int(Rj)) # 0

Lemma 4.3.1. Suppose x E Ri n f-'(Rj) and Aij = 1. Then fV8(x, Ri) V"(f (x),Rj) and fVU(x,Ri) 3 VU(f(x), Rj). Proof. This is the same as the last part of the proof of Theorem 4.2.8.

c

Lemma 4.3.2. Let D c V{ (x) n R, and C c Vc (x) r l as. Then the rectangle [C, Dl is proper if and only if D = cl(int(D)) and C = cl(int(C)) as subsets of V{ (x) n R, and Vz (x) n R, respectively.

Proof. This follows from (B) (b). Let R, S be two rectangles in 0,. Then S is called a u-subrectangle of R if (a) S # 0, S C R and S is proper, (b) Vu(y, S) = VU(y,R) for y E S. Lemma 4.3.3. Suppose S is a u-subrectangle of Ri and Aij = 1. Then f (S) n R j is a u-subrectangle of R,.

Proof. Take x E Ri n f-l(Rj) and put D = Va(x, Ri) n S. Since S is a u-subrectangle, we have

S = U{vU(., Ri) : y E D ) = [VU(z,R;), Dl. Since S is proper and nonempty, by Lemma 4.3.2 we have 0 # D = cl(int(D)) and f ( s ) n R, = U { f v U ( ~R) , n R~ : v E D). Since Aij = 1, we have that f (y) E R j for y E D and y E Ri n f -'(Rj). Thus fVU(y,R;) n R j = Vu(f (y), Rj)

54.3 Symbolic dynamics

by Lemma 4.3.1 and so

Since R, = [ V u ( f ( z )R, j ) ,V 8 ( f ( z )R, j ) ] by Lemma 4.1.6 and R j is proper, V u ( f( a ) ,R j ) is also proper. Since f maps V 8 ( x R , i ) homeomorphically onto a neighborhood of V a ( f( z ) ,R j ) , we have f ( D ) = cl(int(f ( D ) ) )and so f ( S )n R j is proper by Lemma 4.3.2. Since f ( D ) # 0, we have f ( D ) n R j # 0. If y" E f ( S )f l R j , then y" E VU(y',R j ) for some y' E f ( D ) and thus

Therefore

V u ( ~ "R,j ) = Vu(y",f ( S )n R j ) for yl' E f ( S )n R j . This implies that f ( S )n R j is a u-subrectangle of R j .

Theorem 4.3.4. Let XA be the compact subset of Y; defined by

C A = {x = ( x i ) : AaiZi+,= 1 for i E Z} and o : XA -+ XA be the shift map defined by a ( ( z i ) )= as usual. For each a E CA,the set n { f - j ( ~ , ~:)j E Z} consists of a single point which is denoted by ~ ( a ) The . map T : XA + R, is a continuous sujection such that the diagmm

ZA

7 '

XA

and a is injective on the Baire set Y = R,

Proof.If A,,,,,, have that

= 1 for 1 5 i

\ U(fj(87Z):j

5 n - 1, by using Lemma 4.3.3 inductively we

n

is a u-subrectangle of R,,.

E Z).

Thus

is nonempty. Since Kn(a) > Kn+l ( a ) >

..,we have

CHAPTER 4

142

If x, y E K(a), then fj(z), fj(y) E Raj are close for all j E Z and so z = y by expansivity. Thus K(a) is a single point. Here we define n(a) = K(a). Since K(u(a)) = f -j(~,,+,) = f f - j ( ~ , , ) ) = f (K(a)), we have n o a = f o n. Continuity of n is easily checked. Since BR is nowhere dense, Y is a Baire set. For x E Y take a j E Y, with fj(x) E Raj for all j E Z. Then f j ( x ) E int(Raj) and so A,,,,,, = 1. Thus a = (aj) E C A and x = n(a). If x = ~ ( b ) , then fj(x) E Raj and bj = a j since f j ( x ) $ BR. Thus T is injective on Y. Since n('CA) is a compact subset of a, containing Y,we have = a,.

n

(n

Theorem 4.3.5. The shift map a : C A -+ C A is topologically tmnsitive. If fin, is topologically mixing, so is a : C A + C A .

Proof.Let U and V be nonempty open subsets of CA. Then there exist a, b E C A and N

> 0 such that

U 3 Ul = {Z E C A : xi = a;, lil 5 N), V > Vl = {x E C A : X; = bi, lil I N), and

n n N

0 # int(KN(a)) =

f-j(int(~.,)) = Uz,

-N N

0 # int(KN(b)) =

f-j(int(~b,)) = ~

2 .

-N

If ~ ( x E) U2, then f j o ~ ( x 6 ) R,, and f j o ~ ( x E) int(Ra,), which implies X j = a j for I jl 5 N . Thus, n-'(U2) C Ul. Similarly, r-l(V2) C Vl. Since fin, is topologically transitive, we have fn(Ua) n V2 # 0 for some n. Then

By the same argument we can prove that u is topologically mixing if fla, is. Theorem 4.3.6 (Bowen [Bo~]). Under the notations and assumptions of Theorem 4.3.4, there exists an integer d such that n : C A + R, is at most d-to-one map, i.e. card(n-'(x)) 5 d for all a: E R,. Remark 4.3.7. C A 3 x is a periodic point if and only if n(2) is. Since f o n = n o a, it is clear that if x is a periodic point then n(x) is. If fn(y) = y where y = n(a), then n-'(y) > {a,an(a), uzn(a), Since n-'(y) is finite by Theorem 4.3.6, a is a periodic point of a.

.

a * ) .

For the proof of Theorem 4.3.6 we need some notations and lemmas.

$4.3 Symbolic dynamics

As above let 72 = {R1,. Ri, R j E R we define t(Ri, Rj) =

. - ,R,) 1 0

143

be a Markov partition for f)n,. For if f (int(Ri)) r l int(Rj) otherwise

# 8,

and

E = {(Rni)Zm : Rni E R and t(Rni,R,i+l)

= 1,i E Z).

Then Z is a compact metric space under the topology defined as usual. The shift a : E + X defined by U((R,,))~= Rni+,, i E Z, is a homeomorphism. Obviously (E, a) is topologically conjugate to (EA,o). Thus we identify (ZA,a ) with (Z, a ) under the conjugacy. Define Set(x) = {R E R : x E R) for x E as. Lemma 4.3.8. (a) Set(,) = {R E R ' :R = xo for some (xi) E x-'(x)), (b) Set([%,y]) 3 Set(x) n Set(y).

Proof. (a): If x = ?r((xi)), then xo = R E Set(x). That there is such a (xi) for any member of Set(x) was proved in Theorem 4.3.4. (b): If x, y E R, then [x, y] E R since R is a rectangle. Lemma 4.3.9. Let z be near enough x so that [x, z] is defined. (a) If r $! BUR, for 6 > 0 there exists

such that Set(zl) = Set(%)and [x, z'] # BUR. (b) If z # BsR, for S > 0 there exists

such that Set(%')= Set(z) and [zl,x] # BdR. Proof. Remember that 3"R = U{Bu R : R E 72) where

BUR= [vU(x,R), a v S ( ~R)], , BVs(x, R) = Vs(x, R) \ int(Vs(x, R)). Since BVs(x, R) is closed and nowhere dense in V:(x), then V= int(Vs(z, R)) \

we see that if z

U

RESet(z)

RgSet(z)

R

# BUR

144

CHAPTER 4

is an open neighborhood of z in V;(z). Obviously, Set(zt) = Set(%)for all z' E V and V n a u R = 0. Since a u R n V;(z) is contained in a finite union of nowhere dense set of the form aVs(y, R), it is clear that a u R n V;(x) is nowhere dense in v ( x ) . Since each f-,(v(z)) can be covered by finitely many V:(x), we have that

is nowhere dense in v ( z ) . By Baire's theorem, U, f n ( 8 " ' ) ~ V ~ ( z )is nowhere dense in V'(z). By Lemma 4.1.3, [z, ] : V;(z) n N, -+ V;(x) n N, is a homeomorphism. Therefore, [z, z'] @ a U R for some z' E V;(z) \ U, f "(auR).

Lemma 4.3.10. Let R1, Rz, R3 E R . Suppose that

Then x E P R . Proof. Lemma 4.3.1 gives

Since Vu(f (x), R3) contains an open subset contained in every neighborhood of f (x) in V;( f (x)), V contains an open subset contained in every neighborhood of x in Vt(z). As the proof of Lemma 4.3.9 we can find z' E V such that x' @ asR. Since R1 # Rz, we have V c R1 r l R2 C aR1. Since aRl = auR1U asR1, we have x' E aUR1and then Vu(x', R1) c BURl c auR. Therefore, x E auR.

Lemma 4.3.11. (a) Suppose 2 @ a U R . For R; E Set(f(z)) there is a unique R j = T,(Ri) E Set(z) such that A;j = 1 . The map T , : Set(f (2)) + Set(z) is surjective. If y E V;,(z)\dUR andSet(y) = Set(z), then Set(f(y)) = Set(f (z)) andT, =T,. ( b ) Suppose z $! 8%. For R; E Set(f-'(z)) there is a unique R j = T:(R;) E Set(z) such that Aij = 1. The map T: : Set(fw1(z)) + Set(z) is surjective. If y E Vc(z) \ a S R and Set(y) = Set(z), then Set(f-'(y)) = Set(f -l(z)) and Tb = T i . Proof By Lemma 4.3.8, for Ri E Set(f(z)) there is (wi) E .~r-'(f (2)) with wo E R; and so a-'((w;)) E n-l(z). Thus z E R,-,. Since (w;)E XA,we have A,-,,, - A,-,; = 1. By Lemma 4.3.10, w-1 is a unique since 2 $! dUR. Thus T , is well defined. If R; E Set(%),then there is (z;) E n-'(z) with zo E R;. Then R,, E Set(f (2)) and A,, = 1. Thus T,(R,,) = Rz0 and T, is surjective.

$4.3 Symbolic dynamics

145

Take y E V&(z) \ BUR with Set(y) = Set(z). Then y E Vd(z,Ri) for Ri E Set(z). If R j E Set(f(z)) and Ri = T,(Rj), then Lemma 4.3.1 shows that f ( y ) E fVd(z,Ri) C Vd(f(z),Rj) C RjThus Set(f (a)) C Set(f (y)). Symmetrically, Set(f (y)) c Set(f (2)). Thus Set(f (y)) = Set(f (2)). Since the definition of T, depends only on the sets Set(r) and Set(f(z)) (not z itself), we can easily check that T, = T,. (b) is proved in the same way. Let 61 > 0 be as in (B). For z E R, define J,(z) = {V c R : 61 > V6,3z E Vt(z) \ BURwith Set(z) = V), JU(z) = {V C R : 61 > V6, 32 E Vt(x) \ BsR with Set(z) = V), Note that we can choose 6 = 6(z) so small that

.

Let N > 0 and KN be the set of all n-tuples (al, az,. ,an) of integers with 1 5 n 5 N, 1 5 ai 5 N and a1 a2 a,. Define a partial ordering 5 on KN by (al, ,an) 2 (bl,. ,b,) if either n > m or N = m and bi ai for all 1 5 i 5 m (note that this ordering is not the natural condition ai bi). We write (al,... ,a,) > (bl,..- ,b,) if (al,... ,a,) (bl,-.- ,b,), but (al,... ,an) # ( b ~ , . . .,bm). Let #E denote the cardinality of E and P(B)denote the family of subsets of B. F i x N 2 2'Ita. For L E P(P(R)) we define L* E KN to be (al,. ,an) where n = flL and a l , ,a n denote the cardinalities of the n-sets in L.

> >

> > ... > ..

-.-

>

-.

..

Lemma 4.3.12. J,(f (x))* 5 J,(s)* and Ju(f-'(2))'

5 Ju(x)*.

Proof. Define the map R i : J,(z) -+ J,(f(z)) by Ri(V) = Set(f(z)) for V E J,(z). Lemma 4.3.11 ensures that Rz is well defined. We show that R i is surjective. If E E J,(f (z)), by Lemma 4.3.9, for arbitrarily small 6 > 0 we can find w E VdJ(f (z))\BURwith Set(w) = E and w $! f (8"R). Then E = R:(Set( f -l(w))). Let J,(z)* = (al,. a,) where ai = #Vi and J,(z) = {Dl,. ,V,), and let J,(f (x))* = (bl,. ,b,) where bi = #Ei and Js(f (2)) = {El,. ,Em). Then Rz induces a surjection g: [l,n] -, [1,m] by Ri(Vi) = Eg(i). Here we can consider the two cases :

-.

n = m and g is bijective

..

.

Considering the second case, for a certain z we have the surjection T, : &g(i) -+ Vi and thus bg(i) ai. For any i there is j E [l,i]with g(j) 2 i. Otherwise

>

CHAPTER 4

146

g([l, i]) c [I,i - 11 which contradicts that g is one-to-one. Then b; 2 bg(j! 2 a j 2 ai. Here we use the fact that all ai and bi are arranged in descending order. The second statement is proved in the same way. We say that x E 0, is an s-branch (u-branch) point if there exist (xi), (yi) E x-l(x) with x0 = yo but z1 # yl (but x-1 # Y - ~ ) .

Lemma 4.3.13. If x E 0, is an s-branch point, then Js(x)* > Js(f (x))'. If x E 0#is a u-branch point, then Ju(x)* > ~,(f-'(x))*. Proof. Let x be an s-branch point. It is enough to show that one of T, with z E VdiT1(x)\ BURis not bijective. Indeed, if (al, -.- ,an) = (bl, .. ,bn), then g is a bijective and so

We here use that ai 5 bg(i). To have the equality we need ai = bg(i) for all i, i.e. T, is always a bijection. Since x is an s-branch point, we can find Ri E Set(%) and Rjl Rk E Set(f(x)) with R j # Rk and Aij = Aik = 1. By Lemma 4.3.9 we can choose

and since

f vs(xl Ri) C Vs(f (x),Rj) n Vs(f (z),Rk) (by Lemma 4.3.1), we have Rj, Rk E Set(f(z)) and Tz(Rj) = T,(Rk) = Ri. Therefore T, is not bijective. Proof of Theowm 4.3.6. Let c = #'R and e be the length of the longest chain in the partially ordered set (KN, I).Let 0 5 n l < na < . < nk be all the nonnegative integers such that fnd(x)is an s-branch point. By Lemmas 4.3.12 and 4.3.13 Js(f "'(x))* > Js(f na (x))* > -. > Js(f nk (x))*, which shows k e. Let An be the set of all sequences (Rko,... ,Rk,) of elements of 'R such that there is (xi) E x-'(x) with xi E Rki for all 0 5 i n. By the definition of s-branch points, #An+l = #An unless fn(x) is an s-branch point. Thus gAn+l 5 c#An for every n. Since gAo c, we have

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