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

-

.

<

<

<

for all n 2 0. Thus there exist at most ce+' possibilities for ( x i ) r with ( ~ i €) r-l(x). Similarly there exist at most ce+' possibilities for (xi)!, with (xi) E x-l (x). Therefore, #n-l(x) 5

CHAPTER 5 Local Product Structures

In the previous chapter we have seen that every TA-homeomorphism of a compact metric space has a local product structure, and such a homeomorphism is represented as a factor of a symbolic dynamics. For a general TA-map of a compact metric space the shift map of the inverse limit system is a TA-homeomorphism (by Theorems 2.2.29 and 2.3.8) and hence it follows that every TA-map is represented as a factor of a symbolic dynamics through the inverse limit system. The purpose of this chapter is to establish a local product structure theorem for TA-maps under some assumptions. The structure will play an important role for a deep understanding of dynamics of TA-maps, as well as TA-homeomorphisms. In the final section we divide the collection of TA-maps into five classes, each of which exhibits a topological characteristic. $5.1 Stable sets in strong sense Let f : X + X be a continuous surjection of a compact metric space. As before, for x E X and E > 0 we denote as W,S(x) the local stable set { y E X : d ( f i ( x ) ,f i ( ( y ) 5 E , i 2 0). A finite sequence x = 20, X I , ,x , = y of points in X is called an €-chain of s-direction from x to y if xi+l E W,S(xi) for all 0 5 i 5 n - 1. It is clear that if a: = x,-,,zl, ,x , = y is an &-chainof s-direction, then the sequence is also an &'-chain of s-direction for E' > 6. We write x y if for any E > 0 there is an €-chain of s-direction joining x and y. It is easily checked that N is an equivalence relation in X. For x E X we denote as w 8 ( x ) the equivalence class of x and call it to be the stable set in strong sense. By definition the family

- .-

-..

N

is a decomposition of X. If f is c-expansive, from Lemma 2.4.1 it follows that w d ( z )is actually a subset of the stable set W d ( x )= { y E X : d ( f i ( x ) ,f i ( y ) ) --, 0 as i -+ 0 0 ) . By the fact that f (W,b(x)) c W,d(f ( x ) ) for all x E X, clearly f ( x ) f ( y ) if x y. Thus an inclusion f ( w d ( z )C) w d ( f( x ) ) always holds.

-

Theorem 5.1.1. Let f : X + X be a positively expansive map of a compact metric space. Then w d ( x )= { x ) for all z E X .

-

Proof. Let e > 0 be an expansive constant for f . By definition it follows that W,"(x) = { x } for all x E X, which implies that x = y if x y. Thus the conclusion is obtained.

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T h e o r e m 5.1.2. Let f : X -+ X be a continuous surjection of a compact metric space. I f f is a local homeomorphism, then for x E X there is k 2 1 such that the restriction f: wS(x) + w S ( f (x)) is a 12-to-one surjection. Proof. Let (A,8) be Eilenberg's constants for f . For 17 > 0 take 0 < E < min{A,q} a s in Theorem 2.1.1 (4). Suppose z E w S ( f (x)). By definition there is an €-chain yo, yl, ,y, of s-direction from f (3) to z. Since diam(W,d(yo)) < A, by Theorem 2.1.1 the set W,6(yo) can be lifted by f to a set Bo with the property that x E Bo and diam(Bo) < 7. Then Bo C W,"(x). Since yl E W,d(yo) and f (Bo) = W,d(yo), there is a point xl E W,d(x) with f(x1) = yl. Next we lift by f the set W,d(yl) to a set B1 such that xl E B1 and diam(B1) < q, and take a point xz E W;(xl) with f (x2) = y2. Continue this process, a sequence x = XO, X I , xz, -. ,x, is constructed, so that f (x,) = z and xi+l E W;(xi) for 0 5 i 5 n - 1. Therefore we have an q-chain of s-direction from x to x,. Since f-l(z) is finite, we can assume that x, is independent of the choice of 17 > 0. Hence x, E wS(x). Since z is arbitrary, we obtain that f: w S ( x ) + w S ( f (2)) is surjective. Let {xl, ,zk} be the inverse image of f (x) by f : w S ( x ) -, w(f(x)). For a E w S ( f (3)) there is an €-chain f (x) = yo, yl, ,y, = z of s-direction where 0 < E < min{A, q}. .In the same way as above, for each 1 5 j k . construct an pchain x j = x i , xi, ,x i of s-direction with f (xi) = z . Then x i E @(xi) = @(x) for 1 5 j 5 k. By the choice of 8 we have that d(xi, x i ) 2 20 if j # j'. Therefore the cardinal number of the inverse image of z by f : *(x) + w S ( f (x)), say 1, is not less than k. Changing roles of z by f (x), we have f!< k, and thus !f = k.

-..

-

-..

--

<

T h e o r e m 5.1.3. Let f : X -+ X be a TA-covering map of a compact metric space. If X is locally connected, then for every x E X (1) w b ( z ) is the path connected component of x in Wb(x), (2) letting F = {y E X : 3i 0 s.t. fi(y) = f"(x)) one has

>

We recall that if a compact metric space is locally connected, then the space is locally path connected (see Theorem 2.1.4). For the proof of Theorem 5.1.3 we prepare the following proposition. Proposition 5.1.4. Under the assumptions of Theorem 5.1.3, there is eo > 0 such that W,o(x) C ~ ' ( 2 for ) x E X . Furthermore, for 0 < E 5 eo there is 0 < 6 < E such that each point x in X has a path connected set C satisfying the following relation : W,S(x) c c c WEd(x).

.-

For E > 0 a finite sequence x = s o , X I , . ,x, = y of points in X is an &-chainfrom x to y if ~ ( X ~ , X<~E + for~ all ) 0 5 i 5 n - 1. For a continuous

$5.1 Stable sets in strong sense

149

surjection f : X -t X let X denote the space { ( x i ) E X Z : f ( x i ) = %;+I,i E Z) of the inverse limit system. The following lemma is easily checked (c.f. Theorem 2.1.1). Thus we leave the proof to the readers. Lemma 5.1.5. Let f : X + X be a local homeomorphism of a compact metric ,xm space. T h e n f o r ~ > Oa n d n > O t h e r e i s a > O s u c h t h a t i f x O , x l , ~ ~ ~ is an a-chain of points in X , then for (24) E X f with xg = xO there is a sequence (x:), ( x t ), ,( x r ) of points in X f such that

--

( 1 ) xj = x i for O 5 j 5 m, ( 2 ) d(xi,xi+') < E for -n 5 i 5 0 and 0 Ij 5 m - 1. Lemma 5.1.8. Let X be a compact locally connected metric space and f: X -t X a local homeomorphism. If u : [O, 11 + X is a path, then for x = ( x i ) E X f with so = u ( 0 ) there is a path v : [O, 11 -t X f such that v ( 0 ) = x and po ov = u where po : X f + X is the natural projection to the zero-th coordinate.

Proof, Notice that f is a covering map (c.f. Theorem 2.1.1). By the following Theorem 5.1.7 it follows that for each i 5 0 there is a path ui : [O, 11 -+ X such that u;(O) = xi and f o u; = U . For i > 0 let u; = f' o u : [O, 11 + X , and define v : [O, 11 + X f by v ( t ) = ( u i ( t ) ) . Then it is easily checked that v satisfies the desired conditions.

-'

Let X , Y be topological spaces. A continuous surjection p : Y + X is called a covering map if each point x E X has an open neighborhood U evenly covered by p, i.e. there is a collection { ~ x ) x EofAdisjoint open subsets of Y such that

-

(i) p-'(UL= Uxe* U x , (ii) pluA : U A+ U is a homeomorphism for all X E A. In this case Y is called a covering space of X and the neighborhood U is sometimes called a canonical neighborhood of x. Let p : Y + X be a covering map and let Z be a topological space. For a continuous map f : Z -+ X we say that a continuous map 7 : Z + Y is a lift of f by p if a relation f = p o 7 holds. Theorem 5.1.7 (Path lifting property). Let p : Y -t X be a covering map of topological spaces. If u : [O, 11 -, X is a path, then for y E Y satisfying p(y) = u ( 0 ) there is a lift : [O, 11 + Y of u by p such that E(O) = y

Proof. We define a subinterval J c [0,1] as follows: t E J if and only if of u1[0,~j: [0,t ] + X by p such that Eiil[o,tl(0) = y. Let there is a lift a = sup J . Since p is a covering map, we can take an open neighborhood U of u ( a ) which is evenly covered by p. Since u is continuous, there is 6 > 0 such that u ( [ a- 6, a 61) C U . Take t E [a - 6, a 61 with t < a. By definition, t E J and u ~ [ ~has , ~al lift by p such that El[o,tl(0) = y. Since - u(t) E U , and plu : U + U is a there is an open set 0 of Y such that Ello,tl(t)E homeomorphism. Let v : [0,a 61 -t Y be defined by v ( s ) = El[o,tl(s)for

+

+

+

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+

0 5 s 5 t and v(s) = p--' o u(a) for t 5 s 5 a 6. Then v is a continuous map IU satisfying p o v = u on [O, a 61. Thus a 6 E J. This implies J = [O, 11.

+

+

Figure 18 As before, for E > 0 and (xi) E Xfdefine the local unstable set W,U((xi)) = {yo E X : 3(yi) E Xf s.t. d(x;, y;) 5 E , i 5 0). Compare the following lemma with Lemma 2.2.49.

Lemma 5.1.8. Let f : X + X be a continuous surjection of a compact metric space. If f is c-expansive and e > 0 is an expansive constant for f , then for 0 < E < e there is 6 > 0 such that for all x E X and (xi) E Xf with xo = x

where B6(x) = {y E X : d(x, y) 5 6). Proof. By Lemma 2.4.1 there is n > 0 such that fn(W,"(x)) c W,"(fn(x)) for all x E X , and fn(W,U((x;-,))) > W,U((xi)) for d (xi) E Xf.Since f is uniformly continuous, clearly there is 6 > 0 such that if d(x, y) 5 6 then d(fi(x), fYy)) 5 E for 0 5 i 5 n. Then we have the above first equality. The second equality is checked as follows. By definition for yo E W,"((xi)) there is (yi) E Xf such that d(z,, y;) 5 e for i 5 0. Since e is an expansive constant, by c-expansivity it follows that each y; is unique for yo, which implies that for i 5 0 an inverse map of f-' is defined by yo I+ yi on W,"((xi)). Then, by c-expansivity the continuity of the inverse map is easily obtained. Since W,"((X~))is compact, for 6 > 0 small if yo, ro E W,U((xi)) and d(yo, z0) 5 5 then there is (2;) E Xf such that d(yi,zi) 5 E for -n 5 i 5 0. Thus the second equality also holds. Let e > 0 be an expansive constant for a TA-map f and put EO = e/4. Since f has POTP, there is 0 < 60 < €012 such that every 60-pseudo orbit

$5.1 Stable sets in strong sense

151

of f is co/2-traced by some point in Xf. For (xi) E Xf and y E X suppose d(x0, y) < 60. Then a sequence

is a 60-pseudo orbit of f , and hence it is €012-traced by some (zi) E Xj. By expansivity it follows that such a tracing point is unique, which implies that W,U,((xi)) n Wio(y) consists of exactly one point zo. We write [(xi), y]' = (zi) and [(xi),y] = zo. Thus the following maps are defined.

where A(60) = {((xi), y) E X f x X : d(xo, y) in $4.1 of the previous chapter.

< 60).

Compare with the maps

Lemma 5.1.9. Under the above notations one has the following (1) [[(xi),Y]', z] = [(xi), z] and [[(xi),Y]', [(ti), w]]' = [(xi), w]' whenever the two sides of the relations are defined, (2) [ , 1' : A(hO)+ X f is continuous, (3) [ , ] : A(&) + X is continuous, (4) for E > 0 there is 0 < 6 4 60 such that d(xo,y) < 6 implies diam{xo, y, [(xi), YI) < € 9 (5) if ((xi), z), ((yi), z) E A(bo), then for E > 0 there is n > 0 such that

whenever d(xi, yi)

< €0 for -n

5 i < 0.

Proof. By c-expansivity (1) is clear. (2) is checked as follows. Suppose a sequence {(x,, y,)):==, of A(6o) converges to (x, y) E A(60). Let z, = [x,, y,]' and let z, = (zi,,) and x, = (xi,,). Since X f is compact, there is a subsequence {znj) that converges to some z = (xi) E X f . By the definition of [ , ]I, we have d(xi,,,, zi,nj) €0 for i 5 0 and nj, and so d(xi, zi) 5 €0 for i 5 0. Similarly we have d(fi(y), zi) 5 €0 for i 2 0. Thus {z,) converges to [x,y]. Therefore (2) holds. Since [ , ] = po o [ , ]I, we have (3). (4) is obtained by using POTP.(5) is checked as follows. Let uo = [(xi), z] and vo = [(yi), z]. By definition UO,vo E W,d,(z). Also, there are (ui), (vi) E X such that d(xi, ui) 5 €0 and d(yi, vi) 5 €0 for i 5 0. For y = E take a number n > 0 as in Lemma 2.4.1. If d ( ~yi). , 5 €0 for -n i 5 0, then by the above facts it follows that d(fj(u-,), fJ(v-,)) 3eo < e for j 2 0. Therefore v-, E W,d(u-,), and by Lemma 2.4.1 (I),vo E W,d(uo).

<

<

<

Proof of Proposition 5.1.4. Let [ , ] : A(bO)+ X be the map a s in Lemma 5.1.9 and suppose X is locally connected. Let U(x) be the connected component of x in the 60-open neighborhood Ua,(z). Then there is y > 0 such that U(z) > U,(z) for all x E X. For E > 0 take 0 < 6 5 60 and n > 0

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152

as in Lemma 5.1.9 ( 4 ) and ( 5 ) respectively. For eo and n let a > 0 be as in Lemma 5.1.5. Suppose z E W,do( x ) n U ( x ) . Since U ( x ) is connected, there is an a-chain x = u ~ , u ~ , ., u- .r = z in U ( x ) . By Lemma 5.1.5 we (uf ),. ,( u r ) E X f such that d(u{,u{+') < EO for can take points -n 5 i 5 0 and for 0 5 j m - 1. Put y, = [ ( u i ) , ~ ]Then . yj E W:,(z), and hence yj+l E W&,(yj). By Lemma 5.1.9 ( 5 ) we have d ( y j ,yj+l) < e, and by Lemma 5.1.8, yj+l E W,d,(y,) for some E' > 0. Thus the sequence x = yo, 91,. . ,y,,, = s is an &'-chain of s-direction. Since e' can be taken arbitrarily small, we obtain z E w 8 ( x ) ,and therefore W,d2n U ( x ) c w 8 ( x ) . Since U ( x ) > U,(x), it follows that W,do(x)n U,(x) c W d ( x ) . By Lemma 5.1.8 we have W,d,(x)c w ( x ) for some 0 < eo EO. Notice that the number eo does not depend on a point x E X. Next we prove that there is a path connected set C ( x ) such that W,,(x) c C ( x ) c W&,(x). To do this, let U ( x ) be as above. Since U ( x ) is path connected, for z E W:,(x) f l U ( x ) we can take a path u : [O, 1) + U ( x )joining x and z. By Lemma 5.1.6 there is a path v : [0,1] + X f such that v(0) = x and PO o v = u. Define w, : [0,1] + X by w,(t) = [v(t),x].Then w,(O) = x , w , ( l ) = z and w,(t) E WZd,,( x ) for t E [O, 11. Letting

(UP),

<

<

we have that W,",( 3 )C C ( x ) c Wie0( x ) . Let 0 < E 5 eo. Then by Lemma 2.4.1 there is n > 0 such that fn(W;",,(x)) C W , d ( f n ( x ) )for all x E X . Since f n is a local homeomorphism, it is easily checked that there is 6 > 0 such that fn(W,b,(x)) 3 W ; ( f n ( x ) ) for all x E X. For x E X take y E f - n ( x ) and put C = fn(C(y)). Then it is clear that C c W,d(x). Since C ( y ) > W:,(y), it follows that W ; ( x ) c C . L e m m a 5.1.10. Let f : X -t X be a c-expansive continuous surjection of a compact metric space and let e > 0 be an expansive constant for f . For x E X f and x E X the intersection WG2(x)n W 8 ( x )is at most countable.

Proof. If z E W,U12(x) n W 4 ( x ) ,by Lemma 2.4.3 there is n > 0 such that f n ( z ) E W , d / 2 ( f n ( x ) )Put . Fn = fn(WG2(x)).To show that for each n 2 0, Fn n W,"12(fn ( x ) ) is finite, it suffices to prove that Fn is contained in a finite union of some local unstable sets W,"12(yi). Since f is uniformly continuous, there is e > 0 such that if d(a, b) 8 then d( f ' ( a ) ,f '(0) j e / 2 for 0 5 i n. Then we have f "(W:(y)) C W G 2 ( u n ( y ) ) for all y E X f , where u : X f + X f is the shift map. Let 6 > 0 be as in Lemma 5.1.8. By compactness there is a finite 6-dense subset { y l , y2,. ,y k } of W , j 2 ( r ) . Let x = (ri)and take y j = (y:) E X f such that d = yj and

<

<

--

$5.1 Stable sets in strong sense

d ( y f ,x i ) 5 e/2 for i 5 0. Then we have

and hence

W22( x )C u:=, W,"(yj). Therefore, Fn C

W,Y~~(~"(Y')).

Proof of Theorem 5.1.3. Let eo > 0 be as in Proposition 5.1.4 and let y E W d ( x ) .Then by Lemma 2.4.3, f n ( y ) E Wed0(fn(x))for some n > 0. By Propo, so w d ( f n ( y ) )= w d ( f n ( x ) ) . sition 5.1.4 we have f n ( y ) E w d ( f n ( x ) ) and Since f n : * ( y ) + w d ( f n ( y ) )is surjective by Theorem 5.1.2, there is z E w d ( y )such that f "(8) = f " ( x ) , and hence z E F. Therefore ( 2 ) holds. To prove ( I ) , we show first that each of w d ( x )is path connected. By Proposition 5.1.4 there is 6 > 0 such that for each x E X we can take a path connected set C ( x ) satisfying W i ( x ) c C ( x ) c W,d,(x). Let z E w d ( x ) . By definition there is a 6-chain x = xo, X I , . . . ,xn = z of s-direction. Then xi+l E C ( x i ) c W:o(xi) c w d ( x i )for all 0 5 i 5 n - 1. Therefore there is a path u in X joining x and z such that u([O,11) c W d ( x ) . Let u : [0,1]+ X be a path such that u ( 0 ) = x and u([0,1])c W d ( x ) .Let [ , ] : A(bO)+ X be as in Lemma 5.1.9. Clearly there is 0 < t 5 1 such that u([O,t])is contained in the b0-open neighborhood of x. Take x = ( x i ) E Xf with xo = x and define v : [O,t] + W,",(x) by v ( s ) = [x,u(s)].Then v is continuous by Lemma 5.1.9. By the definition of [ , ] clearly v ( s ) E W ~ o ( u ( s ) ) . Since u ( s ) E W d ( x ) ,we have v ( s ) E W d ( x ) ,and by Lemma 5.1.10 it follows that v ( s ) = x. Hence u ( s ) E W,do(x). Since u is continuous, by applying Lemma 5.1.8 we can take 0 < to 5 t such that u ( s ) E W:o(x) C w d ( z )for 0 5 s 5 to. Hence, letting I = { s E [ O , l ] : U ( S ) E w d ( x ) } we , have [O,to]C I. In the same way, it is easily checked that I is open and closed in [ O , l ] . Thus u([O,I ] ) c w d ( x ) . Let f : X + X be a TA-covering map of a compact locally connected metric space. Let eo > 0 be as in Proposition 5.1.4. Then W:o(y) c w s ( x ) for y E w d ( x ) .By this fact together with the following general properties of local stable sets we can introduce a topology in w d ( x )such that for y E w d ( x )the collection {W,d(y): 0 < e 5 e0) is a fundamental neighborhood system on the base of neighborhoods of y.

( 1 ) y E W,d(Y), ( 2 ) W,d(y)n W $ ( y ) = W $ , ( y ) for 0 < e' < eo where (3) if z E W:/,(v), then w:/,(z) c W,"(Y).

E"

= min{e,el),

The topology is called the intrinsic topology of w d ( x ) .Notice that the intrinsic topology is not weaker than the restriction of the topology of X.

Lemma 5.1.11. Let eo > 0 be as in Proposition 5.1.4. Then for 0 < E 5 eo and x E X the restriction to W,d(x)of the intrinsic topology is consistent with

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154

the restriction to W,d(x) of the topology of X . Moreover, there is 6 > 0 such that W , d ( x ) n U 6 ( x is ) an open set of ~ " ( xunder ) the intrinsic topology, where U 6 ( x )denotes the 6-open neighborhood in X . Proof. Suppose a sequence { y i } converges to a point y on W,d(x) under the topology of X. Let 7 > 0. By applying Lemma 2.4.1 we can take 6 > 0 such that W,"(y) > W,d(x)n U6(y)(see Lemma 5.1.8). Hence yi E W,"(y) for large i. Conversely, if {yi} converges to y under the intrinsic topology, then the convergency under the topology of X is easily obtained. By Lemma 5.1.8 there is 6 > 0 such that W,d(x)n U 6 ( x )= W+,(x) for d l x E X . Then it follows that W,"(y) c W,d(x)n U 6 ( x )for sufficiently small 7 > 0 if y E W,d(x)n Us(%). Theorem 5.1.12. Let f : X -, X be a TA-covering map of a compact locally connected metric space. Then for x E X the intrinsic topology of ~ " ( x ) satisfies the following properties: ( 1 ) locally path connected, (2) path connected, ( 3 ) first countable, ( 4 ) second countable. Proof. ( 1 ) follows from Proposition 5.1.4 and Lemma 5.1.11. (2) is easily checked from Theorem 5.1.3. ( 3 ) follows from the definition. To prove ( 4 ) ,let eo > 0 be as in Proposition 5.1.4 and let x E X . Since f has POTP, there is 6 > 0 such that every 6-pseudo orbit of f is eo/2-traced by some point in X f. Let {xl, x 2 , ,x k } .be a 6-dense subset of X and choose x j = ( x i ) E X (1 j k ) such that x i = ? j . By the choice of 6 for y E ~ " ( xthere ) is x j such that W,o12(y)nW,Uo12(x') consists of one point, say I . Then y E W:o12( I ) . Therefore, by Lemma 5.1.10 it follows that ~ " ( xis )expressed as the union of at most countable local stable sets WebOl2(z).Therefore ( 4 ) is obtained from Lemma 5.1.11.

< <

-

We say that a TA-map f : X -, X of a compact metric space is s-injective if the restriction f : ~ " ( x-, ) ~ " ( x f) ) is injective for d l x E X. If f is a (x)) self-covering map, then by Theorem 5.1.2 we have that f : w a ( x )-+ w8(f is bijective whenever f is s-injective. For this case, if X is locally connected, it follows that f: ~ " ( x-+) w"( f ( 3 ) )is a homeomorphism under the intrinsic topologies.

Remark 5.1.13. If a compact metric space X is locally connected, then clearly X splits into a finite union X = X I U X z U . U Xt of connected components. In this case we have that f n ( X i ) = Xi ( 1 i 5 t ) for some n > 0 if f : X -, X is a continuous surjection. In the next Chapter 6 we will introduce fundamental groups and universal covering spaces. It can be proved that a TA-covering map f: X -, X of a

..

<

$5.2 Local product structures for TA-coveiing maps

155

compact locally connected metric space is always s-injective in the case where X has the universal covering space (see $6.4 of Chapter 6). However, it remains a problem of whether it is true for general cases.

$5.2 Local product structures for TA-covering maps In this section we shall establish a local product structure theorem for sinjective TA-covering maps of compact locally connected metric spaces. Let f : X -+ X be a continuous surjection of a compact metric space. As before, we denote as u : Xf + X f the shift map of the inverse limit system and as po : X f + X the natural projection to the zero-th coordinate. Let A(€) = {((xi), y) : d(x0, y) 5 E) for E > 0. We say that f has a local product structure if the following conditions (A), (B) and (C) are satisfied : (A) There is So > 0 and a continuous map [ , ] : A(SO)-,X such that (1) [(xi), XO] = SO for all (xi) E X f , (2) f ([x, y]) = [u(x), f (y)] whenever the two sides of the relation are defined. (B) For each x = (xi) E X f there is a subset Nx of X with xo E Nx, and a continuous section S, : Nx + X f for the projection po, that is, po o Sx is the identity map on Nx, such that

(3) Sx(x0) = x, (4) [SX(X),Y] E Nx forX,Y E Nx, (5) for x, y,z E Nx

(C) There is e

> 0 such that for x = (xi) E Xf, letting

the following property holds: (6) ax : Ds(xo) x DU(x)-, Nx is a homeomorphism.

Theorem 5.2.1. Let X be a compact connected locally connected metric space. If a TA-covering map f : X + X is s-injective, then f has a local product structure satisfying the following conditions (1) {[(~i),Yl)= W,U((zi)) ~ W , " ( Yfor ) x xi),^) E A ( ~ o ) , (2) each of Nx is a connected open set of X ,

CHAPTER 5

( 3 ) there is p > 0 such that Nx > B,(xo) for x = ( x i ) E X f where B P ( x o ) = {Y E X :~ ( x o , Y5) P ) , ( 4 ) for x = ( x i ) E X l

Furthermore, i f f is expanding then D d ( x o ) = { x o ) and D u ( x ) = Nx for x = (xi)E X l , and otherwise ( 5 ) D8(xo) 2 { x O )and D u ( x ) 2 { x o ) for x = ( x i ) E X f .

L

--------L -----------', Figure 19

Before starting the proof we prepare the following theorem. Theorem 6.2.2 (Uniqueness of lifting). Let p : Y -t X be a covering map and let Z be a connected topological space. For a continuous map f : Z -t X --I suppose f , f : Z -+ Y a m lifts o f f by p. ~f T ( x 0 ) = T1(xO)for some point xo 4 in 2, then 7 = f

.

Proof. Let 0 = { x E Z : 7 ( x ) = f l ( x ) ) and 0' = { x E Z : f ( x ) # ~ ' ( 2 ) ) . It is sufficient to show that 0 and 0' are both open in Z. Let x E Z. Since p is a covering map, by definition there is an open neighborhood-U of f ( x ) such that p-'(U) splits into a disjoint union p-'(U) = UxGnU x of open sets of Y. Then T ( x ) 6 and T 1 ( z ) E for some X,X1 E A. Clearly --1 ( U x ) n f " ' ( ~ x t ) is an open neighborhood of x in Z. It is easily V =f checked that X = A' and V c 0 if z E 0, and X # A' and V c 0' if x $! 0. Therefore 0 and 0' are open in Z.

vx

vxt

Proof of Theorem 5.2.1. Let f: X -t X be as in Theorem 5.2.1 and choose eo > 0 as the number satisfying the property of Proposition 5.1.4. Notice that eo is an expansive constant for f . As in the previous section, we take 60 > 0 small for EO = eo/4 and define a map [ , 1' : A(bO) + X l by assigning

55.2 Local product structures for TA-covering maps

157

((xi), y) to a unique point (zi) with the property that d(xi, z;) 5 €0 for i 5 0 and d(f"y), zi) 5 EO for i 2 0. By Lemma 5.1.9 (2) it follows that the map is continuous. Letting [ , ] = po o [ , ]I, we have a continuous map [ , ] : A(SO)+ X. Since

(1) of Theorem 5.2.1 holds. From definition the condition (A) is easily checked. To show (B) and (C), we prepare three claims. First, by Lemma 5.1.9 (4) choose 0 < €1 < 6014 such that if d(xo, y) 5 2 ~ 1then diam{[(xi), y],xo, y) < 6014. Next take and fix 0 < el < €1 as in Proposition 5.1.4. Then, for each x E X there is a connected set C(x) satisfying W,.,(x) c C(x) c W,",(x) c ~ " ( x ) . Let x = (xi) E Xj and C(xo) be as above. Since f is s-injective, for each i E Z the restriction w6(xi) + w6(x0) is bijective. Since C(xo) c w6(xo), we define a map yi : C(zo) + w6(xi) by the formula f -i(yi(y)) = y. Then (Y~(Y)) E Xj for Y E C(XO)-

Claim 1. For each i E Zthe map yi : C(xO)+ w6(xi) is continuous (under the topology of X). Indeed, since f-' : w d ( x i ) + wd(xo) is a homeomorphism under the intrinsic topology, it follows that yi : C(xo) + w6(xi) is continuous under the intrinsic topology. Since C(xo) c W,&xO), by Lemma 5.1.11 the conclusion is obtained

.

For z E W,U,(x), by definition there is (z;) E Xf such that zo = z and d(xi, zi) 5 el < EO for i 5 0. Since f is c-expansive, such a point (zi) is unique for z. So we write (zi(z)) = (zi), and for i a map z; : W,U,(x) + X is obtained. By c-expansivity it is easily checked that each the map zi is continuous. Let (y, z) E C(xo) x W,U,(x). Then d(y, z) < 6. Since (yi(y)) E Xj and yo(y) = y, a point [(yi(y)), z]' in Xj is defined. We write (wi(y,z)) = [(yi(y)),z]' and consider for each i a map wi : C(xo) x W,U,(x) -t X. From Claim 1 together with the fact that [ , ] is continuous, it follows that each w; is a continuous map. By definition wo (y ,z) = [(yi(y)), z], and hence by (5.1) we have for z E W,U,(x)

Claim 2. For (y, z) E C(x0) x W,U,(X)one has wi(y, t.) E wd(zi(z)) for all i E Z. Fix z E W,U,(x). Since xo E C(xo), by definition wi(zo,z) = zi(z). Clearly f -i o wi(y, z) = wo(y, z). Thus each wi(y, z) is a lift of wo(y, z) by f -"such that ~ ~ ( 3t )0 = , z~(z).

158

CHAPTER 5

On the other hand, since D(z) c w8(z), by s-injectivity for each point wo(y, z) E D(z) there is a unique point w:(y, z) in w8(zi(z)) satisfying the formula f - ' ( ~ : ( ~ , z ) )= wo(y,z). Thus a map w: : C(XO) w8(t.i(z)) was defined. Notice that the point z is fixed. Since wo is continuous, as in Claim 1 the continuity of w: is easily checked. Since f-i o w:(y, z) e= wo(y, z), the map w: is also a lift of wo by f-'. By definition w:(xo, z) = zi(z). Since C(x0) is connected, by the uniqueness theorem for liftings we have wi(y, z) = w:(y, z) for all y E C(xo). Thus wi(y,z) E w8(zi(z)).

Claim 3. Fix z E W,Ul(x). Then the map wo : C(xo) -t D(z) is injective. Indeed, suppose wo = wo(y, z) = w0(yt, z) for some y, y' E C(xo) and let (pi)= [(yi(y)), z]' and ( w : ) = [(yi(yl)), z]'. By Claim 2 we have wi, wi E W8(zi(z)). Since wo = wb, by s-injectivity of f it follows that wi = w: for all i, and so (wi) = (w:). By the choice of €1 we have d(wo,xo) < 60. Hence by Lemma 5.1.9

and similarly [(wi), xo]' = (yi(y1)). Therefore y = y'. Let (y, z) E W,b,(xo) x W,Ul(x). Since W:(xo) c D(zo), it is clear that (Y,z) E D(xo) x W,", (x). Now define ax(y, z) = [(yi(y)), z](= wo(y, z)) and let

Since el < E I , by the choice of we have diam(R,) < 60. The following properties are easily checked by definition : for all y E W&(xo) and all z E w,Ul( 4 ax(x0,z) = z, ~ X ( Y , X= O )Y. Since a, = wo, clearly

is a continuous map. If a,(y, z) = aX(yt,a') = w, by c-expansivity it follows that z = a', and hence y = y' by Claim 3. Therefore a, is injective, which implies that a, is a homeomorphism. For a E Rx, take (y, z) E W,d,(xo) x W ~ ( X such ) that a = ax(y, z), and ) ) , (wi(y, z))). From the above results it follows define S,(a) = [ ( ~ ~ ( y z]'(= that S, : R, 4 X f is a continuous map. Since

S, is a section on Rx for po. Hence, by Lemma 5.1.9 we have that X f satisfies the condition (B).

S, : R,

4

55.2 Local product structures for TA-covering maps

Next, we prove that there is p'

> 0 such that

R,

> Bp,(xO). Since f

159

has

POTP,there is p' > 0 such that every pl-pseudo orbit is el-traced by some point in Xf. Suppose w E Bpl(x0) and let (zi) = [(xi),w]' E Xf. Then {ZO) = W,U,((xi)) n W l: (w). Since W:(w) c w8(z0), by s-injectivity we can take a unique (wi) E Xf such that wo = w and wi E w8(zi) for i E Z. Since d(w, xo) < p', it follows that W,Ul((wi)) n Wl: (xo) consists of one ponit, say yo. Since yo E W,d,(xO)C w8(x0), we choose (yi) E Xfsuch that yi E w8(xi) for all i. In the same way as the proof of Claim 2, we obtain d(w;, yi) 5 el for all i 5 0. Hence w E W,Ul((yi)), and so w E W,Ul((yi)) n W,dl(zO). This implies w E Rx. Let M, denote the interior of R, in X and suppose a, b E M,. Then there are y, y' E W,b,(xo), z,zl E W,U,(x) such that a = a,(y,z) and b = aX(y1,2'). Letting c = aX(y1,z)(= wo(yi(yl), I)), we can show c E M,. Indeed, for c = S,(c) = (w;(yl,z)) E Xf take el > E > 0 sufficiently small and in the same manner as above construct R,(E) for E. Since diam(R,) < 60, we have diam(R, U R,(E)) < 60 for E small. Since a E M,, by applying Lemma 5.1.9 we have [x,W,"(c)] C W,U,(x) whenever E is small. This implies W,U(c) c aX(y1,W,U,(x)). On the other hand, since b E M,, there is 7 > 0 such that Wyd(b) c M,. Then [S,(w), c] E R, for w E W,d(b). Since [S,(c), b] = [c, b] = b, if E is taken small, by the continuity of [ , ] we can define a map W," (c) + W," (b) by v I+ [S,(v), b] In the same way as the proof of Claim 2, we have [S,[S,(v), b], c] = v, which implies W,"(c) c a,(W,"l (xo), z). From this result together with the above fact it follows that R,(E) C R,. Since R,(E) contains a neighborhood of c in X , we obtain c E M,. By the above result the inverse image of M, by a, can be written as V8(xo) x Vu(x). We denote as N, the connected component of xo in M,. Then it follows that a, : D8(xo) x DU(x) + N, is a homeomorphism for some D8(xo) c V8(x0) and Du(x) C VU(x). Since X is locally connected, (2) of Theorem 5.2.1 is obtained. Clearly (3) of Theorem 5.2.1 follows from definition. Restricting the section S, to N,, we have (B). (C) is clear from the construction. To prove (4) of Theorem 5.2.1, first we show that D8(xo) is the connected component of xo belonging to the interior of W,d,(xo) c w8(zo) under the intrinsic topology. Since D8(xo) = W:d,(xo) n N, and N, is open in X, for y E D8(xo) we have Wydly) c D8(xo) if 7 is sufficiently small, which means that D8(xo)is open in W8(xo). The connectivity follows from the fact that N, is connected and a, : D8(xo) x Du(x) + N, is a homeomorphism. Hence D8(xo) is contained in the connected component, say Dl, of xo belonging to the interior of W,b,(xo)under the intrinsic topology. Conversely, let c E D'. Then D' > W,"(c) if E > 0 is small. As above define R,(E) where c = S,(c). Then it is easily checked that R,(E) C R,. Thus c is a point in M, which is the interior of R,, and so c E W&(xo). This implies D' C D8(xo). Therefore D' = D8(x0).

.

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160

Since f (Wll (30)) c Wi, (xl) and f :wd(x0)-' ws(xl) is a homeomorphism under the intrinsic topology, by the above result f(Dd(xo))is connected and open in Wll (xl), and hence f (Dd(xo))c Dd(xl). From this fact it follows that N = a,(,)( f (Dd(xo))x DU(o(x)))is a connected open subset of Nu(,). Take a lift R of N by f such that xo E W. Then we have Da(xo) = W : (20) f l and where f (o'(o(x))) = Du(o(x)) and f

(TI ( ~ ( x ) )= ) W,",( ~ ( x ) ) Therefore .

is connected and open in X. Thus from which we have W c Nx because D (u(x)) c Du(x), and so we obtain f (Du(x)) > Du(o(x)). To prove the last part of Theorem 5.2.1, let Xd = { x E X : Dd(x) = {x)). Then X d is open in X. Indeed, if x E X d then

1 1

and for y E Nx

Since Dd(y)nNxis open in Dd(y)and Do(y)is connected, we have Dd(y) = {y) and hence y E X d , i.e. N, c Xd. Since B,(x) C Nx for x E X (by (3)), X" is closed and therefore X" = X unless Xd = 0. If Xd= X , then we have Nx = DU(x)for x E Xf.Then, by using c-expansivity, we obtain that f : M + M is positively expansive, that is, expanding (because f is an open map). Otherwise, Xd= 0. It is easily proved that

is open and closed in X. Supposing X U #

0, we have X u = X and then

..

for some X I , . ,x, E X. It follows from Lemma 2.4.1 that f n(X) # X for some n > 0, thus contradicting. 17 Let f: X + X be an s-injective TA-covering map of a compact connected locally connected metric space. For x = (xi) E Xflet ax : Dd(xo)x DU(x)-+ Nx be as in Theorem 5.2.1. Then we have the following lemmas.

55.2 Local product structures for TA-covering maps

161

Lemma 6.2.3. Let S, : N, + Xf be a continuous section for po such that Sx(xo) = x . For k 2 0 let p-k : Xf + X be the natural projection to the -12-th coordinate and define

Then x-k E N,(-k) phism.

and the restriction f k : Nx(-k) + N, is a homeomor-

Proof. This is easily checked from the definition of Sx.

Lemma 6.2.4. Let p x = ( x i ) E Xf

>0

be the number in Theorem 5.2.1 (3). Then for all

Proof. Let w E W;(xo). Since w E BP(xo)C N,, there is ( y , z ) E D8(xo)x D u ( x ) such that a&, z ) = w. Then w E Wt,(z). Since p < E O , we have z E W&,(xo)n W,",((xi)),and z = xo by c-expansivity. Thus w = a,(y, x o ) = y E Dd(xo).In the same way, we have W,U(x)C Du(x).

Lemma 6.2.5. The set D 8 ( x )is a connected open subset of ~ ' ( xunder ) the intrinsic topology. Proof. Since D8(x) = W:(x) n N, and N, is open in X , for y E D a ( x )we have W,d(y)C D8(x)i f 7 is sufficientlysmall, which means that D a ( x )is open in ~ ' ( x(see ) Lemma 5.1.11). The connectivity follows from the fact that N, is connected and a , : D 8 ( x )x D U ( x )+ Nx is a homeomorphism.

Theorem 6.2.6 (Uniformly local splitting theorem). Under the notations and assumptions in Theorem 5.2.1, for any 7 > 0 there is m > 0 such that for each x = ( x i ) E Xf if one let K 8 ( x m ) = fm(D'(xo)) and f m ( K U ( x )= ) DU(om(x)),then ( s l ) K a ( x m )C B7(xm), (92) K a ( x m )is open in D8(xm), (93) f m : D d ( z o )+ K a ( x m )is a homeomorphism, ( ~ 1K)U ( x )C B,(xo), (u2) K U ( x )is open in D u ( x ) , (u3) f m : K u ( x ) + D u ( o m ( x ) )is a homeomorphism, (su) let N 1 = a x ( D 8 ( x x) K U ( x ) )and N 2 = ( Y ~ ~ ( ~ ~ ( Kx' D (X U (, o) m ( x ) ) ) , then N 1 and N 2 are open subsets of X, and a,: D8(xo)x K U ( x )-,N ' , a,=(,) : K8(x,) x Du(am(x))+ N~

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162

are both homeomorphisms and the diagnzm

N1

I

-

am"(x)

commutes.

N2

f"

Furthermore, there is 0 < p' for all x = (xi) E Xf.


such that N1

> Bpt(xO)and N2 > Bp~(xm)

Proof. By Lemma 2.4.1 we choose m > 0 such that f "(W;(x)) C W,d(f m(z)) for all x E X and fm(W;(x)) > W:(um(x)) for all x E Xf.Then (sl) and (ul) are clear. Since f : w a ( z o ) + w 8 ( x m ) is a homeomorphism under the intrinsic topology, by Lemma 5.1.11 we have (s3). (s2) follows from Lemma 5.2.5. (u2) and (u3) are obtained from the fact that f : NUm(,)(-m) + N,,(,) is a homeomorphism. By Theorem 2.5.1 we have f '(N1) c NUi(,) for 0 5 i 5 m. From this fact together with the conditions (A) (2) and (C) we obtain (su). The last part is easily checked by uniform continuity o f f .

Figure 20

$5.3 Expanding factors of TA-maps

We recall that for a TA-map f : X + X of a compact metric space the nonwandering set R(f) is decomposed into a finite union of basic sets, each of which splits into a finite union of elementary sets (see Theorem 3.4.4). In the case when f is a TA-homeomorphism, the stable and unstable sets of f are dense in the elementary sets respectively (see Theorem 3.2.6). In this section we discuss density of the stable sets in strong sense in the elementary sets.

$5.3 Expanding factors of TA-maps

163

Remark 5.3.1. Let f : X -,X be a TA-map of a compact metric space and a : X f -+ X f the shift map of the inverse limit system. Let po : Xf + X be the natural projection to the zero-th coordinate. Then for x = ( x i ) E X f one has po ( W 8 ( x ) = ) W 8 ( x 0 )and the diagram

I

-

W8(x0)

f

1.0

commutes.

W8(x1)

Indeed, let (yi) E W 8 ( ( x i ) ) .Then d ( ~ ~ ( ( x ~ ) ) , a " + ( ( 0~ as ~ ) n) )+ oo where d denotes a metric for X f . This implies d(f " ( x o ) ,f " ( y o ) )= d(zn,y,) + 0 as n -+ oo. Therefore, yo E W 8 ( x o ) .Conversely, let yo E W 8 ( x o )and take (yi) E X f . Since d(fn(yo),fn(xo)) + 0 as n + m, we have (yi) E W 8 ( ( x i ) ) . Thus, p o ( W 8 ( ( ~ i )=) )W 8 ( x o ) .Since f 0p0 = po o a on X f , we have the above commutative diagram. For x E X denote as W U ( x f; ) and W U ( xa; ) the unstable sets of f and a respectively. Then in the similar way as above we have that po(Wu(x;a ) ) = W u ( x ;f ) and the diagram

commutes.

Remark 5.3.2. Let f : X 4 X be a TA-map of a compact metric space and let C be an elementary set of f . Then ( 1 ) if x E C then W 8 ( x )n C is dense in C , (2) if ( x i ) E X f and xi E C for all i E Z then W U ( ( x i )n) C is dense in C . Indeed, since a is a TA-homeomorphism, by Theorem 3.1.11 the nonwandering set R ( a ) splits into a finite union R ( a ) = B1 U U Be of basic sets (see Theorems 3.1.2 and 3.1.8), and each of Bi is decomposed into a finite U U Bi,,i of elementary sets. Since C is an elementary set union Bi = : C + C is topologically of f , there is a > 0 such that f a ( C ) = C and mixing. By Theorem 3.5.3 it follows that the space of the inverse limit system f a : C -,C coincides with one of Bi,j ( 1 5 i 5 L, 1 5 j 5 ai). Therefore, from Theorem 3.2.6 together with Remark 5.3.1 we have ( 1 ) and (2). Let f : X + X be a TA-map of a compact metric space. We say that g: Y -+ Y is an expanding factor of f if ( 1 ) g:Y + Y is an expanding map of a compact metric space, (2) Y is an uncountable set, ( 3 ) g is a factor of f , i.e. there is a continuous surjection h: X 4 Y such that g o h = h o f .

.

fb

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164

It is clear that every expanding map of a compact metric space with uncountable elements is itself an expanding factor. Notice that every expanding map of a compact metric space is a TA-covering map (see Theorem 2.3.10). Let f : X + X be an expanding map of a compact metric space. If e > 0 is an expansive constant for f , then (1) there is 6 > 0 such that U6(xo) c W,U((x;)) for all (xi) E X f where U a ( ~ o= ) {y E X : d ( ~ o , < ~ 6). ) Indeed, since f has POTP, there is 0 < 6 < e such that every 6-pseudo orbit of f is e-traced. Fix (xi) E X f and let y E Ua(xo). Then we have a &pseudo orbit

Remark 5.3.3.

... ,X-~,X-~,Y,~(Y),~~(Y),...,

>

which is e-traced by (2;) E X j. Clearly d( f '(z0), f '(y)) < e for i 0. Since f is positively expansive, zo = y and therefore y E WZ((x;)). Thus (1) holds. Since f is a TA-map, by Theorem 3.4.4 the nonwansering set n ( f ) is decompsed into basic sets, R(f) = R1 U n nt, each of which splits into elemetary sets, ni = C;,l U U Ci,ai. Here f (Ci,,) = Ci,,+l (1 j ai) and Ci,ai+l = Ci,1. The following are easily checked: (2) if (x;) E X j and xo E Cij, then Ci,, C WU((xi)), (3) if U is a non-empty open set of C;,,, then there is n 2 0 such that f ""i (U) = C;,. Indeed, let e > 0 be an expansive constant for f and let 6 > 0 be as in (1). Let (xi) E Xf and suppose xo E Cij. Then, Ua(xo) n C;,, c C;,,. Since f a i : Ci,, -t C i j is topologically mixing, there is n > 0 such that f nai(U6(xo)) is 6-dense in C;,,, that is, for w E C;,, we can take z E fnai(U6(x0))with w E U6(z). Hence there are finite points (zf),... ,(E!) E X f such that zb E Ua(xo) & and d(z:, xi) + 0 as i -t -m for (1 5 j 5 k), and Uj=l u6(z!,) = Ci,,. This implies that Wu((x;)) n CiVj = Cij. Therefore (2) holds. To show (3), let U be a non-empty open set of Cilj and take (xi) E Per(o) with xo E U. Then, W,U((x;)) f l C i j C U for some e with 0 < E < e. If p is a period of (xi), then by Lemma 2.4.3 we have

...

...

< <

which implies (3) because Cif is compact. T h e o r e m 5.3.4. Let f : X -+ X be a TA-covering map of a compact connected locally connected metric space. Let C be an elementary set for f and let a > 0 be a natural number such that fa(C) = C. Then among the following statements one has the implications (2) =S (1) and (1) + (3). (1) w8(xo) n C is dense in C for some xo E C, (2) w8(x) n C is dense in C for all x E C, (3) f a : C + C has no expanding factors.

$5.3 Expanding factors of TA-maps

165

Proof. (2) + ( 1 )is clear. To show ( 1 ) + (3),suppose g: Y + Y is an expanding factor of f a : C + C and let h: C + Y denote a continuous surjection such that g o h = h o f a . By Proposition 5.1.4 we can take eo > 0 satisfying the property that We,(%)c w n ( z )for all x E X . If x E C, clearly h(C n Wi",(x))C Y . Since h is uniformly continuous and g is expanding, the diameter of h(C n W,b,(x))is less than an expansive constant of g for all x E C whenever eo is sufficiently small. From this and the fact that f "(W:",x) C W i o ( f a ( x ) it ) , follows that h(C n W,",(z)) consists of one point, ) C ) is also a set of one point. Since w n ( x o )n C which implies that h ( w a ( x n is dense, Y = h(C) must be a set of one point, thus contradicting.

Remark 5.3.5. It seems likely that in Theorem 5.3.4 the implication (3) + (2) is true. However this is left as a problem. Let X be a compact connected l o c d y connected metric space and let f : X + X be a TA-covering map. For ( x i )E Xfwrite

lim(w8(xo); t ( x i ) )= { ( p i ) E Xf : y; E * ( x i ) for all i E Z). Then we have the following theorem.

Theorem 5.3.6. Under the above assumption, let C be an elementary set for f and let C denote the elementary set for the shift map o : Xf + Xf such that po(C) = C . Suppose f is s-injective. Then the following properties hold: ( 1 ) i f ( x i ) E C then l&(wn(zo); ( x i ) )is the path connected component of ( x i ) in the stable set W n ( ( x i ) ) , ( 2 ) w n ( x ) n cis dense in C for allx E C if and only if1@(wn(xo);(xi))rl C is dense in C for all ( x i ) E C . Proof. (1): First we show that ~ @ ( w ~ ( x(~z )i );)is path connected. Let ( y i ) E l & ( ~ ~ ( x(xi)). ~ ) ; Then, yo E ~ " ( x o )By . Theorem 5.1.3 (1) we can take a path u in w n ( x o )from xo to yo. Since f is s-injective, the restriction f : ~ ~ ( x+- w~n ( )z o )is a homeomorphim (under the intrinsic topologies). ~ ) 3 - 1 to y-1 such that f o u-1 = u. Hence there is a path u-1 in ~ " ( x -from Similarly, for i < 0 we can find a path ui in w n ( x i )from xi to yi satisfying f oui = ui+l. Put ui = fiou for i 2 0. Then ( u i )is a path in l @ ( ~ ' ( x ~()x;i ) ) joining ( x i ) and (yi). Therefore the conclusion is obtained. If ii is a path in W n ( ( x i ) starting ) at ( x i ) ,then po o ii is a path in W "( x o ) from xo to yo, and so it is a path in w n ( x o ) .Let u = po oTi and construct ( u i ) as above. Then we have ( u i )= ii. Thus (1) holds. ( 2 ) is easily checked.

CHAPTER 5

166

55.4 Subclasses of the class of TA-maps

In the previous chapters we have seen that the collection of all TA-maps of a compact metric space contains special TA-maps, expanding maps and TAhomeomorphisms. Moreover the collection contains important maps which are called strongly special TA-maps. Let X be a compact metric space and f : X -t X a continuous surjection. We say that f is a strongly special TA-map if f satisfies the conditions: (i) f is not injective, (ii) f is a special TA-map, (iii) for all x E X the path connected component of x in the stable set W6(x) is dense in X.

Remark 5.4.1. The following can be easily checked from definition. (1) Every toral endomorphism of type (11) is a special TA-map, but not a strongly special TA-map. (2) The product of a toral endomorphism of type (11) and a toral endomorphism of type (111) is a special TA-map, but not a strongly special TA-map. (3) A necessary and sufficient condition for a toral endomorphism to be a strongly special TA-map is that it is a toral endomorphism of type (111) without the component of type (11). (4) Every expanding map is not strongly special. (5) The product of a TA-homeomorphism and an expanding map is a special TA-map, but not a strongly special TA-map. Let X be a compact metric space and denote as C(X) the set of all continuous surjections of X. Define here the following subclass of C(X) 'TA = { all TA-maps ), S'TA = { all special TA-maps },

7 ='TA\S'TA, 'TAN = { all TA-homeomorphisms ),

PEM = { all expanding maps ), S S T A = { all strongly special TA-maps }. Then we have C(X) 3 S'TA 3 I A N U PEM

USS'TA.

$5.4 Subclasses of the class of TA-maps

Figure 21

CHAPTER 6 TA-Covering Maps

We have studied in the preceding chapters the behaviours of the orbits of continuous surjections of a compact metric space. To make a step forward the study of topological dynamics we shall discuss fundamental groups, universal covering spaces and covering trnsformation groups, and then apply them to dynamics of TA-covering maps. In the last section we shad discuss an interesting result on the classification up to topological conjugacy of TA-covering maps on tori. The readers may skip the sections 6.1, 6.2 and 6.3 if they like. $6.1 Fundamental groups

In this section we will introduce fundamental groups of topological spaces. Let X be a topological space. For two points xo, xl in X denote as R(X;xo, X I ) the set of all paths from I = [O, 11 into X which join XO and xl. Let u, v E R(X; xo, xl). If there exists a continuous map F : I x I + X such that

Figure 22

u

then F is called a homotopy from u to v . For this case u and v are said to be homotopic and we write u v (or u v (F)). N

N

Lemma 6.1.1. Let X be a topological space and let zo,xl E X . Then the relation i s an equivalence relation in R(X; xo, xl ). N

--

-

-

Proof. To see u u, define a homotopy F : I x I + X by F(t, s) = u(t). Then we have u u (F). If u v (F),then v u (F') under a homotopy F' : I x I + X defined by F1(t, s) = F(t, 1 - 8). To see the transitive relation

56.1 Fundamental groups

169

-

-

suppose u v (Fl) and v w (F2).Then it is easily checked that a continuous map F : I x I t X defined by

is a homotopy joining u and w. Therefore, u

N

w.

For x E X we can define a map u : I + X by u ( t ) = x. Clearly u is a path in X and for this case we write u = 0, = 0. Let u : I + X be a path from xo to xl, and v : I + X a path from x1 to 2 2 . Then we can define a path from xo to xa. For example, we set

and denote by w = u s v. For a path u from x0 to path from x1 to xo by ii(t) = u ( l - t ) , t E I.

XI,

we define the inverse

Lemma 6.1.2. Let X be a topological space. For u , v E a ( X ;xo, XI) and u', v' E n ( X ;X I , x 2 ) the following properties hold : u u' v v', ( 1 ) u v and u' v' ( 2 ) U N V =+ i i - 8 , (3) ( u . v ) . w - u . ( v . w ) forw E ~ ~ ( X ; X Z , X ~ ) , ( 4 ) u O,, u and O,, .u u , ( 5 ) u ii O,, and t i . u O Z 1 .

-

-

-

--

Proof, (1): I f u

-

--

v ( F ) and u'

-

-

v' (F'), then we have u u'

-

v . v' (Fl) by

(2): If u -- v ( F ) ,then we have ii 6 ( E ) by B ( t , s ) = F ( l - t , 8 ) . (3): ( u .v ) . w and u .( v w ) satisfy N

CHAPTER 6

170

respectively. Thus we define a continuous map F : I x I

.

-+

X by

.

Then F is a homotopy joining ( u s v ) w and u ( v w ) . (4): Since (U

.OXl)(t)=

a continuous map F : I x I

-+

112 5 t 5 1,

X is defined by

Then F is a homotopy joining u.OXl and u. Similarly we can prove Ox, .u (5): Since

a continuous map F : I x I

-+

u.

X defined by

u(2t(l - 8 ) ) u ( ( 2- 2 t ) ( l - s ) )

F(t,s)=

-

0

I t 5 112

112

is a homotopy joining u . ii and Ox,. Similarly .ii.u

-


Let X be a topological space. Take and fix xo E X. We say that a path from xo to xo is a closed path. If u and v are two closed paths from xo to xo, then the product u v and the inverse .ii are also closed paths from xo to s o . Obviously 0 = Ox, is a closed path. For simplicity we write R ( X ; x o )the family of all closed paths from xo to xo.

.

L e m m a 6.1.3. Let X be a topological space. For u , u', v , v' E R ( X ;x o ) the following properties hold : ( 1 ) u v and U' v' ju .u' v v', (2)U N V c-e, (3) ( U . V ) . u . ( V W ) for E n(x;x O ) , ( 4 ) u - i i = O and.ii-u=O.

-

-

-

-

.

.

Proof. This follows from Lemma 6.1.2.

-

-

Let X be a space and an equivalence relation in X . For x E X let [x] be the equivalence class of x. We denote as X / the family {[XI: x E X ) and say that X / is the identifying space with respect to -.

-

$6.1 Fundamental groups

-

171

-

Let X be a topological space and R(X,xo)/ the identifying space with respect to the equivalence relation by homotopty. We write this set

For [u], [v] E nl(X, s o ) define a product of [u] and [v] by [ul [vl = [u . vl The product [u][v] is well defined by Lemma 6.1.3. In fact, ?rl(X,xo) becomes a group with this operation, i.e. (1) if [u],[v],[w] are any three elements in ?r1(X, xo), then [u]([v][w])= ([uI[vI1[wl, (2) there exists the identity element [0] in ?rl(X, xo) with the property that [u][O]= [O][u]= [u] for every [u] in ?rl (X, xo), (3) to each [u] in ?rl (X, xo) there corresponds another element [a] in al(X, xo), called the inverse of [u],with the property that [u][a]= [G][u] = [O] The group 7rl(X,xo) is called the fundamental group at a base point xo of X . A fundamental group ?rl (X, xo) is not abelian (i.e. [u][v] # [v][u] for every [u],[v] in ?rl (X, xo)) in general. If ?rl (X, xo) is abelian, then we shall use additive operation as the operation of ?rl (X, xo). If, in particular, ?rl(X, xo) is a group consisting of the identity, then X is said to be simply connected with respect to a base point xo.

.

Figure 23 Let xo and xl be points in X. If there exists a path w joining xo and then we can define a map

-.

.

by wu(u) = ( w u ) for u E R(X,xl). For U , V E R(X,xl) suppose u Then wl(u) wp(v) by Lemma 6.1.2 and thus wp induces a map

XI,

N

v.

172

CHAPTER 6

Lemma 6.1.4. w, is an isomorphism. Proof. To simplify notations we write u-v-w = (u.v).w since (u.v).w By Lemma 6.1.2 we have

-

u.(v.w).

from which w*( [uI [vI1 = w*( IuI )w* ( [vI). Thus w, is a homomorphism. Since 27) is a path from homomorphism a, : nl (X, 20) + n1 (X, XI) by

31

to xo,

27)

induces a

and then 27)*w*= w*a* = 1. Indeed, since 27),w,([u]) = 27),([w.u.27)]) = [a.w.u.27).w] = [OX1 .u.Ox1] = [u],we have a,w, = 1. Also we have w,27), = 1. Therefore w, : nl(X,xl) -+ nl(X, xo) is an isomorphism. Let X be a path connected space. Take two points x and x' in X. Then the fundamental groups nl (X, z ) and nl(X, x') are isomorphic (written r l ( X , x) Z r l ( X , 2')). Thus we can consider the fundamental group as the group that does not depend on base points of X , so that we write this nl(X). nl(X) is called the fundamental group of X . If a topological space X is path connected and nl(X) is a single point (nl(X) = 0), then X is said to be simply connected.

Remark 6.1.5. An n-dimensional euclidean space Rn is simply connected. Indeed, let u : I + Rn be a closed path at the base point 0. Then F(t, s) = s (1 - s)u(t), (t, s ) E I x I, is a homotopy joining u and 0, and so u 0 (F). Therefore [u] = 0 and Rn is simply connected. Similarly, for z E Rn and E > 0 the &-open neighborhood U,(x) is also simply connected.

-

+

Lemma 6.1.6. Let X be a simply connected space and let xo,xl be arbitrary points of X . Then arbitrary paths u and v from xo to zl am homotopic.

-

Proof. Since u - 6 is a closed path at the base point xo and X is simply connected, we have nl (X, xo) = 0. Thus u @ O,, and so

.

Let X and Y be topological spaces and f: X -t Y a coritinuous map. Take xo E X and let yo = f(xo). It is clear that f u = f o u E n(X, yo) for u E n ( X , xo). Thus we can find a map f n : R(X,xO)-t n(Y, yo) by

$6.1 Fundamental groups

-

173

-

If u v ( F ) for u , v E R ( X , xo), then we have f u f v ( f o F ) , from which a map f* : T l ( X ,5 0 ) + X l (Y,Y o )

.

.

is induced by f8. Since f ( u v ) = ( f u ) ( f v ) for u , v E R(X,xo), it is easily checked that f,([u] [ v ] )= f,([u])f,([v]). Thus f* is a homomorphism. We say that f , : ?rl ( X ,xo) + a1 (Y,yo) is a homomorphism induced fmm a continuous map f : X + Y .

Lemma 6.1.7. Let X , Y and Z be topological spaces and let xo be an arbitmry point in X . ( 1 ) For the identity map id : X -+ X the induced homomorphism id, : ?rl ( X ,xo) -+ T I ( X ,x o ) is the identity map. ( 2 ) If f : X + Y and g: Y -+ Z are continuous maps and f ( x o ) = yo and g(yo) = zo, then (g o f ) , = g, o f,, that is, the diagram

f.

.~rl(x,xo) ( g f)+ \

nl(z,~

~~(Y,Yo)

J g*

commutes.

0 )

Proof. This is clear from the definition of homomorphisms induced by continuous maps. Let f,g : X -+ Y be continuous maps of topological spaces. If there is a continuous map F : X x [O,1]-+ Y such that F(x,O) = f (x) and F ( x , 1) = g(x), then we say that f is homotopic to g, and write f g (or f g (F)). Here F is called a homotopy from f to g. The following lemma is easily checked as the proof of Lemma 6.1.1, and so we omit the proof. N

N

Lemma 6.1.8. Let X and Y be topological spaces. Then the relation equivalence relation in the class of all continuous maps from X to Y .

-

is an

Let f , g : X -t Y be homotopic and F a homopoty from f to g. Then for xo E X we can define a path w E R ( Y ;f ( Z O ) , ~ ( X O ) )by

and the relation between homomorphisms f , : q ( X ,X O ) g* : ?rl ( X ,so) ?rl (Y,g(x0)) is expressed as follows.

-+

?rl(Y, f ( 5 0 ) )and

-+

Lemma 6.1.9. g, = ti3, o f , . Proof. For each t E I we define a path wt E R ( Y ;F(xo,t),g(xo)) by

and put ~t

= 2Tlt(Ft o u)wt

t EI

CHAPTER 6

174

.

( f u ) w to ul = gu, for [u]E ?rl(X,xo). Then ut is a homotopy from uo = and hence w* 0 f*([u]) = [ w .( f u ).w] = [gu]= g*[u]. Thus a, o f , = g, holds.

A continuous map f: X + Y is called a homotopy equivalence if there is a continuous map g: Y -+ X such that g o f idX and f o g idy where idx and idy are the identity maps of X and Y respectively. Here g is called a homotopy inverse of f . It is clear that f is a homotopy equivalence if it is a homeomorphism. N

N

Proposition 6.1.10. I f f :X -+ Y is a homotopty equivalence, then for xo E X , f* : T I( X ,xo) + nl(Y,f (xo)) is an isomorphism. Proof. Let g : Y -+ X be a homotopy inverse of f . Since g o f i d x and f og i d y , by Lemma 6.1.9 there are paths w E R ( X ; xo,g o f ( x o ) ) and w' E R ( Y ;f (so),f 0 g ( f ( x o ) ) )such that

-

N

From Lemmas 6.1.4 and 6.1.7 it follows that f , is an isomorphism.

$6.2 Universal covering spaces The aim of this section is to establish the existence theorem of universal covering spaces for a class of topological spaces that contains the class of topological manifolds. We recall that a continuous surjection p : Y 4 X is a covering map if for x E X there is a canonical neighborhood U of x in X , which is evenly covered by p (see 55.1 of Chapter 5). T h e o r e m 6.2.1 (Homotopy Lifting property). Let p : Y -+ X be a covering map of topological spaces and let f: Z -+ Y be a continuous map. If a continuous map F : Z x [O,1]-+ X satisfies

then there is a continuous map G : Z x [O,1]-+ Y such that Namely, if the square part of the following diagram commutes, then there is a continuous map G represented by the right up arrow that makes the two triangle parts commutative :

$6.2 Universal covering spaces

where i : X

-+

X x [O,1] is an inclusion map defined by i(x) = (x, 0).

Proof. First, we show that for z E Z there are an open neighborhood N, of z and a continuous map G, : N, x [O,1] + Y such that

Since [O,l] is compact, F ( z x [O,l]) is covered by a finite number of canonical neighborhoods Ul, Uz,. ,Uk.We may suppose that for some finite sequence 0 = to < tl < < t k = 1 in [O,l], F ( z x [ti,tj+l]) c Ui for 0 i 5 k - 1. Since each Ui is open, there is an open neighborhood N, of z such that F(N, x [ti, ti+l]) C U;(0 i 5 k - 1). If continuous maps G; : N, x [ti, ti+l] -, Y with the following properties are constructed : (1) each of G; is a lift of FIN, x [ t i , t i + l ~ by p, (2) Go(zl, 0) = f (z') for z' E N,, (3) G;(zl,t;+l) = Gi+l(z',ti+l) for z' E N, and 0 5 i 5 k - 1, then the desired map G, is gived by

.-

<

<

Figure 24 To construct Gi we use induction on i. Since Uo is a canonical neighborhood, disjoint open sets of Y, say by definition p-'(Uo) is decomposed into Xo E Ao, such that each restriction p : UAo -, Uo is a homeomorphism. Let Vxo = f -'(VA,) n N,. Then {Vxo} covers N, and Vxo n V', = 0 if Xo # po. Hence, since F(N, x [to,tl]) c Uo, there is a unique continuous map Go : N, x [to, tl] + Y satisfying the following conditions

uxo,

176

CHAPTER 6

Suppose Gi-1 is constructed. Since F ( N z x [ti,ti+l]) c Ui and p-l(Ui) splits into - disjoint open sets Uxi (Xi E Ai), we let VAi = (2' E NZ : Gi-l(z'lti) E Uxi). In the same way as above, we can choose uniquely a continuous map Gi : N, x [ti,ti+l] -+ Y such that

From the construction it follows that the maps Gi satisfy the conditions (I), (2) and (3). Next, for z1, z2 E Z let (N,, ,G,,) and (N,,, G,,) be as above. If y E Nz1 n Nza 1 then Gzl larx[o,l]and G,,lvxIo,ll are continuous maps from y x [0, 11 to Y and

= G,, Ivx[o,ll.Since y is arbitrary, Thus, by Theorem 5.2.2 we have G,, larx[o,ll it follows that G,, is consistent with G,, on (N,,n N,,) x [O,l]. Thus we can define G : Z x [O,1] -+ Y by GIN,x[o,ll = G, (z E Z), and G satisfies all the conditions in the theorem.

Lemma 6.2.2. Let p : Y -+ X be a covering map. Let xo E X and bo E p-l (xo). Suppose that u, u' are paths of X starting at s o and v, v' are paths of Y starting at bo satisfying p o v = u and p o v' = u', if u and u' are homotopic, then the terminal point of v coincides with that of v', and v is homotopic to vl. Proof. Let F be a homotopy from u to u'; i.e. F : I x I -+ X is a continuous map satisfying F(t1O) = ~ ( t ) ,F(t, 1) = ~ ' ( t ) , F(0, s) = s o , F(1, s) = u(1) = ul(l). By the homotopy lifting property there exists a continuous map G : I x I -+ Y such that G(t, 0) = v(t) and p o G = F. Since G(0 x I ) = so, it follows that G(0,s) = v(0) = bo. Similarly, G(1,s) = v(1). Put h(t) = G(t,l) for t E I , then h: I -, Y and

Thus h is a path of X starting at bo and p o h = u'. By Theorem 5.2.2 we have h = v', and therefore

and G is a homotopy from v to v'.

56.2 Universal covering spaces

177

Lemma 6.2.3. Let p : Y + X be a covering map. Then for y E Y, p, : nl(Y, y) + ?rl(X,p(y)) is injective. Proof. For [u], [v] E rl (Y, y) suppose p, ([u]) = p, ([v]). Then po u is homotopic to p o v. Therefore [u] = [v] by Lemma 6.2.2.

Lemma 6.2.4. Let p : Y + X be a covering map. If X is connected, then the cardinal number of p-'(x) is constant. Proof. Since p : Y + X is a covering map, by definition each point x in X has a neighborhood evenly covered by p. Hence the cardinal number of p-'(x) is locally constant. Since X is connected, the conclusion is obtained. For a covering map p : Y + X the cardinal number of p-'(x) is called the covering degree of p if it is constant.

Lemma 6.2.5. Let p : Y + X be a covering map. If Y is path connected, then the index of p, ( r l (Y, b)) in ?rl (X, p(b)) is equal to the covering degree of P. Proof. For each y E ~ - ' ( ~ ( b )take ) a path u, in Y from b to y. Then p o u, is a closed path from p(b) to p(b), and so p+(?rl(Y,b))[p o uy] is a right coset in nl(X, p(b)). Let [v] E ?rl(Y,p(b)) and u : [O, 11 -t Y be a lift of v by p such that u(0) = b. Then y = u(1) E P-'(P(~)) and u By is a closed path from b to b. Therefore, [v] = [p o (u ~ , ) ] [ po u,] E p,(?rl (Y, b))[p o u,]. On the other hand, let [v] E p+(rl(Y,b))[po u,] np,(rl(Y, b))[po u,~]. Then [v] = [P o u][p o u,] = [ p o u'][p o U ~ I ]for some [u],[u'] E rl(Y, b). Since (p o U ) . (p o uy) is homotopic to (p o u') (p o uyr), by Lemma 6.2.2 we have u uy(l) = u' U,I (1), and so y = u,(1) = u,~(1) = y'.

.

.

.

.

Let p : Y + X be a covering map. If Y is locally path connected and simply connected, then Y is called a universal covering space of X and p :Y -t X is said to be the universal covering. Let p : Y + X and p' : Y' + X be covering maps. If there is a homeomorphism h: Y' + Y such that p o h = p', that is, the diagram

I P'

J X

then p :Y

+X

commutes,

P

is equivalent to p' : Y'

-, X.

Lemma 6.2.6. Let p : Y + X be a covering map and let X be path connected. Suppose Y' is locally path connected and simply connected. If xo E X and bo E p-'(xo) and if yo E Y', then for a continuous map h:Y' + X with h(yo) = xo there ezists only one lift : Y' + Y satisfying z(yO) = bo and

z

4

CHAPTER 6

Proof. We first define a map h : Y' + Y as follows. Take y E Y'. Since Y' is path connected, yo and y are joined by a path w . Put v = h o w. Then u is a path in X starting at X O . Choose a path v of Y starting at bo such that p o v = u. Let b = v ( 1 ) and define h(y)= b. To show that h is well defined, let w and w' be paths in Y' joining yo and y. Since Y' is simply connected, w and w' are homotopic ( w w') by Lemma 6.1.6. Thus h o w h o w'. Let v and v' be paths in Y starting at bo such that p o v = h o w and p o v' = h o w'. By Lemma 6.2.2 we have v ( 1 ) = v l ( l ) . Thus is well defined. From the construction it follows that = bo and p o h = h. Next we show that h : Y' -,Y is continuous. Fix yl E Y'. Put x l = h ( y l ) and take a canonical neighborhood U of xl for p. Since h is continuous, clearly V = h-'(U) is an open neighborhood of X I . Hence, since Y' is locally path connected, we can choose a neighborhood W of yl such that W c V, and such that for any y E W there is a path in V joining y and yl.

-

z(yo)

-

x

Figure 25 Let bl = h(yl).Since p(bl) = xl and U is a canonical neighborhood of there is an open neighborhood U ( b l ) of bl such that p : U ( b l ) -, U is a homeomorphism. Now, define g: W + Y by g ( y ) = ( p l u ( b l ) ) - lo h ( y ) . Clearly g is continuous. We claim that h = g on W . Indeed, take a path w joining yo and yl in Y and a path w' joining yl and y in V. If v is a path starting at bo satisfying p o v = h o w, then by definition v ( 1 ) = b1. Since h o w' is a path in U starting at X I , clearly v' is a path in U(b1) starting at bl where v' = ( P I U ( b l ) ) - l o h o w'. By definition v l ( l ) = g ( y ) , and so v .v l ( l ) = g ( y ) . On the other hand, since p o ( v v') = h o ( w e w'), we XI,

-

$6.2 Universal covering spaces

have v v l ( l ) = E(y). Therefore, E = g on W , from which we obtain that continuous. The uniqueness of E follows from Theorem 5.2.2.

179

is

Proposition 6.2.7. Let p : Y + X be a covering map and let p' :Y' + X' be the universal covering. Let xo E X , bo E p-'(zo) and xb E X1,bb E p'-'(xb). If X is path connected and h: X' + X is a continuous map with h(x6) = xo, then there is a unique continuous map E : Y' + Y satisfying x(bb) = bo and hop1.

Proof. Since Y' is locally path connected and simply connected, by applying Lemma 6.2.6 to a map h o p' : Y' -t X we obtain the conclusion. Theorem 6.2.8 (Covering theorem). Let p : Y + X be a covering space and let p' : Y' + X be the universal covering. Let xo E X , bo E p-'(xo) and bb E p'-'(xo). If Y is path connected, then there exists a unique continuous map h:Y1+Y

satisfying E(b&)= bo and p o h = p', and h : Y'

4

Y is the universal covering.

Proof. By choosing the identity map instead of h of Proposition 6.2.7 there exists a unique : Y' + Y satisfying h(bb) = bo and p o x = p'. We show that h is surjective. Indeed, for b E Y let v be a path joining bo and b and put u = p o v. Take a- path v' of Y' starting at bb such that p o v' = u and set b' = v l ( l ) . Then h(bl) = b and therefore h is surjective. For b E Y let U be a path connected neighborhood of p(b) in X which is evenly coverd by both p and p'. Then there exists a neighborhood U(b)of b in Y such that p : U(b)+ U is a homeomorphism, and for each 6 E h-'(b) there is a neighborhood ~ ( 6of) 6 in Y' satisfying

x

6 E X-'(b), h : ~ ( 64) U ( b )is a homeomorphism, ( 2 ) X-' ( ~ ( b = ) )U { U ( ~: 6) E X-' (b)), ( 3 ) if 6,8 E h-'(b) and 5 # 8, then ~ ( 6n)~ ( 8 = 0. )

(1) for each

Therefore I;:: Y'

-,Y

is a covering space.

Theorem 6.2.9 (Uniqueness theorem). Let p : Y + X and p' : Y' + X be the universal coverings. Then p : Y + X is equivalent to p' : Y' + X .

CHAPTER 6

180

Proof. Take xo E X. For bo E p-l(xo) and bh E pt-l(xo) there exist continuous maps

satisfying the conditions that z(bh) = 4 , 3(bo) = bh, p o We consider two the continuous m a p

= p' and p' o 3 = p.

Since p' o 3 o X = p' = p' o id and 3 o x(bh) = bh, by_Theorem 6.2.8 it follows o = idyl. Similarly, o 3 = i d y . Therefore h is a homeomorphism. that ?j

x

x

A topological space X is said to be semilocally 1-connected if for each x E X there is a neighborhood U of x such that the homomorphism i , : ?rl(U,x) -t ?rl (X, x) induced by the inclusion map i : U + X is trivial. A topological space X is said to be locally contractible if for each x E X and for a neighborhood N of x there is a neighborhood U of x with U C N such that the inclusion U + N is homotopic in N to a constant map U + {x). Remark 6.2.10. By definition it follows that every topological manifold is locally contractible and every locally contractible space is semilocally 1connected. Theorem 6.2.11 (Existence theorem). Let X be a locally path connected, semilocally 1-connected and path connected space. Then there ezists the universal covering p : Y -t X . Proof. Take and fix xo E X. We define R(X; xo, X ) = {u : u is a path from [O,l] to X such that u(0) = 30).

-

-

For u,ut E n(X; xO,X), if u(1) = ul(l) and u is homotopic to u', then we write u u'. Clearly is an equivalence relation in R(X; xo, X). Denote as [u] the equivalence class of u for u E R(X;xo,X) and define p([u]) = u(1). Then p is a map from Y = {[u] : u E R(X; 30,X)) to X. In fact p :Y -t X is surjective. This follows from the fact that if u is a path joining xo and x then P([UI)= 3. To introduce a topology of Y we define a fundamental neighborhood system as follows. Since X is locally path connected and semilocally 1-connected, for x E X and for any neighborhood 0 of x there exists a neighborhood U of x with U c 0 such that U is path connected and ?rl(U,x) + ?rl(X, x) is trivial. Hereafter, for x E X we choose such a neighborhood U. To define a fundamental neighborhood U(b) of b = [u] take a neighborhood U of u(1) = x and put U' = {u u : a is a path in U starting at x),

56.2 Universal covering spaces

and let

U ( b ) = { [ u .u] : U u E U'). Then {U(b)} is a fundamental neighborhood system of b. Indeed, ( 1 ) it is clear that b E U(b). ( 2 ) Two sets U ( b ) and V ( b ) are constructed by certain neighborhoods U and V of x = u ( 1 ) respectively. Now take a neighborhood W such that W c U n V . Then W constructs a set W ( b ) satisfying

(3) Take b' E U(b). Then b' = [u'] and u' = u s a for some path in U. Let V be a neighborhood of x' = u l ( l ) such that V C U . Construct a set V ( b l )by V . Then we have b' E V ( b t ) C U(b). We show that p : Y + X is a continuous open map. Let p(b) = x and take U a neighborhood of x. Then we have p(U(b)) = U . Indeed, clearly p(U(b)) c U . Since U is path connected, for x' E U let u be a path in U from z to x'. Then p ( [ u . u ] )= x' and thus U C p(U(b)). Therefore, p(U(b)) = U , which shows that p is continuous and open. We prove that p : U ( b ) --,U is a homeomorphism. To do this it suficies to see that p is injective. Let p(bl) = p(b2)(= x') for bl, b2 E U(b). If ul and 0 2 are paths of U from p(b) = x to x', then b = [u],bl = [u ul]and bz = [us uz]. Since ?rl (U,x ) -+ ?rl ( X ,z ) is trivial, it follows that a1 uz (c.f. Lemma 6.1.6). Therefore u . u1 u uz and bl = [us ul] = [u u2]= bz. We check that if b, b' E p-'(x) and b # b', then U(b)n U(bl)= 0. Indeed, suppose U ( b )n U(bl) # 0, from which take a point c. If b = [u]and b' = [u'], then c = [usu]= [u' -0' 1 for some paths u and a' in U starting at x. Thus we have u u u 0'. Since u a', we have

-.

N

0

-

-

-

.

--

-

( u . u ) . ( u l - a ' ) = ( u . 0 ) . ( 2 . i ' ) ( u . ( 0 . ; ' ) ) .ti'

from which

u

-

N

U.i',

u',

and so b = [u]= [u']= b', thus contradicting. It is checked that p-l(U) = U { U ( b ) : b E p-'(x)}. Indeed, it is clear that p-l ( U ) > U { U ( b ) : b E p-l ( x ) ) . Take b' = [u'] E p-I ( U ) and put x' = p(bl). Let a be a path of U from a: to x', and define u = u' a. If b = [u],then p(b) = x and U' U' (a O ) (u' .a ) ' 6 = '11.6.

.

N

.

N

Thus we have b' = [u'] = [ u u ] E U(b). We have seen that p : Y + X is a covering map. To show that p : Y -,X is the universal covering, it sufficies to show that (i) Y is path connected and (ii) Y is simply connected. Proof of (i) : Let 4 = [Om,]and take any point b E Y. To show the existence of a path of Y joining b and bo we choose a path u of X joining xo and p(b)

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such that [u]= b, and define paths u , : I -t X by u,(t) = u ( s t ) for 0 5 s 5 1. If b(s) is an equivalence class containing a path u,, then a map v : I + Y is defined by v ( s ) = b(s),and we have that v(0) = bo and v(1) = b. Continuity of v is proved as follows. Take a neighborhood U of u,,(l) in X and let U(b(s0)) be a fundamental neighborhood of b(so) in Y. Since u : I -+ X is continuous, we choose 6 > 0 such that

For s with 1s - sol < 6 let u be a path in U from u ( s O )to u ( s ) . Then the path u , is homotopic to u,, u. Therefore, [u,] = b(s) E U(b(s0)). Proof of (ii): To show ?rl (Y, bo) = 0 let v be a closed path from b~ to 4. Put u = p o v and for s with 0 5 s 5 1 define a path u , : I -,X by u,(t) = u(st). Then a map v' : I -+ Y defined by v t ( s ) = [u*]is continuous, that is, a path in Y. Since

.

by Theorem 5.2.2 we have v' = v , and in particular

[u]= v'(1) = v ( 1 ) = bO = [O,,], which implies that u and O,, are homotopic, and by Lemma 6.2.2, v is homotopic to Obo. Thus Y is simply connected. As an easy corollary we have the following proposition.

Theorem 6.2.12. Let X be a locally path connected, semilocally 1-connected and path connected space. For xo E X let p be a subgroup of 7 ~ 1 ( X2,0 ) . Then there ezist a covering space p : Y + X and bo E p-'(so) such that

Proof. Let u, U' E R ( X ;x O , X ) . We write u u' if u ( 1 ) = u'(1) and [u.;'] E p. Put Y = R ( X ;x o , X ) / N and define a map p : Y + X by p([u])= u(1). By introducing an adequate topology in Y we see that p : Y 4 X is a covering map. The proof is very similar to that of Theorem 6.2.11 and so we omit the proof. N

56.3 Covering transformation groups

Let X be a topological space and let be an equivalence relation in X . As before, the identifying space X / is a family {[XI : s E X} of all equivalence classes. Let p : X -, X / be the natural projection defined by p(x) = [XI. Clearly p is surjective. We define a topology of X / as the strongest topology such that p is continuous, i.e. a subset 0 of X / is open if and only if p-'(0) is open in X .

-

N

N

N

N

56.3 Covering transformation groups

183

Lemma 6.3.1. Let X be a topological space and p : X + X/ the natural projection. Let Y be a topological space and suppose f : X + Y and g: X/ w-+ Y satisfy a relation f = g o p. Then g is continuous if and only if f is continuous. X P1

XI-

-< B

Y

Proof. Suppose j is continuous. To see continuity of g it suffices to show that g-'(0) is open when 0 is an open subset of Y. Since f = g o p, we have p-'(g-l(0)) = j-'(0). Since f is continuous, p-'(9-'(0)) is open if 0 is open, so that g-l(0) is open. Conversely, if g is continuous, then f = p o g is continuous since p is continuous. Let X be a topological space and let G be a group. We say that G acts (continuously) on X if to (g, x) E G x X there corresponds a point g(x) in X and the following conditions are satisfied: (1) e(x) = x for x E X where e is the identity, (2) g(gl(x)) = ggl(x) for x E X and 9, g1 E G, (3) for each g E G a map x H g(x) is a homeomorphism of X. When G acts on X , for x, y E X letting x

N

y

(jy

= g(x) for some g E G,

an equivalence relation in X is defined. Then the identifying space X/ N, denoted as X/G, is called the orbit space by G of X. I t follows that for x E X , {g(x) : g E G) is the equivalence class. N

Lemma 6.3.2. The natural projection p : X

-t

X / G is a continuous open

surjection. Proof. Let 0 be an open subset of X. Then p-'(p(0)) = U{gO : g E G). Here g 0 = {g(x) : x E 0). Since g 0 is open in X , P-'(~(O)) is open in X. By the definition of topology of X/G, we have that p ( 0 ) is open in X/G, and so p is an open map. The continuity of p follows from the definition of topology. Let p : Y -+ X be a covering map. A homeomorphism a : Y -+ Y is called a covering transformation for p if p o a = p holds. We denote as G(p) the set of all covering transformations for p. Since (1) the identity map i d y of Y belongs to G(p), (2) if a', a2 E G(p), then a1 o a2 E G(p), (3) if a E G(p), then a-' E G(p), it follows that G(p) is a group, which is called the covering transformation group for p.

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L e m m a 6.3.3. Let p : Y * X be the universal covering. If a continuous map h: Y -+ Y satisfies p o h = p, then h E G ( p ) . Proof. Suppose a continuous map h: Y + Y satisfies p o h = p. Take b E Y and let c = h(b). By Lemma 6.2.6 there is a continuous map h' : Y -+ Y such that h l ( c ) = b and p o h' = p. Since h o hl(c) = c and p o ( h o h') = p, by Theorem 5.2.2 we have h o h' = i d y . Similarly, h' o h = i d y . Thus h is a homeomorphism. We say that an action of G on X is said to be properly discontinuous if for each b E X there exists a neighborhood U ( b ) of b such that U ( b ) n gU(b) = 8 for all g E G with g # 1. Here gU(b) = { g ( x ) : x E U ( b ) ) .

Theorem 6.3.4. If p : Y --+ X is the universal covering, then for each b E Y ( 1 ) the map a H a ( b ) i s a bijection from G ( p ) onto p-'(p(b)), ( 2 ) the map $Jb : a H [p o u,(~)]is an isomorphism from G ( p ) onto the fundamental group .rrl(X,p(b)) where u,(b) is a path from b t o a(b). Furthermore, the action of G ( p ) o n Y is properly discontinuous and Y / G ( p ) is homeomorphic to X . Proof. (1): If z E p-l(p(b)), then by Lemma 6.2.6 there is a unique continuous map h: Y + Y such that h(b) = z and poh = p, and by Lemma 6.3.3, h E G ( p ) . Thus ( 1 ) holds. ( 2 ) : Let u and u' be paths from b to a(b). Since Y is simply connected, by Lemma 6.1.6 u and u' are homotoptic, which implies [p o u] = [p o u'] E .rrl(X,p(b)). Thus $b is well defined. For a,@ E G ( p ) suppose & ( a ) = $Jb(@). Since [p o u,(~)]= [JI o it follows that p o u,(b) and p o up(b) are homo( l~)p ( ~ ) by ( l )Lemma 6.2.2. By uniqueness a = @. topic, and hence ~ , ( ~ )= Therefore $Jb is injective. If [v]E s l ( X , p ( b ) ) , then there is a path u starting at b such that p o u = v, and u ( 1 ) E ~ - l ( ~ ( b )From ) . this and ( 1 ) we have that $Jb is surjective. To show that $b is a homomorphism, let a,@E G(p). By definition @ou,(b) is a path starting at @(b)and so

from which we have o @)= $ J ~ ( ~ ) T , ! Thus J ~ ( / $b~ ) is . a homomorphism. ( 2 ) was proved. For b E Y let U be a canonical neighborhood of p(b) and take an open neighborhood U ( b ) such that p : U ( b ) -+ U is a homeomorphism. If a E G ( p ) and a ( U ( b ) )n U ( b ) # 0, then there is x , y E U ( b ) such that a ( x ) = y, and so p ( a ( x ) ) = p ( x ) = p(y). This implies z = y, and so a is the identity map. Thus the action of G ( p ) is properly discontinuous. To see that Y / G ( p )

56.3 Covering transformation groups

185

is homeomorphic to X, define a map f: Y/G(p) -t X by f (G(p)b) = p(b). Clearly, f is surjective. By (1) we have that f is injective. The continuity o f f follows from that of p. Since the natural projection Y -, Y/G(p) is continuous, we obtain the continuity of the inverse map f-'.

x

Remark 6.3.5. Let p : -+ X be a covering map and let Y be a connected locally path connected semilocally 1-connected space. Suppose f: Y -+ X is a continuous map and let f (yo) = xo and % E p-l(xo). A necessary and sufficient condition that f has a lift g:Y -+ X satisfying g(yo) = Zo and P 0 g = f is that f*?rl(Y,YO)c p*?rAX,Zo). Indeed, if there is a lift g: Y -+ X, then and hence the condition is necessary. To see the converse, let q : Y -+ Y be the universal covering- and - take go io qq-l(yo). Then by Proposition 6.2.7 there is a continuous map f : Y -+ such that p o 'j = f o q and T(go) = 5. For ti E F put y = q(g) and 5 = 7(g). If Ti7 is a path in P from g to go, then w = q o Ti7 is a path in Y joining y and yo, and hence

x

c ( f 0 w)* 0 p*.1(X,%) = p* 0 ( 7 0 E ) * ? ~ I ( ~ , Z O ) = p*7rl (X, 5). Let cr be a covering transformation for q and take a path Z in F from to a@). Since q oii; is a closed path from y to y, it follows that [f o q 04E f,?rl(Y, y) c p,al(x,f). ~ i n c e ~ o T of oqoii;, ~ = we have Ipof OZ] ~ ~ , ? r l ( x , ~ ) ~ w h i c h implies that 7 o ii is a closed path in X (see Lemma 6.2.2). Therfore, - - f (g) = T(a(p)). Since cr is arbitrary, by Theorem 6.3.4 we can project f : X -+ F to a map g from Y to X. Thus the condition is sufficient. Let G be a group and S a subgroup of G. The largest subgroup of G containing S as a normal subgroup is called the normaliaer of S in G. We denote this subgroup as N ( S ;G).

Remark 6.3.6. Let X be a connected locally path connected semilocally 1-connected space and let p : -t X be the universal covering. Suppose Y is path connected and q : Y -+ X is a covering map and let r : X i? Y be the universal covering that satisfies p = q o r (see Theorem 6.2.8).

x

186

CHAPTER 6

Then a covering transformation a E G(p) can be projected to a map in G(q) by T if and only if a E N(G(r); G(p)). Moreover G(q) is isomorphic to the factor group N(G(r); G(p))/G(r). Indeed, suppose a E G(p) can be projected to a map of Y by r. Let x E X. Then for P E G(r) we have r o a o P(x) = r o a(%),which means that a o /3 = Pt o a for some p' E G(r), and hence aG(r)a-' c G(r). Therefore a E N(G(r); G(p)). It is easy to check the converse. Thus we leave the proof to the readers. -

x".x

1 Y-Y &(a)

Since every member in G(q) can be lifted to a map in G(p) by Proposition 6.2.7, a map 4 : N(G(r); G(p)) -t G(q) defined by assigning each a E N(G(r); G(p)) to the projection by r is surjective. Obviously 4 is a homomorphism. By definition it follows that the kernel Ker(4) coincides with G(T). Therefore the second statement is obtained from the homomorphism theorem described as follows: if A and B are groups and if 4 : A + B is a homomorphism, then the quotient group AIKer(4) is isomorphic to the image Im(q5). The following theorem describes the converse of Theorem 6.3.4.

Theorem 6.3.7. Let G be a group and X a topological space. Suppose that G acts on X and the action is properly discontinuous. Then the following (1) and (2) hold : ( 1 ) the natural projection p : X -,X/G is a covering map, (2) if X is simply connected, then the fundamental group al(X/G) is isomorphic to G. Proof. (1) : By Lemma 6.3.2 the projection p : X + X/G is continuous and it is an open map. Let b E X. Since the action of G is properly diicontinuous, there is a neighborhood U(b) such that gU(b) n U(b) = 8 if g E G is not the identity. Put U = p(U(b)). Then U is open in X. For b', b" E U(b) let p(bt) = p(bf'). Then there is g E G such that g(bf) = b". This implies g = 1, and so b' = b". Thus p : U(b) + U is a homeomorphiim. Similarly, we have that p :gU(b) + U is a homeomorphism for g E G. Since p-'(U) = U{gU(b) : g E G) and gU(b) n gtU(b) = 0 for g # g', it follows that U is evenly covered by p. Therefore p : X -+ X/G is a covering map. (2) : Take xo E X/G and bo E p-'(xo), and define a map $ao : G -, nl (XIG, xo) as in (2) of Theorem 6.3.4. In fact, for a E G let u be a path in X from bo to ( ~ ( 4 )Put . v = p o U. Then [v] E al(X/G, 30). Since X is simply connected, by Lemma 6.1.6 it follows that [v] is uniquely determined for a. Thus we define $bo(a) = [ v ] . Then, in the same way as the proof of Theorem 6.3.4 (2), we obtain that $b0 is an isomorphism.

56.3 Covering transformation groups

187

Let Tn = Rn/Zn be an n-torus and p :Rn -+ 'P the natural projection. Then p : Rn + Tn is the universal covering and nl ( W )E Zn holds. Since a map p : Zn x Rn -+ Rn is defined by p(n, x ) = n x , the group Zn acts on Rn. For x E Rn let U ( x ) = {y E Rn : ( ) x- yll < 1/21. Then p(n, U ( z ) )n U ( z ) = 8. Hence the action p is properly discontinuous, and by Theorem 6.3.7 ( 1 ) the natural projection p : Rn + W is a covering map. Since Rn is simply connected, it follows that Rn is the universal covering space of Tn and .rrl(W) Zn by Theorem 6.3.7 (2).

Remark 6.3.8.

+

Let p : X -1 X be a covering - - map and f : X + X a continuous map. We say that a continuous map f : X + X is a lift of f by p if p o 7 = f o p holds.

-

-

X - Xf

-

x-x

f

x

Theorem 6.3.9. Let p : -+ X be the universal covering and f : X + X a continuous map. Let xo E X , bo E p-'(30) and b6 E p-'(f (zo))Then there is - a unique lift f : X + of f by p such that T(bo)= b;. If continuous maps f ' , f 2 : X + are lifts of f by p, then f1= a o T2 for some a E G ( p ) .

x

x

- Proof. The first statement follows from Proposition 6.2.7. Suppose f ', f : X + X are lifts off by and let z E X.since p o T l ( x ) = f o p ( x ) = ~ o T ~ ( z ) , there is a E G ( p )such that f ( x ) = a0T2( x ) . By Theorem 5.2.2 the conclusion is obtained. Let :55 -+ X be the universal covering and f : X + X a continuous map. - pFor f : X + a lift of f by p we define a homomorphism % : G ( p ) -+ G ( p ) by

x

where b E X , c = T ( b ) , f , : .rrl ( X , p ( b ) ) -, .rrl ( X , p ( c ) )is the induced homomorphism; and +b and are isomorphisms as in Theorem 6.3.4. Then we obtain the following lemma.

+,

Lemma 6.3.10. a E G(P)

7, does not depend on the choice of b.

Furthermore for every

Proof. For b' E X and c' = f(b1) define a homomorphism : G(p) + G(p) as in (6.1). To show that f, = take a path w in from b to b' and let

T*,

x

CHAPTER 6

188

a E G(p). Then a o w is a path from a(b) to a(bl). By the definition of f , and f, it is easily checked that f o a o w is a path from j,(a)(c) to T:(a)(c'). On the other hand, since there is p E G(p)such that f o a o w = /3 o ( f o w). This implies f , ( a ) = /3 = f,(a). Thus 'j, = Next, we show (6.2). Let a E G(p) and u,(b) be a path from b to a(b). Since ~ ~ ( ~ = a(b), ~ ( clearly 1 ) ( f 0 ~ , ( ~ ) ) (=1 )f 0 a(b). Since f o u,(b) is a lift of f o ( p o u,(b)) by p starting at c, by definition f,(a)(c) = (70u,(b))(l). Therefore, 7 o a(b) = f,(a) o f(b). Since the two sides of (6.2) are both lifts of f by p, by Theorem 5.2.2 we obtain the equality of (6.2).

-I

7:.

Under the assumptions of Remark 6.3.6, let f : X --,X be a continuous map and f' : Y -, Y a lift of f by q. Then a homomorphism f: : G(q)-t G(q)is defined and for every t E G(q),f' o 1 = f : ( t ) o f' holds. Indeed, let p : -+ X be the universal covering and let f : -+ X be a lift of f by p. Then 'j is also a lift of f' by r. Here r : -t Y is the universal covering such that p = q o r. Let 7, : G(p)+ G(p)be as in (6.1). If a E G ( T ) , then we have

Remark 6.3.11.

x

x

) G(r). From this fact it is from which ?,(a) E G(T). Hence f , ( ~ ( r ) c easy to see that ~ , ( N ( G ( T G(p))) ); c N(G(r);G(p)).Therefore f , induces a homomorphism f: : N ( G ( r ) ;G(p))/G(r)-t N(G(r);G(p))/G(r).Since G(q) is isomorphic to N(G(r);G(p))/G(r), we define a homomorphism f: : G(q)-t G(q)by the following commutative diagram:

Then, for t E G(q)we have

where a E G(p) is a covering transformation satisfying t o r = T o a. Thus f 1 0 e = f: (el 0 f l .

$6.4 S-injectivity of TA-covering maps

x

189

Theorem 6.3.12. Let-p :- -, X be the universal covering and f:X -t X a self-covering map. Let f : X -+ be a lift o f f by p. Then (1) 7 is a homeomorphism, (2) the cardinal number of G ( ~ ) / ~ , ( G ( ~is) )equal to the covering degree of f 9 (3) f is a homeomorphism i f and only if T,(G(~))= G(p). Proof. Since p : X -+ X is the universal covering, by Proposition 6.2.7 it follows that f : + is a self-covering map, and by Lemma 6.2.5 it must be a homeomorphism because is simply connected. Thus (1)holds. (2) and (3) are obtained from Lemma 6.2.5 and the definition of 7,.

x x

x

x

$6.4 S-iqjectivity of TA-covering maps In the previous chapter we have seen that if a TA-covering map of a compact locally connected metric space is s-injective, then the local product structure theorem is established. In this section we may find a sufficient condition for TA-covering maps to be s-injective.

x

Theorem 6.4.1. Let X be a compact metric space with metric d and let be topological space. Let p : + X be a covering map. If X is locally connected, and a constant 60 > 0 such that then there are a compatible metric for ( 1 ) for 0 < 6 5 bo and x E

x

x

x

is an isometry where U6(x) = {y E : Z ( X , ~< ) 6 ) and Ua(p(x)) = {Y E X : ~(P(x),Y)< 61, ( 2 ) all covering tmnsformations for p are isometries, (3) is a complete metric space with respect to 2.

x

Proof. Let {Ui) be a finite cover of X by connected open sets such that each Ui is evenly covered by p. Then p-l(Ui) is decomposed into the disjoint union UxEAUi,x of connected open sets in such that each restriction p :Ui,x -+ Ui is a homeomorphism. Let T > 0 be a Lebesgue number of {Ui), and choose a finite cover {V,) of X by connected open sets such that the diameter of is contained in some Ui, the inverse image each Vj is less than r/2. Since p-l(Y) also splits into disjoint connected open sets Vj,x, X E A. Let 60 > 0 be a Lebesgue number of (5). To define the metric for for x, y E X put

x

x,

~ ( x , Y=)

x, y E VjlAfor some Vj,x, otherwise.

CHAPTER 6

190

fact the following is easily checked: z = y if and only if z(x, y) = 0, and d(x, y) = 2(y,x) holds. The triangle inequality is proved as follows. Suppose d(a, z) d(z, y) < 60. Then d(p(x),p(z)) d(p(z),p(y)) < 60 and hence d(p(x),p(y)) < 60. It suffices to show that d(p(x),p(y)) = z(x, y). Since d(x,z) < 60, by definition there are Vj,x such that z, z E vj,x. Similarly, z and y are points in some V ~ I , ~Since I. In -

+

+

it follows that we - have &UVjt c Ui for some Ui E {Ui). Since z E Vj,xnVj~,x~, Vj,x U V , I , ~ is I contained in some Therefore, 3, y E Gitx. On the other hand, since d(~(x),p(y))< 60, clearly p(x) and p(y) are points in some & I # , and then &#I c Ui. This implies z, y E rjtt,xtt. Thus, a(x, y) = d(p(z),p(y)). By the definition of 2, the conditions (2) and (3) are easily proved. To show (I), let 0 < 6 5 60. It is clear that p(U6(x)) C U6(p(x)). Let d(p(x), y') < 6. Then p(a), y' E Vj for some Vj E {Vj). Hence there is y E p-'(y') such that x, y E Vj,x, and we have z(x, y) = d(p(x), y'). Therefore, P(U~(X))= Ua(p(x)). To see that p : U6(z) -t U6(p(z)) is injective, let z(x, z) = d(p(x), y') for ~I x and z, from which p(x), y' E some z E p-'(y'). Then some V ~ I ,contains p(vJ2) n P(Vj',Al). Hence p(Vj,x),p(vj~,x~) c Ui for some Ui E {Ui). Since ~ , is g;,x which contains vj,x and vjt,x1, and therefore x E Vj,x n v j t , ~there x, y, z E Gi,x. This implies y = z since p(y) = p(z). Therefore p : U6(x) + U6(p(x)) is bijective, hence an isometry.

vivx.

Let X be a metric space and d a metric for X. If for every E > 0 and x E X the open ball U,(x) of radius E centered at x is connected, we say that d is a connected metric for X . I t follows that X is connected and locally connected whenever it admits a connected metric. Conversely, the following holds.

Lemma 6.4.2. Let X be a compact metric space. If X is connected and locally connected, then there exists a connected metric for X . Proof. Let d' be a metric for X. Since X is path connected by Theorem 2.1.4, for x, y we define d(x, y) = inf{diam(u([O, I])) : u is a path in X from x to y) where diam(u([O, 11)) denotes the diameter of u([O, 11) with respect to d'. It is x = y, (2): d(x, y) = d(y,x) and (3): d(x, y) 5 clear that (1): d(x, y) = 0 d(x, z) d(z, y). Therefore d is a metric for X. We show that d is compatible with the topology of X. Since X is locally path connected (Theorem 2.1.4), for E > 0 we can choose 6 > 0 such that for x E X the path connected component of x in U,(x; d') contains U6(x; dl). Here U,(x; dl) = {y E X : dt(x, y) < E). Then, if d1(x,y) < 6 then d(x, y) < E. Since dt(x, y) 5 d(x, y) for x, y E X by definition, the metrics d and d' are equivalent. Connectivity of U,(x; d)

+

$6.4 S-injectivity of TA-covering maps

191

is checked as follows. If y E U,(x;d), by definition there is a path u in X joining x and y such that the diameter of u([O, 11) is less than E . This implies u([O, 11) C Ue(x; d). From the following theorem together with Theorem 5.2.1 we have that if f: X -+ X is a TA-covering map of a compact connected locally connected metric space and if X is semilocally 1-connected, then f has a local product structure.

Theorem 6.4.3. Let X be a compact connected locally connected metric space. Let f : X + X be a TA-covering map. If X is semilocally 1-connected, then f is s-injective. For the proof we prepare the following lemma.

x

Lemma 6.4.4. Let and X be metric space with metrics 2 and d respectively. Let 7 : X + W and f: X + X be continuous sujections and let p :X -+ X be a covering map. Suppose p o 7 = f o p and X is compact. If there exists 6 0 > 0 such that for each x E X and 0 < 6 5 60 the open ball U6(x) of x with radius S is connected and

is an isometry, then there is EO > 0 such that for 0 restriction P : W:(x) W,"(P(X)) is an isometry. Here

< E < EO

and x E

x the

+

7

x x

Proof. As we saw in the proof of Lemma 2.2.34, the map : -t is uniformly continuous. Hence we take 0 < EO < do such that if z(x, y) < €0 then Z ( ~ ( X ) , ? ( ~ )<) 6 0 . Let 0 < E < € 0 . If y E w:(x) and i 2 0, then we have E 2 ~ ((x), 7 T ( Y ) ) = d(f 0 P(x), f 0 P(Y)) and hence p(y) E W:(p(x)). To see that p,~;(=)is surjective, if y' E W:(p(x)) then - there is y E such that y' = p(y) and d(y, x) 5 E . Since E < € 0 , we have d(f ( x ) , ~ ( Y )< ) 6 0 and

By induction

CHAPTER 6

192

and so y E w : ( x ) . Since w : ( x ) The proof is complete.

c US0(x),it follows that p,w::(=)is injective.

Proof of Theorem 6.4.3. By Theorem 6.2.8 there is the universal covering p : -t X . Let d be a connected metric for X . Apply Theorem 6.4.1 to the and a constant covering map p : -1 X . Then there exist a metric ;ifor 60 > 0 such that the properties of Theorem 6.4.1 hold. Let 7 : + be a lift of f by p (see Theorem 6.3.9). Since f is a self-covering map, by Theorem 6.3.12 it follows that 7 is a homeomorphism. Take 0 < eo < 60 as in Lemma 6.4.4 and let 0 < eo < eo be a number with the property of Proposition 5.1.4. Notice that eo is an expansive constant for f . For x E X let y E w d ( x ) .Since w d ( x )is path connected by Theorem 5.1.3, there is a path w : [O, 11 + w d ( x )from x to y. Then by Lemma 5.1.11 for some < tk = 1 we have ~ ( [ t ~ , t i +C ~W]f ), ( w ( t i ) ) finite sequence to = 0 < tl < for 0 5 i 5 k - 1. Take b E pql(x) and choose a path u in starting at b such that p o u = w.Then, by Lemma 6.4.4 it follows that for 0 5 i 5 k - 1 the restriction p : w:,,(u(ti)) + Wedo(w(ti))is a homeomorphism, and so ti,ti+l])c w;,(u(ti)). Let E > 0 be a number such that ke < 60. By Lemma 2.4.1 there is n > 0 --n d such that f "(W,",( z ) ) C W , d ( f n ( z ) )for all z E X . Then, f ( W e O ( z t ) C ) 3 - d d 11 w : ( f n ( z l ) )for all z1 E In particular, f W e 0 ( u ( t i ) )C W,(f ( u ( t i ) ) )for 0 5 i 5 12- 1. Hence

x

x

x

x x

x

x.

-*

from which we have d ( f n ( x ) ,f n ( y ) ) = d(f ( u ( ~ ) ) , T ( u ( l ) ) On ) . the other hand, since 7 is a homeomorphism, f"(u(0)) # T ( u ( 1 ) ) . Therefore, f " ( x ) # f n ( y ) and so f ( x ) # f ( y ) . Since y is arbitrary in w d ( x ) ,it follows that the restriction f: w d ( x )+ wd(f( 2 ) )is injective. Thus f is s-injective.

Theorem 6.4.5. Let X be a compact connected locally connected metric space. Suppose X is semilocally 1-connected. Then ?rl ( X ) is finitely generated. Proof. Let {Ui) be a finite open cover of X such that each Ui is path connected and if Ui n U j # 0 for i and j then the inclusion Ui U U j 4 X induces the zero map on the fundamental groups. W e and fix p E X . Let xi E Ui for i and denote by ui a path from p to xi. If Ui n U j # 0, then we can choose a path ui,j in U; U U j from xi to xj. Then it is not difficult to show that the finite family of closed paths vi,j = ui ui,j i i j at a base point p generates the fundamental group.

.

-

$6.5 Structure groups for inverse limit systems

56.5 Structure groups for inverse limit systems

In this section we will introduce a transformation group for the inverse limit system of a self-covering map, and show that the transformation group possesses similar properties as covering transformation groups. For the case where the space is an n-torus, in the next chapter (Chapter 7) we shall discuss in more detail the geometric structure of the inverse limit systems of the selfcovering maps, by making use of the algebraic structure of solenoidal groups. We say that (Y, X, F,p) is a fiber bundle if the following conditions are satisfied: (1) Y,X and F are topological spaces, (2) p : Y -+ X is a continuous surjection, (3) there exists an open cover {Ux : X E A} of X such that for X E A there is a homeomorphism

satisfying p o cpx(x,y) = x. Here Y, X, F and p are called a total space, a base space, a fiber and a bundle projection respectively. And each homeomorphism cpx is called a coordinate function. For x E X we say that p-'(2) is the fiber over x, and write F, = p-'(x). Note that F, is homeomorphic to F. Indeed, let x E Ux, define px,,(y) = cpx(x,y) for y E F. Then cpx,, : F -, F, is a homeomorphism. If a fiber F has the discrete topology, then it follows that a fiber bundle (Y, X , F,p) means a covering map p : Y + X with the covering degree equal to the cardinal number of F. Let f : X -t X be a continuous surjection of a compact metric space and let o : Xf-+ X f be the inverse limit system. As before, we denote as po : X f -+ X the natural projection to the zero-th coordinate.

Theorem 6.5.1. Let X be a compact connected locally connected metric space. Suppose X is semilocally 1-connected. I f f : X + X is a self-covering map and the cove~ingdegree is greater than one, then ( X f , X, C,po) is a fiber bundle where C denotes the Cantor set. Proof. Let p : + X be the universal covering. Given x E X , we choose a canonical neighborhood U of x for p such that U is connected. Then for each n 0 the neighborhood U is evenly covered by fn. Indeed, let z E f -"(x) and take b, c E Si? such that p(b) = t and p(c) = x. Since U is evenly covered by p, we can find an open neighborhood V(c) of c and a homeomorphism h: U -+ V(c) such that p o h is the identity map idu on U. By Theorem 6.3.9 there is a lift jj : + of f n by p and J(b) = c. Since is a homeomorphism by Theorem 6.3.12, we put 1, = pojj-' o h. Then f n o kZ = idv, i.e. k, is a lift of idv by f n. Hence, letting U(z) = k,(U) we have that U(z) is an open neighborhood of z and k, : U -, U(z) is a

>

x x

194

CHAPTER 6

homeomorphiim. If U(z) n U(zl) # 0, then by Theorem 5.2.2 k, = k,,, and so z = 2. This implies that U is evenly covered by f n . For each point x = (xi) E Xf with xo = x, by the above result we can take a continuous map Sx : U t Xf such that Sx(x) = x and po o Sx = idv. It follows that Sx(U) n Sy(U) = 0 if x # y. Now, we define a map $ : U x prl(x) t p r l ( u ) by $(y,x) = S,(y). It is clear that $ is a bijection. If pi : Xf -t X is the projection to the i-th coordinate, then pi o $ is a map sending (y,x) to fi(y) if i 2 0, and sends (y, (xi)) to k,,(y) if i 0, where k,, is a lift of idv by f -i such that k,,(x) = xi. This implies the continuity of $. If the closure cl(U) of U is also evenly covered by p : + X, then in the same way, we obtain a continuous bijection from cl(U) x pr1(z) onto p i l ( c l ( ~ ) ) , which is a homeomorphism because cl(U) x p i 1 (x) is compact. Therefore $ is a homeomorphism whenever U is taken small. Since the covering degree of f is greater than one, it follows that p,'(x) is compact and totally disconnected, and has no isolated points. Combining this fact with Remark 2.2.38 we have that p,l(x) is homeomorphic to the Cantor set. The proof is complete.

<

Figure 26 Let X be a topological space and let O be a family of subsets of X. If each point of X has a neighborhood which intersects at most finite members belonging to 0, then O is said to be locally finite. We say that X is paracompact if for any open cover V of X there is a locally finite open cover U of X such that U is a refinement of V.

Remark 6.5.2. It is known that for a fiber bundle (Y, X, F,p) if a base space X is a paracompact Hausdorff space then the bundle projection p : Y + X is a fibration, that is, p : Y + X possesses the homotopy lifting property stated in Theorem 6.2.1. See Spanier [Sp]. Let X be a topological space and f: X + X a self-covering map. As usual, define the space of the inverse limit system as a subspace Xf = {(xi) E XZ : f (xi) = xi+l, i E Z) of the product topological space xZ.Then the

$6.5 Structure groups for inverse limit systems

195

natural projection po : X f + X satisfies the following properties (compare to Theorems 5.1.7, 5.2.2 and 6.2.1.): (1) Path lifting property : If u : [O, 11 + X is a path, then for x E X f with po(x) = u(0) there is a path v : [O, 11 + X f such that v(0) = x and po o v = u. (2) Uniqueness of lifting : Let Z be a connected topological space and let f: Z + X be a continuous map. If continuous map g,gl : Z + X f satisfy po o g = po o g1 = f and there is a point EO € Z such that g(z0) = gl(zo), then g = g'. (3) Homotopy lifting property : Let Z be a topological space and f : Z + X f a continuous map. If a continuous map F : Z x [O,1] + X satisfies F(E, 0) = po o f (z) for z E Z, then there is a continuous map G : Z x [O,l] + X f such that po o G = F and G(z, 0) = f (z) for E E Z. Indeed, given a path u : [O, 11 + X and a point x = (xi) E X f with po(x) = u(0), by Theorem 5.1.7, for each i 5 0 there is a path vi : [O, 11 + X starting at the point xi such that f-i o vi = u. Letting vi = f i o u for i > 0, we define v : [O, 11 + X f by v(t) = (vi(t)). Then v satisfies the conditions in (1). Similary, (2) and (3) are easily checked from Theorems 5.2.2 and 6.2.1 respectively. Let f : X + X be a self-covering map of a compact connected locally connected metric space. For x = (xi) e Xf we define a subgroup H,(x) of the fundametal group ?rl (X,xo) as follows: [u] E H,(x) if and only if for each i < 0 the lift of u by f -' starting at xi is a closed path from xi to xi. From Lemma 6.2.3 it follows that if f, : ?rl(X,xo) + ?rl(X,xl) is the induced homomorphism, then the restriction f, : H,(x) + H,(u(x)) is an isomorphism where u : X f + X f is the shift map. A homeomorphism a : X f + X f is called a transformation for the natural projection po : X f -, X if po o a = po holds. We denote as F(po) the set of all transformations for po. Then (1) the identity map of X f belongs to F(po), (2) if a1,az E F(Po), then a1 0 E F(Po), (3) if a E F(po), then a-' E F(po). Hence F(p0) is a group. In fact, F(p0) is a topological group equipped with the C0 topology. F(p0) is called the structure group for the inverse limit system u : X f+ X f .

Theorem 6.5.3. Let X be a compact connected locally connected metric space and let f: X + X be a self-covering map with the covering degree greater than one. Suppose X is semilocally 1-connected and H,((ci)) is nornaal in ?rl (X, co) for some (ci) E X f . If the factor group nl(X,co)/H,((ci)) is abelian, then the structure group F(po) for the inverse limit system u : X f + X f possesses the following properties: (1) for x = (xi) E X f the map a H Q(X) is a homeomorphism from F(po) onto the fiber Pi1 (xo), (2) the orbit space X /F(po) is homeomorphic to X ,

CHAPTER 6

( 3 ) there is a (homeomorphic) automorphism u , : F(po) + F(po) such that for every a E F(po) U O Q

= u,(a) o u ,

( 4 ) let p : X

+ X be the universal covering, G(p) the covering tmnsgroup for p, and for a lift 7 : + formation of f by p let f a : G(p) + G(p) be the induced homomorphism, then there is a homomorphism r, : G ( p ) -t F(po) such that r,(G(p)) is dense in F(po) and u* o 7, = r, o 7, holds, i.e. the following diagmm commutes :

x

Moreover for every ( x i ) E Xf the subgroup H,((xi)) possesses the following properties: ( 5 ) if xo = co then H w ( ( x i ) )= Hm((ci)), ( 6 ) if w is a path i n X from co to xo then w*(H,((x~))) = H,((ci)) where w, : nl ( X ,x o ) -+ nl ( X ,co) denotes the induced isomorphism in Lemma 6.1.4. For the proof we need some lemmas. Hereafter, let X be a compact connected locally connected metric space with metric d and let f : X -+ X-be a self-covering map. Suppose X is semilocally 1-connected and let p : X -+ X be the universal covering. We take and fix a metric 2 for and a constant 60 satisfying the properties of Theorem 6.4.1. Also, we denote as G ( p ) the covering transformation group for p and as O ( f ) the family of all lifts of f by p. Notice that every element in O ( f ) is a homeomorphism of (Theorem 6.3.12). A compact subset C of is called a compact covering domain for p if p(int(C)) = X where i n t ( C ) denotes the interior of C in X. Since X is compact, it follows that a compact covering domain for p always exists.

L e m m a 6.5.4. all x , y E

x

For

E

> 0 there is 6 > 0 such that for all 7j E O ( f ) and for

Proof. Let m be the covering degree of f . Fix 7 an element in O ( f ) . By Theorem 6.3.12 (2) there is a finite sequence al , ,am-1 in G(p) such that

..

$6.5 Structure groups for inverse limit systems

197

-

If C is a compact covering domain for p, then Co = C U ~ , '(c)u- - u a ; i l ( ~ ) is compact, and hence there is 6 > 0 such that for 2, y E Co

-

For z,y E X suppose a ( x , y) < 6. Since p(int(C)) = X, if 6 is small then a(x), a(y) E C for some a E G(p). Since a = a; o f , ( ~ )for some P E G(p), we have 7 , ( ~ ) ( z )?,(~)(y) , E Co. Hence, by Lemma 6.3.10 it follows that

and

(Theorem 6.3.9). Thus, For any 3 E O(f) there is 7 E G(p) such that 3 = 3(y)) = Z ( ~ ( X 7(y)) ), < E and Z(3-' (z), g-'(y)) = 07-1 (z), j-' o 7-'(y)) < E because 7 is an isometry. -

-2

X

a(?-'

Define a product set XZ= {(ui) : uj E X,i E Z) and a shift map T :XZ-, as usual by 3((u;)) = (u;+'). Write

It is - clear that ~ ( f f=) its

xfholds. Let u = (ui) E f f . For each i E Z denote

f ui,ui+l the element 7 in O(f) such that 7(u;) = u;+l and define

7"

Then for each i E Z, is a homeomorphism from X onto itself and by Lemma 6.5.4 it is bi-uniformly continuous with respect to the metric 2. Given 3 E O(f) we have i f u = a; o 2 for some a; E G(p) if i 2 0. If, in particular, f is a homeomorphism, then = a; o 3 for all i E Z where a; E G(p). We define a map rU: X -,X f by

7;

CHAPTER 6

198

By definition the following properties are easily checked:

where pi : X + X denotes the natural projection to the i-th coordinate. From now on, let us suppose that the covering degree of f is greater than one. By Theorem 6.5.1 it follows that ( X j , C , X , p o ) is a fiber bundle where C is the Cantor set. We note that a coordinate function cp : U x C 4 p ; ' ( ~ ) for ( X j ,C, X,po) exists whenever U is a connected open set of X with small diameter.

xj

Lemma 6.5.5. For u = ( u i ) E the following properties hold: ( 1 ) rU : X + X is continuo~s, ( 2 ) rU(X)is dense in X f , ( 3 ) r U ( X )is the path connected component of rU(u0)in X j , ( 4 ) if U is a connected open set of X and cp : U x C + p , ' ( ~ ) as a coordinate function for ( X j , C , X , p o ) ,then there is a dense subset C' of C such that cp(U x C') = p ; l ( ~ )n r U ( x ) . Proof. (1) is clear. To show (2),let ( x i ) E X j and let n > 0. Take a point y in p-l(x-,) and choose x E such that T i n ( z ) = y. Then pi o T,,(x) = si for i 2 -n. Since n is arbitrary, the point ( x i ) must belong to the closure of r U ( X )in X I . Thus ( 2 ) holds. Since X is path connected, by (1) it follows that r U ( X )is path connected. Conversely, let u be a path starting at ru(uO)in X f . Then po o u is a path in X starting at p(uo). Lift po o u by p to a path v with the initial point uo. Then ru o v is a path in X j starting at rU(uo).Since po o ru o v = po o u, by Theorem 5.2.2 we have T, o v = u. This means that r U ( X )contains the path connected component of rU(u0)in X j . Therefore ( 3 ) holds. Since U is path connected and C is totally disconnected, by ( 3 ) we have r U ( x )n p , l ( ~ ) = cp(U x C') for some subset C' of C , and by ( 2 ) it follows that C' is dense in C. Thus ( 4 ) holds.

x

xj.

Let u E We define a family Tu of subsets of r u ( X ) as follows: V E Tu if and only if there is a connected open set U of X such that V is expressed as V = cp(U x { a ) ) by a coordinate function cp : U x C -,p ; ' ( ~ )for ( X j ,C, X,po), where a is a point in C . I t is easily checked that (1) any point in r U ( x )belongs to some V E Tu, ( 2 ) if V',V2 E Tu and x E Vl n V2, then there is V3 E Tu such that XE

v3c v1nv2.

Therefore, the family Tu generates a topology of r U ( x ) .We call this topology the intrinsic topology of

rU(x).

$6.5 Structure groups for inverse limit systems

af

199

-

Lemma 6.5.6. For u E the map T, : X + T,(Z) and the restriction po : T,(Z) --r X are both covering maps under the intrinsic topology of %(XI, and the following diagram commutes.

Proof. By defintion it is clear that the diagram commutes. Let x E X and let U be a connected canonical neighborhood of x in X for p. By Lemma 6.5.5 (4) it follows that the U is evenly covered by po : T,(Z) -+ X if diam(U) is small, from which po : T,(X) + X is a covering map. Let V be an open set of r U ( Z ) such that po : V + U is a homeomorphism. Since U is evenly covered by po, by the above commutative diagram we see that V is evenly covered by T,, and so T, is a covering map.

Lemma 6.5.7. If po, : ?rl(~,(X), T,(x)) -+ nl(Xlp(x)) is the induced homomorphism of fundamental groups, then po, (TI(T,(X), T,(x))) = H ~ ( T ~ ( x ) ) . Proof. If [u] E Hoo(~,(x)),then by definition for each i 5 0 there is a closed path ui from pi(~,,(x)) to pi(~,(x)) such that f -i o ui = u. Letting ui = f i o u for i > 0, we have a closed path v = (ui) in T,(X) from T,(x) to T,(x). Then [v] E ?r1(rU(x),~,(x)). Since po o v = u, it follows that po,([v]) = [u]. Conversely, if [v] E ?rl(~,(X), T,(x)), then for each i 5 0, pi ov is a closed path in X from pi(7,(x)) to pi(7,(x)) and f-i o pi o v = po o v, and so po*([v]) = [POO w ] E Hoo(~u(x))Lemma 6.5.8. Let h, h' E F(p0). If h(x) = ht(x) for some x E Xf,then h = h'.

at

Proof. Choose points u = (ui) and v = (vi) in such that (p(ui)) = x and (p(vi)) = h(x). Then x E r u ( Z ) and h(x) E T,,(X). Since T,(X) is the path connected component of x in Xf by Lemma 6.5.5 (3), we have that ~(T,(X)) is path connected, and so h ( ~ , ( x ) ) c T,,(Z). Since h is a homeomorphism, it follows that ~ ( T , ( X ) )= T,,(X). Next, we show that h: T,(X) + T,,(X) is a homeomorphism under the intrinsic topologies. Let U be a connected open set of X and suppose diam(U) is small. Since h E F(po), it follows that h ( p i l ( ~ ) )= p , l ( ~ ) . Notice that p , l ( ~ ) is expressed as p i l ( ~ )= cp(U x C) by a coordinate function cp. If V = cp(U x {a)) c T,(X), then we have h(V) = cp(U x {a')) for some a' E C. This means that h: T,(Z) + T,,(X) is a homeomorphism. Since h(x) = ht(x), in the same way as above we have that ht(rU(X)) = T,,(X) and h' : T,(Z) -+ T,,(X) is a homeomorphism. Since pooh = po = pooh', we obtain h = h' (see Remark 6.5.2 (2)).

CHAPTER 6

200

af,

Lemma 6.5.9. For u E o(rU(x)) = r=(,)(X) and the restriction a : + T * ( ~ ) ( Xis) a homeomorphism under the intrinsic topologies. Furthermore the following diagram commutes:

rU(x)

Proof. Since a is a homeomorphism, in the same way as the first part in the proof of Lemma 6.5.8 we have a ( r U ( X ) )= r q U ) ( x ) . I t is clear that the diagram commutes. Let U be a connected open set of X with small diameter. Since pooa = f opo, clearly ( T ( ~ ; ' ( U ) )c p;'(f (27)). If (P : U x C + p ; ' ( ~ ) is a coordinate function and V = cp(U x { a ) ) for a E C, then u ( V ) = cpt(f(U)x {a')) for some coordinate function cp' : f ( U ) x C -, po(f ( U ) ) and some a' E C, because po o o ( V ) = f o po(V) = f(U). This implies the continuity of a : r U ( X ) ?(u) By the commutative diagram together with Lemma 6.5.6 the conclusion is obtained.

-

(x).

Now, we suppose that H,((ci)) is normal in n l ( X ,co) for some (ci) E X f , and choose b = (b;) E such that (p(bi))= (ci). Since q, : X + is a covering map by Lemma 6.5.6, we denote as G(q,) the covering transformation group for q,. By Lemma 6.5.6 it is clear that G(q,) is a subgroup of G(p). Apply Theorem 6.3.4 (2) to p : -+ X and q, : + q,(X). FYom Lemma 6.5.7 together with the assumption that H , ( ( s ) ) is normal in n l ( X , c o ) ,it follows that G(q,) is normal in G(p). Hence we can consider G(p)/G(q,) as the covering transformation group for po : -+ X (see Remark 6.3.6).

af

q,(x)

x

q,(x)

Lemma 6.5.10. Let x E X . Then G(p)/G(q,) is transitive on q,(X) n PO'(X). Proof. This follows from Theorem 6.3.4 (1). Moreover we suppose that ?rl(X,co)/H,((ci)) is an abelian group. Then G(p)/G(q,) is abelian. Lemma 6.5.11. G(p)/G(q,) is a uniformly equi-continuous family with respect to the compatible metrics for X f . Proof. We recall that X f has a metric

J defined by

56.5 Structure groups for inverse limit systems

Fix 6 > 0 small and suppose J((x~),(yi)) < 6 for (xi), (yi) E

201

%(x).

Then d(xo,yo) < 6. Since 6 is small, there is a connected canonical neighborhood U for p such that xo and yo belong to U. Notice that U is evenly covered by f -i for i 5 0 (see the proof of Theorem 6.5.1). Hence for each i 5 0 we can take ) X such that xi E U(xi) and f-i : :(xi) + U a connected open set U ( X ~of is a homeomorphism. Since d((xi),(yi)) < 6, it follows that for some large n(6) > 0 if i 2 -n(6) then yi E U(xi). Take x, y E such that (xi) = q,(x), (yi) = q,(y). We claim that for each a E G(p), if -n(6) 5 i 5 0 then f(b(a(x)) and o f.;(a(y)) are both points in the same component of fi(U). Indeed, let u be a path in from x to a(x). Then p o u is a closed path from xo to xo. If v denotes a path form x to y, then p o v is a path from xo to yo. Put w = p 6 v p o u p o v. Then w is a closed path from yo to yo and its lift stnrting at y by p has a(y) as the terminal point. Let xi = ffb(a(x)) and y: = fb(a(y)). Then p(x{) is the terminal point of the lift of po u starting at xi by f-' and p(y:) is that of the lift of w starting at yi by f-i. Since G(p)/G(q,) is abelian by assumption, this implies that p(x{) and p(y:) are both points in same component of f i ( u ) whenever yi E U(xi). Therefore the above claim holds. Let E > 0. By the above result it follows that d(pofb(a(x)),pof(b(a(y))) < E for -n(6) 5 i 5 n(6) if d(xo, yo) = d(p(x),p(y)) is small, and then d(%(a(z)), %(a(y))) < E because n(6) is taken large. Hence J ( a ' ( ( ~ ~ ) ) , a ' ( ( ~ i )= )) d(q,(a(x)), q,(a(y))) < E for a E G(p) whenever d ( ( ~ ~ ) , ( < ~ ~6 )and ) 6 is small. Thus G(p)/G(%) is uniformly equi-continuous.

x

x

.

.

%(x)

Since is dense in Xf by Lemma 6.5.5 (2), we have by Lemma 6.5.11 that each a' E G(p)/G(q,) is extended to a continuous map of X f , written as %,(a). From the fact that G(p)/G(a) is a group, it follows that %,(a) is a homeomorphism. Since po o a' = po holds, clearly %.(a) is a member in the structure group F(po).

Lemma 6.5.12. q,, : G(p) -t F(po) is a homomorphism. Proof. This is clear from definition. We denote as %(X f ) the set of all homeomorphiims of X f . Then a metric d for 'H(Xf) is defined by d(h, h') = max{sup{d(h(x), h1(x)) : x E Xf),

'H(Xf) is a complete metric space. Since q,,(G(p)) is a uniformly equicontinuous subgroup by Lemma 6.5.11, it follows that q,,(G(p)) is relatively compact in 'H(Xf), that is, the closure F of q,,(G(p)) is compact. Notice that each element in F belongs to F(p0).

CHAPTER 6

202

.

Proof of Theorem 6.5.3. Let x = ( x i ) E X f Since q,,(G(p))x is dense in p,l(xo) by Lemmas 6.5.5 ( 4 ) and 6.5.10, we have F x = pr'(xo). Here F x = { h ( x ) : h E F ) . Hence by Lemma 6.5.7, F = F(po) and therefore ( 1 ) holds. To show (2), define g: X / F ( p o ) -+ X by g(F(p0)x)= po(x). By Lemma 6.3.1 it is clear that g is continuous. Since g is bijective, ( 2 ) holds. From ( 1 ) together with Lemma 6.5.7 we obtain (5). To see ( 6 ) , let w be a path in X from co to 20. We can take a path iij in X f starting at the point c , such that po o Ti7 = w (see Remark 6.5.2 (1)). Since c = q,(bo) E the path Ti7 is in from which an isomorphism Ti7, : n l ( q , ( x ) , E ( l ) )-+ r1(q,('jli),C ) is induced as in Lemma 6.1.4. Project this isomorphism by po and use Lemma 6.5.7. Then w,(Hm(Ti7(1))) = H,(c). Since po(G(1)) = 30, by ( 1 ) and Lemma 6.5.7 we-have - H,(Z(l)) = H,((xo)). Therefore ( 6 ) holds. To show (3) and (4),let f : X -+ 'jlibe a lift off by p. Since nl ( X ,xo)/Hm( ( z i ) ) is abelian for all - ( x i ) E Xf by ( 6 ) , we may suppose 7; = (Vi E Z). Then q, = rz(bl on X , and hence q,, = ra(b),: G(p) + G(p)/G(q,). Since the shlft map a satisfies po o a = f o po, we have the restriction a : p,l(co) + p;'(f (co)). We calculate for a E G(p)

q,(x),

q,(x),

fi

Since

by uniquenss of lifting (Remark 6.5.2 ( 2 ) ) it follows that a o ~ b , ( a= ) q,+(f, ( a ) )o u for all a E G(p). Since a map from q,,(G(p)) to F(p0) defined by

is uniformly continuous, we have its extension a , : F ( p o )-+ F(po). Therefore ( 4 ) holds. Let $J, : F(p0) -+ p;'(co) be defined by & ( h ) = h(c). By ( I ) , $J., is a homeomorphism. Similary, define $,(,) : F(po) -+ p;l(f(co)). Then we have on p;l(co) = 7Cl,(c) 0 a* 0 ff

$2,

$6.6 Lifting of local product structures

from which a, is a homeomorphism. Therefore (3) holds.

36.6 Lifting of local p r o d u c t s t r u c t u r e s

Let X be a compact connected locally connected metric space with metric d. In $85.2 and 6.4 we have shown that a TA-covering map f : X + X has a local product structure if X is semilocally 1-connected (in other words, X has the universal covering space). This section will discuss that the local product structure can be lifted on the universal covering space, and investigates some properties of the stable and unstable sets on the space. Furthermore, a characterization of special TA-covering maps may be found. Throughout this section we suppose that X is semilocally 1-connected. As before, denote as p : X -+ X the universal covering and as G(p) the covering transformation group for p, and take a metric 2 for X and a constant 6 0 > 0 satisfying the properties of Theorem 6.4.1. we define a Let f : X + X be a self-covering map. For x E and u E local stable set W:(x; u) and a local unstable set Wr(x; u ) by

xf

x

71

= pi o f If we fix an element 7 in O(f), then there is pi E G(p) such that (i 1 0). Hence z(~"(x),f:(y)) = ~ ( y ( x )f (, y ) ) for i 2 0, from which the local stable set w:(x; U) does not depend on the choice of u. For simplicity we write ~:(x)=W;(x;u) ( x ~ x a n d u ~ x ~ ) . I f f , : G(p) + G(p) is not isomorphic, then the local unstable set Wr(x; u) is not necessarily independent of u. Let EO > 0 be a number with EO < 6 0 such that the implication in Lemma 6.5.4 holds for E = 6 0 and d = E O . In the case where f is c-expansive, we take EO as an expansive constant for f because EO is small. We note that for 0 < E < EO the sets m:(x) and m r ( x ; u) are projected isometrically on X by p, because p : Ua,(x) + Ug,(p(x)) is an isometry for x E X. L e m m a 6.6.1. Let x E

x and u E xf. Then for 0 <

E

< EO

(1) p(w:(x)) = W,S(p(x)) and p : W;(x) -,W,S(p(x)) is an isometry, (2) p(W;(x; u)) = W:(T,(X)) and p : r ( x ; u ) + W:(T,(X)) is an isometrgl.

CHAPTER 6

204

Proof. ( 1 ) follows from Lemma 6.4.4. To show ( 2 ) , let y E C ( x ;u). Then for i _< 0

A(?,))

2 J ( T " ( ~ ) , T " ( Y )=) d(p 0 T"(x), P 0 = d ( ~0i ru(x),pi

0

~ U ( Y ) )

where pi denotes the projection from Xf to the i-th coordinate. Therefore po 0 r U ( y )= p(y) E W,U(rU(x)).Conversely, if z E W,U(rU(x)),then there is z = ( z i ) E Xf such that zo = z and d(zi,p 0 T " ( x ) ) 5 L for i 5 0. Since d(p(x),zo) = d(p(z),z ) I E < 60, we can find y E X such that p(y) = z and d(x, y) I E. Since 0 < E < € 0 ,

60

> J ( 7 , 1 ( 5 ) , 7 ; 1 ( ~ ) )= d(p 0 f ; l ( x ) , p = d(p-1

0

0

7;l(y))

ru(x),P-1

0~U(Y)).

Since f (p-1 o r U ( y ) )= p(y) = z and f ( p - ~ ( z )= ) z , it follows that p-l orU(y)=

- --1

;'

Hence d(p-1 0 r U (-x --2 )P-1 , 0 r U ( y ) )I E and so d ( f , ( x ) ,f ( Y ) ) I E. In the same way, we have d ( f u ( z ) , 7 i 2 ( y ) )5 E , and by induction it follows - -that d ( f u i ( x ) , 7 ; i ( y ) ) I E for i 2 0. Therefore y 6 r ( x ; u ) .

P-1 ( 2 ) .

af

L e m m a 6.6.2. Suppose f : X -+ X is c-expansive. Let u E and 0 < E < EO. Then for x , y E X,r ( x ;u ) n W : ( y ) consists of at most one point. Proof. If u # v E E ( x ;u)nW;(Y), then p(u), ~ ( vE)W:(ru(x)) n W , ~ ( P ( Y ) ) , and so p(u) = p(v) by c-expansivity. Since x , y E Ua,(x), we have u = v , a contradiction. From Lemma 6.6.2 we obtain that if f : X

--+

X is c-expansive then for

X,YEX - -4

d(f,(x),X(Y)) I

(6.7)

Eo,

vi E Z

*

2

= Y.

L e m m a 6.6.3. Suppose f : X --+ X is c-expansive. For 7 n, > 0 such that for u E and x E X

xf ( 1 ) x(w:,(x)) c W y ( fu ( x ) ) ,

0 there exists

--n

8

(2)

>

7,n (We,u ( x ;4)C W , ( f--nu --u

z-n(u))

(2);

for all n 2 n,. Proof. This follows from Lemmas 2.4.1 and 6.6.1. Fix u E Xr.Let x E set V ( x ;U ) by

X

-

and define a stable set W s ( x ;u ) and an unstable

X EX

( F i ( x ) , ~ ( -+y )o) as i -, oo},

w S ( x ; u )= {y E

:~

w

:2 ( 7 i ( z ) , 7 i ( y ) )-, as i

Y

( 2 ; ~=){y

o

--+

-oo).

56.6 Lifting of local product structures

205

Then we have for i E Z

Since w d ( x ;U ) is independent of u, we write

It follows that the families

are decompositions of

L e m m a 6.6.4.

x,and for i E Z

Suppose f: X

+X

is c-expansive. For 0 < e < € 0 and x E X

Proof. This is clear by Lemma 6.6.3.

T h e o r e m 6.6.6 (Lifting o f local product structure). If f : X -+ X is a TA-covering map and 0 < e < € 0 is a small number, then for x E and and a u E there is a connected open neighborhood E ( x ; u ) of x in continuous map Eu : N ( x ;u ) x N ( x ;u) + r ( x ;u) such that

=-T(Y;

(1) { ~ u ( Yz ), ) u ) n W : ( z ) for Y , z E E ( x ; u), (2) for-ylzl w E N ( x ;u ) ~ U ( Y , Y )=Y, au(Y,Eu(z,w))=-EU(Y1 w ) = EU(EU(Y1 w ) , ( 3 ) the restriction Eu : D ~ ( Xx )P ( x ;U ) + x ( x ; U ) is a homeomorphism where n d ( x )= W : ( x ) n r ( x ;U ) and s ( x ;U ) = r ( x ;u ) n E ( x ;u), ( 4 ) thereis a-constant p > 0 such that m ( x ; u ) > q ( x ) where K ( x ) = {Y :d ( x l ~ )5 P } , ( 5 ) fU(Dd(2)) C D d ( T u ( x ) )and f U ( r ( x ;u ) )3 3 ( 7 u ( x ) ; 7 7 ( ~ ) ) . Furthermore, i f f is expanding then ?;jS(x) = { x ) and P ( x ;U ) = N ( x ; u), and otherwise ( 6 ) D d ( x )2 { x } and s ( x ;u) 3 { x } .

CHAPTER 6

206

Proof. Since X is semilocally 1-connected, by Theorem 6.4.3 it follows that f is s-injective, and hence by Theorem 5.2.1, f has a local product structure satisfying the conditions stated in Theorem 5.2.1. Fix 0 < e < EO. For x E and u E let NTu(,) be as in Theorem 5.2.1. Since diam(NTu(z))< EO < 60, we define r ( x ; u ) as the inverse image of NTu(,) by the isometry p : Uao(x) + Uao(p(x)). Then, by Theorem 5.2.1 (2) we have that F i x ; u) is a connected open set. (4) follows from Theorem 5.2.1 (3). To show the existence of the map E,, let S,(,) : NTU(,) + X f be the continuous section satisfying the condition (B) of the local product structure. We claim that S,(,) (p(y)) = ru(y) for all y E r ( x ; u). Indeed, consider a map ru o T : N,,(x) -+ X f where T denotes the inverse map of p : p ( x ; u ) + NTU(,). Since S,(,)(p(x)) = ru(x) by definition, by Theorem 5.2.2 we have S,(,) = rUo T. This implies the above claim. Since for a, b E Nr,(p(x))

x

xf

[S,(z)(a),bl = W,"(S,(x)(a))

n W,"(b)

by Theorem 5.2.1 (I), from Lemma 6.6.2 it follows that v ( y ; U) n w:(z) consists - of only one point, say Zu(y, z), for y, z E r ( x ; u). Therefore a map 5, : N(x; u) x r ( x ; u) -+ p ( x ; u ) is obtained, so that (1) holds. The continuity By the condition (B) we have of Eu follows from those of [ , ] and S,(,). (2). (3) is obtained from (2) and the continuity of Eu. Notice that n a ( x ) and ----U D (x; u ) are mapped by p to Dd(x) and DU(rU(x))respectively. Hence (5) follows from Theorem 5.2.1 (4). The last statement is clear from the second part in Theorem 5.2.1. Suppose f : X -+ X is a TA-covering map and fix 0 < e < EO small. Under the notations of Theorem 6.6.5 we have the following Lemmas 6.6.6-6.6.12.

Lemma 6.6.6. Let p

ucXf

> 0 be as in Theorem

6.6.5(4). Then for x E

x and

-

~;(x)cD"(x)

U

~,(x;u)~DU(x;u).

Proof. This is easily checked from Lemma 5.2.4.

af.

x

Lemma 6.6.7. Let x E and u E Then w a ( x ) and w U ( x ;u ) are path connected Proof. Combining Lemmas 6.6.6 and 6.6.4, we have ~ ' ( x )=

-

wU(x;u) =

U7;i(u)(D i20 i

-a

UZ-,(,,(D

--u

i

(fu(2))). --i

(fu (x);~-~(u))).

i20 On the other hand, by Theorem 6.6.5 (3) it follows that Dd(x) and g ( x ; u) are path connected. Thus the conclusion is obtained.

56.6 Lifting of local product structures

L e m m a 6.6.8. For x E

and u E

207

xf,

( 1 ) p ( V ( x ;4 )= W U ( r u ( x ) ) , ( 2 ) p ( W 8 ( x ) )= w 8 ( p ( x ) )where I@(p(z)) is the stable set in strong sense, and p : W 8 ( x )+ w 8 ( p ( x ) )is bijective.

~(E(Z),T"(~))

Pmof. Let y E T ( x ;u). Since -+ 0 as i -+ -m, we have d(p; o r U ( x )pi o r u ( y ) ) -t 0 (i + -00). Thus p(y) E W U ( r u ( x ) )and so p ( V ( x ; u ) )c W U ( r u ( x ) ) . If y E W u ( r U ( x ) )by , definition we can find ( y ; ) E Xfsuch that yo = y and d(y;,pi o r u ( x ) ) -+ 0 (i -+ -00). Let p > 0 be as in Theorem 6.6.5 (4). Then by Lemma 6.6.6 we have

and so

w,"(uio r u ( x ) ) C DU(ui o r U ( x ) )for i 5 0. Hence there is i < 0 such u

-

i

that yi E DU(ui o r u ( x ) ) . Since p(D ( f ,(x);$(u))) = Du(ui o r u ( x ) ) ,we can u -i find jji E D ( f ,(x); $(u)) such that p(jji) = y;, and then

---i -,' fr:(u)(%)E f p c U ) ( D( f u ( x ) ; a i ( u ) ) )

c (W"(T"(x);z+(u))) c W - ( x ;u).

-'

Since p o f;:(,,:;,)(&)= f o p(&) = f -i(yi) (note i 5 0 ) and y = yo = f -i(yi), we have y E p ( w ( z ;u ) ) ,i.e. W U ( r U ( x )C) p ( w ( x ;u)).Therefore ( 1 ) holds. Let y E W 8 ( z ) . Since ~ ( T " ( z ) , 7 " ( ~+ ) ) 0 as i -+ m, it follows that d ( f i ( p ( x ) ) f, ( p ( y ) ) ) + 0 as i -t oo. Hence p ( W 8 ( x ) ) c W 8 ( p ( x ) ) . Since w 8 ( x )is path connected by Lemma 6.6.7, so is p ( w 8 ( x ) ) . F'rom these facts together with Theorem 5.1.3 ( 1 ) we have p ( W 8 ( z ) )C w 8 ( p ( z ) ) .Conversely, let z E w 8 ( p ( x ) ) .Since w 8 ( p ( x ) )is path connected, there is a path u from p ( z ) to z. By Lemm 5.1.11 we have that for some sequence 0 = to < tl < < tk = 1, ~ ( [ t ~ , t c~ W + :~( u]( t)j ) ) (0 I j 5 k - 1 ) where 0 < E < €0. If v is a path in X starting at x such that p o v = u , then by Lemma 6.6.1, ~ ( [ t ~ , t ~c+W~: (]v )( t j ) )for 0 5 j 5 k - 1. Hence ; d ( f f . ( u ( ~ ) ) , f f . ( v ( l-+ ) ) 0) (i -+ m ) by Lemma 6.6.3, and so v ( 1 ) E W 8 ( x ) . Since p(v(1)) = z , we have p ( ~ " ( x )>) w 8 ( p ( z ) ) .Therefore, p ( W 7 x ) ) = w 8 ( p ( x ) ) .Let y,z E W 8 ( x ) and y # z. Then there is i > 0 such that

...

from which p , ~ ' ( , )is injective. Thus ( 2 ) is obtained.

208

CHAPTER 6

af.

Lemma 6.6.9. Let u E For x , y E there are at most countable sets A c D d ( y ) and B c S ( y ;u ) such that ( 1 ) W ( y ;U ) n W d ( x )= E,,(D'(~) x B ) , ( 2 ) W ( y ;U ) n W ( X U );= &,(A x Dl(y; u)), ( 3 ) the connected component of s in X ( x ; u )n W d ( x ) coincides with 8 ( x ) and that of x in W ( x ;u ) n r ( x ;u ) coincides with r ( x ; u). Proof. Letting B = r ( y ;U )nwd( x ) , from Lemma 5.1.10 together with Lemmas 6.6.1 and 6.6.8 ( 2 ) we have that B is at most countable. On the other hand, if w : ( a ) n W d ( b )# 8 for a , b E W ,then w : ( a ) C W d ( b )(by Lemma 6.6.3). Hence, by Theorem 6.6.5 ( 1 ) and ( 2 ) we have

Therefore ( 1 ) holds. To show ( 2 ) , let A = r ( x ;u ) n n d ( y ) . Then A is at most countable. This is checked by following the proof of Lemma 5.1.10 since is uniformly continuous for each i 5 0 by Lemma 6.5.4. (The detail is left to the readers.) In the similar way as above, we obtain ( 2 ) . ( 3 ) follows from ( 1 ) and (2).

f:

--

Lemma 6.6.10. For x E X , w d ( x ) is the path connected component of x in P-l ( J + d ( ~ ( x ) ) ) . Proof. Let u be a path starting at x in p - ' ( ~ ~ ( p ( x ) ) )Then . p o u is a path in w d ( p ( z ) )with p o u ( 0 ) = p(x). This implies u ( 1 ) E w d ( x )(see the proof of Lemma 6.6.8 ( 2 ) ) . Thus the conclusion is obtained.

Lemma 6.6.11. Let G ( p ) be the group of all covering tmnsformations for p. Then ( 1 ) a ( W s ( x ) )= w d ( a ( x ) )for a E G ( p ) and x E (2) if f : X -t X is a special TA-covering map, for a E G ( p ) , u E and x E X the following properties hold: (2.a) T ( x ;U ) is the path connected component of x in p - l ( W U ( r U ( x ) ) ) , (2.b) a ( W ' ( x ; u ) ) = r ( a ( x ) ;u ) .

x,

Proof. ( 1 ) is clear. By Lemma 6.6.8 ( 1 ) we have

If we establish

u

P - ~ ( W " ( ~ U (= X)))

aEG(p)

V ( a ( x ) ;4,

xf

$6.6 Lifting of local product structures

209

then r ( a ( x ) ; u ) is the path connected component of a(x) in p-I (WU(ru (x))). This is checked as follows: since the fundamental group ?rl (X) is at most countable by Theorem 6.4.5, from Theorem 6.3.4 it follows that p-l(~~(x)) is at most countable. Hence, by Lemma 6.6.9 (2) for z 2. E there is at most a countable set A, c n d ( z ) such that Let u be a path in X starting at x such that u([O, 1)) is contained in p-l(WU( ~ ~ ( x ) ) )Then . u([O,tl]) C x ( x ; u) for some 0 < tl 5 1. Since A, is at most countable, it follows that u([O,tl]) c g ( x ; u ) c r ( x ; u ) . Hence, letting J = {t E [O,1] : u(t) E W ( x ; u)), we have [0,tl] c J. In the same way as above, it is easily checked that J is open in [0, 11. Since each W(Z;u) contains the pneighborhood of z (Theorem 6.6.5 (4)), J is closed in [0, 11. Thus J = [O, 11. Combining this fact with Lemma 6.6.7, we obtain that WY(y; U) is the path connected component of y in p-'(WU(ru(x))). On the other hand, since a ( r ( x ; u)) is path connected, we have a(w"(x; u)) c v ( a ( x ) ; u) and therefore a ( r ( x ; u)) = r ( a ( x ) ; u). We must clear away the above assumption to complete the proof of (2). uE and a E G(p). Since Notice that po o T ~ ( X=) p~ o ru(a(x)) for x E f :X X is a special TA-covering map, we have Wu(ru(a(x))) = WU(rU(x)). Since p ( r ( a ( x ) ; u)) = WU(~,(a(x)))for a E G(P) (by Lemma 6.6.8 (I)), we have p(W-(a(x); u)) = WU(ru(x))and hence

x, xf

Let y E p-I (wU(7,(x))). Then p(y) E WU(7,(x)) and so WU(rU(x))= WU(7,,(y)). By Lemma 6.6.8 (I), p ( W ( y ; u)) = WU(~,(y)), from which p ( v u ( y ; u)) = WU(~,(x)). Hence we have

and so a(%)E V ( y ; u) for some a E G(p). Therefore, y E F ( a ( x ) ; u).

Lemma 6.6.12. Suppose f is a special TA-covering map. Let u,u1 E Then for x E W U w (3; u) = W ( x ; u').

xj.

Proof. This follows from Lemma 6.6.11 (2.a). As above, let X be a compact connected locally connected semilocally 1connected metric space. As a characterization of special TA-maps we have then the following Theorem 6.6.13.

CHAPTER 6

210

Theorem 6.6.13. Let f : X -+ X be a TA-covering map. Then f : X -, X is special if and only if for E > 0 there is 6 > 0 such that for x , y E Xf with xo = y,, = x W z ( x )n B6(x) = W:(Y) n B6(x).

where B a ( x ) = {Y E X : d ( x , y ) 5 6 ) . Proof. e): This follows from Lemma 2.4.3. +): Letting e = E , we take p > 0 as in Lemma 6.6.6. By Lemma 6.6.3 u --n there is n > 0 such that 7 ; n (-W , ( a ) ) c W,(f, ( a ) , T n ( v ) )for all v E W f and a E Let z E p - l ( x ) and choose u,u' E X f such that r U ( z )= x and ~,,,(z) = y. Since f is special, by Lemma 6.6.12 it follows that v ( z ;U ) = U W ( z ;u'). Hence

x.

-

n

By the choice of n we have jj(DU(jj-'(z); a n ( u ) ) )c W]r(z;u ) where j j = 7; , and so ij(D (g ( z ) ;a n ( u ) ) )c X ( z ; u') n W u ( z ;u'). Since g ( s ( j j - I ( z ) ;an -U --1 ( u ) ) )is connected, we have jj(D (g ( z ) ;a n ( u ) ) ) C Z ( z ;u') by Lemma 6.6.9 (3), and g ( s ( 7 j - ' ( z ) ; a n ( u ) ) ) c W ; ( z ; u t ) . Notice that T ( z ; u ) C --U --I jj(D (g ( z ) ;a n ( u ) ) )for some 7 = y(e, n) > 0. Hence F ( z ;u ) C r ( z ;u'), and by Lemma 6.6.1 we obtain W,U(x)C W:(y). Similarly, W,U(y) C W:(x). Therefore the conclusion is obtained from Lemma 5.1.8. Theorem 6.6.14. Let f : X -+ X be a TA-covering map. For x E X f let a, : D S ( x o )x D u ( x ) -+ N, be as i n Theorem 5.2.1. Then for x E X there is at most countable subset B c D U ( x )such that

w S ( x )n N, = a,(Ds(xo) x B). Furthermore, i f f is special, then there is at most countable set A such that W u ( y )n N, = a,(A x D u ( x ) ) .

C

Ds(xo)

Proof. The first part follows from Lemma 5.1.10. Combining Lemma 6.6.9 ( 2 ) with Lemma 6.6.11 (2.a), the second part is easily checked. Theorem 6.6.15. Let Y and X be both semilocally 1-connected and let q : Y -+ X be a covering map. Let g: Y -+ Y and f : X -+ X be self-covering maps. Suppose f o q = q o g. Then (1) f is a TA-covering map i f and only i f so is g,

$6.7 TA-covering maps of closed topological manifolds

(2) f is a special TA-covering map if and only if so is g, (3) i f f is a TA-covering map then for y E Y the stable set w ~ ( in ~ )strong ). sense is the path connected component of y in q - ' ( ~ ~ ( q ( ~ ) )Furthermore, i f f is special, then for (yi) € Y, the unstable set WU((yi)) is the path connected component of yo in q-'(Wu((q(yi)))).

Proof. From Lemma 2.2.34 (3) and Theorem 2.3.14 together with Remark 2.2.35 we have (1) (use Theorem 6.4.1 and Lemma 6.4.2). (2) follows from Theorem 6.6.13. As Lemma 6.6.10 we have the first part of (3). The second part is easily checked by Lemma 6.6.11 (2.a).

$6.7 TA-covering maps of closed topological manifolds In the previous sections we have seen the behaviour of orbits of TA-covering maps. In this section we restrict ourselves to closed topological manifolds. Let M be a connected topological manifold without boundary and let 3be a family of subsets of M . We say that 3 is a generalized foliation on M if the following holds; (1)3 is a decomposition of M , (2) each L E 3,called a leaf, is path connected, (3) if x E M then there exist non-trivial connected subsets D,, K, with {x) = D, n K,, a connected open neighborhood N, of x, and a homeomorphism cp, : D, x K, -t N,, called a local coordinate at x, such that (a) ( ~ 4 x2) 9 = x, (b) PE(Y,X) = Y(YE Dx) and VX(X,z ) = ~ ( E zKx), (c) for each L E 3 there is an at most countable set B C K, such that N, n L = cp,(~, x B).

Figure 27 Let 3 be a generalized foliation on M. If D, and K, are manifolds for all x E M , then 3 is a foliation of class C0 in the usual sense.

212

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Remark 8.7.1. If M is two or three dimentional manifold without boundary then 3is a foliation of class C0 in the usual sense. (cf. Bing [Bin] and Wilder [Wll). For the case of d i m M = 2, this is checked as follows. Fix x E M and let cp : D x K --t N be a local coordinate at x. I t suffices to show that D and K are one dimensional manifolds. We show that this is true for D. The conclusion for K will be obtained in the same way. For simplicity, denote by I,p and If the intervals [0, 11, (0,l) and (0,l) respectively. Since D x K is a connected manifold, it is checked that D is arcwise connected. Indeed, let a, b E D. Since D x K is connected, there is a continuous map a : I 4 D such that a(0) = a and a(1) = b. By Theorem 2.1.5, a ( I ) is arcwise connected. Hence we can find an arc from a to b in a(I). Let AD (resp. AK) be defined as the set of all injective continuous maps from I to D (resp. K). For a E AD, a : F' -t a ( p ) is a homeomorphism since a is injective. Moreover a ( P ) is open in D. Indeed, take and fix P E AK. If (U, p) is a chart of D x K , then the map cp 0 (ff x P ) P P n ( a ,)-I(u) u2 u2 is injective and continuous. By Theorem 2.2.18 the image of cp o ( a x P ) is open in W2, and therefore a ( I o ) is open in D. Put VD = UuEAD a(IO). If VD = D, it follows that D is 1-manifold without boundary. For the case where VD # D, we show that D is homeomorphic to I or I1. Take and fix x 6 VD and define A', = {a E AD : a(0) = x). First we claim that there is the implication between a l ( I ) and a z ( I ) for f f l , f f 2 E A',. Indeed, assume that a l ( I ) @ az(I). Then we have that az(I) C a l ( I ) . For, let J = a l ( I ) \ aZ(I). Since al(0) = aZ(0) = x, J a l ( I ) and so a c l ( J ) $ I. Since J # 0, there is a boundary point c of a;l(J). Since J is open in a l ( I ) , a,'(J) is open in I. Hence c 6 a l l ( J ) and so al(c) 6 J. This implies that al(c) E a2(I). But a z ( I O )is open in D. Since c is a boundary point of a,'(J), we have that al(c) = a2(0) or al(c) = a2(l). If al(c) = az(O), then we must have c = 0 (since a l , a z E A',). In this case there is d E (O,1] such that al([O, dl) n az([O, 11) = {x). This shows that there is in D an arc of al(d) to a z ( l ) through the point x. By definition we have x E VD, thus a contradiction. Therefore al(c) = a z ( l ) . Since c is arbitrary in the boundary set of a,'(J), the boundary set of a l l ( J ) is a single ponit. Therefore a l l ( J ) is an open interval in I and a;'(J) 3 1, from which we see that a , ' ( ~ ) = (c, 11 and hence al([O, c]) c a2([0,11) and therefore a 2 ( I ) C a l ( I ) . We proved that { a ( I ) : a E Ab) is a totally ordered set under inclusion.

5

$6.7TA-covering maps of closed topological manifolds

On the other hand, since D is arcwise connected, we have

These two facts implies that D is homeomorphic to either I or I1. For the case of dim M = 3, the proof is done by making use of singular homology. For details, see Wilder [Wi]. If dim M 2 4, then leaves of generalized foliations are not necessarily locally euclidean. Indeed, there is a topological space X without manifold structure such that X x R is homeomorphic to R4. See Bing [Bin]. Let 3 be a generalized folitaion on M. If x E L for some L E 3,then we have D, c L by (b) and (c). For fixed L E 3 let O L be a family of subsets D of L such that there is an open subset 0 of M such that D is a connected component in 0 n L. Then the topology generated by OL is called the leaf topology of L. If D, is a s in (3) and x E L, then D, is open in L with respect to OL. The restriction of the topology on the leaf L to D, is consistent with that of the topology on M to D,. Hence, D is open in L if and only if for x E L, D n D, is open in D,. Since M is locally contractible and has a countable base, the topology on the leaf L has the following properties; (1) path connected, (2) locally contractible, and (3) second countabiiity. Let 3 and 3' be generalized foliations on M and let f: M -+ M be a homeomorphism satisfying f ( 3 ) = 3'. Then, for L E 3 we have that f lL : L 4 f (L) is a homeomorphism with respect to 01, and O f ( L ) .

Remark 6.7.2. Let 3 be a generalized foliation on M and let U be an open subset of M. Denote by L(x) the leaf through x of 3 and put

Then V is open in M. Indeed, for x E V we have L(x) n U # 0. Take w E L(x) n U. Then there is an arc in L(x) with end points x and w. By (3) there is a finite sequence {xi : 1 Ii 5 1) C L(x) such that D,, contains the arc, D,, c U, DXin DXi+,# 0 ( 1 5 i 5 1- I), and x E D,,. Let wi E DXin D,,+, for 1 i 5 1- 1 and let K,, (1 5 i 1) be a connected subset as in (3). Since ( P + , ( W ~ , Z=~w1 ) by (b), we have wl E U and hence there is an open set KL2 c K,, such that cp,, (wl, KL, ) c U. Put NL, = cp,,(D,, x K;,). Then NL, c N,, and L(z) nu # fl for z E N;,. Replace N;, by U and use the above technique. Then we have that there is Ni3 C Nx3bsuch that L(z) n N;, # 0 for z E N;,. Using induction, we have

u:=,

<

<

214

CHAPTER 6

L(B) n NL,-l # 0 for r E NL,. In fact, L(z) n U x E D,, C NL, C V, V is open in M.

# 0 for

t. E

NLt. Since

Let 3and 3' be generalized foliations on M. We say that 3is tmnsverse to 3' if, for x E M , there exist non-trivial connected subsets D,, DL with {x) = D, n DL, a connected open neighborhood N, of x in M (such a neighborhood N, is called a coordinate domain at x), and a homeomorphism $, : D, x DL -, N, (in particular called a canonical coordinate at x) such that (a) 4x(x,x) = 2, (b) $'x(Y,z) = Y(YE Dx)and $'z(z,z) = z(z E DL), (c) for any L E 3 there is an at most countable set B' c DL such that N, n L = $',(D, x Bt), (d) for any L' E 3' there is an at most countable set B c D, such that N, n L' = +,(B x DL). It is clear that if 3 is transverse to 3' then 3' is transverse to 3.

Figure 28 Let 3 and 3' be transverse generalized foliations on M and denote as L(x) and L1(x) the leaves through a point x in M respectively. Given N, a coordinate domain, for y E N, we define D(y) and D1(y) as the connected components of y in N, fl L(y) and N, fl L1(y) respectively. Then, for y, z E N, the intersection D1(y)n D ( z ) is a single point. Hence we can define a map

by (y, z) I+ D1(y) n D(z), and have then

If Nx and Nv are coordinate domains and if N, is a coordinate domain such that N, C N, n N,, then it follows that

$6.7 TA-covering maps of closed topological manifolds

215

Let p : N -t M be a covering map and 3 a generalized foliation on define a family 7 of subsets of N by

M. We

3 = {i/ : 3 L E 3 s.t. i/ is a path connected component of p-l(~)). Then this family is a generalized foliation on N. For ?;E F the image p(z) is an element in 3 and the restriction p : -t p(z) is a covering map under the leaf topologies. We say that 7is the lift of 3 by p. If 3' is another generalized foliation on M and it is transverse to 3,then it follows that the lift 71 by p is transverse to 7. The following is obtained from Theorems 5.1.3 and 6.6.14 together with Lemmas 6.6.7 and 6.6.8 (1).

z

Theorem 6.7.3. Let f : M -t M be a TA-covering map of a closed topological manifold. Suppose f is not eqanding. Then the family 3;= { w S ( x ) : x E M ) of all stable sets in strong sense is a generalized foliation on M . If f is special, then the family 3; = { W U ( x ) : x E M f )of all unstable sets is also a generalized foliation on M and the families 3;and 7 are transverse. Notice that the intrinsic topology for stable sets in strong sense is consistent with the leaf topology. From Lemmas 6.6.7 and 6.6.9 together with Lemmas 6.6.10 and 6.6.11 (2.a), we have the following theorem.

z

Theorem 6.7.4. Under the assumptions in Theorem 6.7.3, let p : +M be the universal covering and let be as in (6.3). For u E M f the families -s = { w ( x ;u) : x E are transverse 3 = {Ws(z) : z E and generalized foliations on The family Fs is the lift of 3; by p and if f is special, then is the lift of by p.

z)

a.

mf

z)

In Chapter 11 we will introduce the concept of orientablility for generalized foliations by making use of singular homology. And we will establish in Chapter 12 a fixed point index theorem for TA-covering maps of closed topological manifolds, which says that any iteration of a TA-covering map has the same fixed point index 1 or -1 at each fixed point if the family of the stable sets in strong sense and the closed topological manifold are both orientable. This will be useful in counting the number of fixed points of TA-covering maps.

Remark 6.7.5. For f : M -t M a TA-covering map the homeomorphism a, : Ds(xo)x DU(x) t N, in Theorem 5.2.1 is a local coodinate (or a canonical coordinate ) at xo for F; ( and 3;). Hence

where yNx is defined as above (see (6.8)). Similarly, the homeomorphism E, : Bs(x)x D U ( x ;U) 3 N ( x ;U ) of Theorem 6.6.5 is a canonical neighborhood at x for Fsand 7;.

CHAPTER 6

We denote subsets of Rn by

and let X be a topological space. Take and fix xo E X . In the family of maps f : I n -t X with f ( a I n ) = xo we introduce the operation of product f g by

In + X denote a homotopy satisfying ft(dIn) = xo between fo and gt : ( I n ,d l n ) -t ( X ,x o ) denote that between go and gl. Then f t gt gives a homotopy between fo . go and f l gl. Thus the product of homotopy classes { f ) and { g ) is defined by { f ) { g ) = { f g}, and so the set of homotopy classes, r n ( X ,xo), becomes a group under the operation. If X is path connected, then T,(X,XO) is isomorphic to r n ( X ,yo) for xo,yo E X . We say that n n ( X ,so)is the n-homotopy group, and write ?rn(X) = ?rn(X,xo)when X is path connected. If in particular n = 1, then r l ( X ) is the fundamental group. Let and

-

ft : fl,

.

-

Proposition 6.7.6. Let p : Y -t X be a covering map of topological spaces and let xo E X and yo E p-'(xo). If n 2 2, then the induced homomorphism p, : rn(Y,yo) + r n ( X ,xo) is an isomorphism. For the proof, see Spanier [Sp].

Remark 6.7.7. uE

Under the assumptions in Theorem 6.7.4, let x E % and

mf.Then we have

This is easily proved as follows. Choose p > 0 as in Theorem 6.6.5 (4). Then we can find a finite open cover {U} of M such that each U is homeomorphic to Rn and diam(U) < p. Then { V ) = { V : V is a connected component of p-l (U)) is an open cover of ;i?. In fact, each V is homeomorphic to Rn and diam(V) < p. Let 6 > 0 be a Lebesgue number of { V ) . For x E ii? and u E %if we have

-

w S ( x )=

(Jf ; : ( u ) ( s ( f : ( x ) ) ) i20

--i

-s

-4

and f,i(,)(D ( f u ( x ) ) )is open in w s ( x ) under the leaf topology. If y : -+ w S ( x )is a continuous map with y ( a I k ) = x , then there is i 2 0 such that

56.7 TA-covering maps of closed topological manifolds

and so

7" $1" c n 8 ( 7 ; ( x ) ) . 0

By Lemma 6.6.3

7" ? ( I k )c B I ( ~ " ( x ) ) 0

for sufficiently large i, and so 6.6.5 (4) we have

2o

c V for some V E {V}.

r ( ~ k )

By Theorem

v c X ( ~ ; ( X ) ; ai(u)). We identify V with Rn and define a homotopy Ft : ( I k ,,Ik)

+ (w8(z), x)

by

for all 0 5 t 5 1 and z E I k where

) Fo(z) is a constant. Thus denotes the projection. Then F l ( z ) = ~ ( zand ?rk(WS(x)) = 0. Similarly we have n k ( T ( x ;u ) )= 0. Let ?r be a group. We say that a path connected topological space X is of type K ( n , 1 ) if n1 ( X ) 2 ?r and n k ( X )= 0 for k # 1.

Proposition 6.7.8. Let N be a topological space of type K ( T , 1 ) and let M be a compact connected topological manifold. Let xo E M and yo E N . Then, given a homomorphism 4 : .lrl(M,x o ) + ?rl(N,yo), there exists a continuous map f : M + N with f ( x o )= yo such that f , = 4. Conversely, i f f , g : M + N are continuous maps with f ( x O ) = g ( x O )= yo and if f,,g, : ?rl(M,xO)+ .lrl(N,Y O ) satisfies f,(a) = pg,(a)p-',Va E ?rl(M,xo)for some p E m ( N ,Y O ) , then f and g are homotopic. For the proof, see Spanier [Sp] and Kirby-Siebenman [Ki-S].

Remark 6.7.9. Let f,g : T n + T n be continuous maps of an n-torus and let f ( x o ) = g(xo) for some xo E Tn. Then f and g are homotopic if and only if f , = g, : ?rl ( T n ,xo) + ?rl ( F ,f ( ~ 0 ) ) .From this fact we have that if f : Tn + T n is a continuous map, then there is a unique toral endmorphism A : F + T n homotopic to f . Indeed, since the natural projection Rn + T n is a covering map and ?rk(Rn)= 0 for all 12, by Proposition 6.7.6 the torus T n is of type K(?r,1). Here ?r = Zn (Remark 6.3.8). Hence by Proposition 6.7.8 we obtain the conclusion.

218

CHAPTER 6

Remark 6.7.10. Let M be a compact topological manifold and let d be a metric for M. Then there is 6 > 0 such that for continuous maps f , g of M if d ( f ( x ) ,g ( x ) ) < 6 for all x E M then f and g are homotopic. Moreover, one can choose a homotopy F : M x [O,1]-+ M from f to g satisfying the property that d(f ( x ) ,F ( x , t ) )is small for all x E M and t E [O,1]whenever so is 6. In Chapter 10 we will show that M is an euclidean neighborhood retract, i.e. there exist an open set 0 of an euclidean space Rn,and continuous maps r : 0 -t M and i : M -, 0 such that r o i is the identity map of M . By making use of this fact, the above statement is easily checked as follows. Define a subset UofMxMby

+

U = { ( x ,y) : ( 1 - t ) i ( x ) t i ( y ) E 0 for all t E [O,l]). Then U is an open neighborhood of the diagonal in M x M. Since M is compact, we can find 6 > 0 such that { ( x ,y) : d ( x ,y ) < 6 ) c U , and define F : M x [ O , l ] -t M by

F ( x , t ) = r ( ( 1 - t)i o f ( x ) + ti o g ( x ) ) . It is easy to see that F satisfies all the desired properties. In the remainder of this section we give the proof of the following statement mentioned in Theorem 2.4.7 of Chapter 2.

Theorem 6.7.11. Let M be a closed topological manifold and f: M -, M a self-covering map, but not injective. If f is a TA-covering map and has topological stability in the class of self-covering maps, then f is expanding. Proof. Suppose that f : M 4 M fails to be expanding. Since f is a TAcovering map, there exists a periodic point po G M with period n > 0. Let /3 be a number such that x # y and f ( x ) = f ( y ) implies d ( x ,y) 2 p. Let e > 0 be an expansive constant for f and choose 6 > 0 in the definition of topological stability. For an n-cyclic sequence ( p i ) E Mf there exist pLl E f - { P - ~ and } a neighborhood U ( p L l ) such that diam(U(pL,)) < P / 2 and diam(f ( U ( p L l ) ) )< 6, and such that (pi) n U(p'_,) = 0 and is a homeomorphism. Since f is not expanding, we have W,"(p'_,) # { p L l ) by Theorem 5.2.1, and so 9-1 E W,d(pi1)- { P L ~ ) . Now construct a self-covering map g such that f = g on M - U ( p L l ) and g(q-1) = po . Obviously d( f , g ) 5 6. Thus there exists a continuous surjection h: M M such that h o g = f o h and d(h,i d ) e by topological stability in the class of self-covering maps. Since ( p i )n U ( p L l )= 0, we have

<

for i E Z , i.e. ( h ( p i ) )E Mf. Since d ( h ( ~ ~ 5) ,e ~for~ i) E Z , by c-expansivity we have ( h ( p i ) )= (pi). On the other hand,

56.8 Classification of TA-covering maps on tori

219

and f (p'_,) = po. Since d(h(q-l),p'_,) < P, we have h(q-,) = p'_, because is bijective. Next take 9-2 E f-'(9-1) fl ( M \ U(p'_,)) and put pL2 = h(q-2). Then f ( p L 2 )= h(q-1) = p'_, and d(pL2,9-2) < e. By induction we have sequences (qi),( p i ) E Mf with d(p:, qi) < e for i E 8. Since 9-1 E W,"(p'_,), we have p'_, = 9-1, thus contradicting.

56.8 Classification of TA-covering maps on tori In $5.4 we have seen that the class ?A of all TA-maps of a compact metric space splits into five subclasses

?A = S 7 A U 7 , STA 3 ?AX u PEM u SSTA. In order to discuss the classification of maps in ?A up to topological conjugacy, we restrict 7 A to the class of all TA-covering maps of an n-torus. We shall show that for a map in each subclass its dynamics is topologically conjugate to an algebraic endomorphism associated with the TA-covering map, which indicates a characteristic of the subclass.

Theorem 6.8.1. Let f: W + T n be a TA-covering map of an n-torus and denote as A : T n -t T n the tom1 endomorphism homotopic to f . Then A is hyperbolic. Furthermore the inverse limit system of ( W ,f ) is topologically conjugate to the inverse limit system of ( W ,A ) . Theorem 6.8.2. Let f and A be as in Theorem 6.8.1. Suppose f is special, i.e. f is a map in STA. Then the following statements hold : ( 1 ) if f belongs to 'TAX, then A is a hyperbolic automorphism and f is topologically conjugate to A , ( 2 ) i f f belongs to PEM, then A is an expanding endomorphism and f is topologically conjugate to A, ( 3 ) i f f belongs to SS'TA, then A is a hyperbolic endomorphism and f is topologically conjugate to A. Fkom the above theorems we obtain the following corollary.

Corollary 6.8.3. Let f and g be TA-covering maps of an n-torus W . Suppose f and g are homotopic. Then ( 1 ) the inverse limit system of ( T n ,f ) is topologically conjugate to the inverse limit system of ( W ,g).

CHAPTER 6

220

Furthermore the following statements hold: (2) if f is a map in TA'H, then so is g, and f and g are topologically conjugate, (3) if f is a map in P E M , then so is g, and f and g are topologically conjugate, (4) iff and g belong to S T A and f is a map in SSTA, then so is g, and f and g are topologically conjugate. From Theorem 6.8.1 we have (1) of Corollary 6.8.3. (2) and (3) follows from Theorem 6.8.2 (1) and (2) respectively. By Theorems 6.8.1 and 6.8.2 (3) together with Theorem 5.3.6, (4) is easily checked. The proofs of Theorems 6.8.1 and 6.8.2 are developed with the help of tools such as local product structure (this chapter), uniform local splitting theorem (Chapter 5), solenoidal groups (Chapter 7), fixed point formula (Chapter 10) and fixed point index (Chapters 10 and 12). Although we do not settle here, to avoid complication, R e m a r k 6.8.4. the case of TA-covering maps of infra-nil-manifolds, it seems likely that the methods and the results presented here could be pushed to give that case. R e m a r k 6.8.5. If f :Tn + W is a TA-covering map but not special, then there exist no toral homomorphisms to which f is topological conjugate. This is clear from the fact that every hyperbolic toral endomorphism is special. See Remark 2.4.2. R e m a r k 6.8.6. It remains a problem of whether Theorem 6.8.2 (3) is true for members belonging to S T A - (TAX U P E M U SSTA). In the remainder of this chapter we mention how to show Theorems 6.8.1 and 6.8.2. T h e outline of proof of Theorems 6.8.1 a n d 6.8.2. In order to capture orbit structures of TA-covering maps, we shall prepare the following two key points. The first point is to introduce orientability for generalized foliations. This is necessary to establish a fixed point index theorem for maps belonging to ?A - PEM. For maps belonging to P E M a fixed point index theorem is found by another technique. These will be useful in counting fixed points of TA-covering maps. The second point is to use solenoidal groups which are a natural extension of tori. In the proof of the theorems two the key points will be used as follows. Let M be a closed topological manifold and let f : M -, M be a TA-covering map. If is the universal - -covering space and we let p : + M denote the covering map, then a lift f : M -t is a biuniformly continuous homeomorphiim which possesses expansivity and POTP. If, in paricular, M is an n-torus Tn then the group of all covering transformations, G(p), is considered as the lattice Zn and there is a homomorphism : Zn + Zn such that

a

a

-

x

f(~+i)=X(i)+j(v)

(vER~,~Ez~).

56.8 Classification of TA-covering maps on tori

221

x,

In order to show hyperbolicity of we shall apply the properties of fixed point indices on euclidean neighborhood retracts, and Lefschetz's formula. As saw in 55.2, a neighborhood of each periodic point of f is uniformly decomposed into the s-direction and the u-direction (a uniformly local decomposition theorem). Since the family 3: of all stable sets in strong sense is a generalized foliation on M , we can define orientability for 3;and prove that the fixed point index at each periodic point of f is a constant whenever M and 3: are orientable (the secret of the proof is in the uniformly local decomposition theorem). If M or 3;is not orientable then we can find a double or double-double covering space of M on which the same result holds. Because the fixed point index at each periodic point off is a constant, the hyperbolicity of 2 is proved by using Lefschetz's formula. Since the distance between and 7 are Co-bounded on Rn, by a modification of topological stability of 2 we-have - that there exists a semi-conjugacy map h : Rn + Rn such that h o 3 = A o h and h is continuous and surjective. The proof of that h is injective will be done as follows. As h can not directly induce a map of Tn, we first find a solenoidal group S and a homeomorphism f : S + S which is topologically conjugate to the inverse limit system of , with (Tn, f), and next show that the non-wandering set of f , ~ ( f ) coincides S. This implies R(f) = Tn. By this fact we have that : Rn + Rn is injective. After that we can conclude that 5;: is biuniformly continuous with respect to the euclidean metric. Then induces a conjugacy map on S between j and the group antomorphism, from which Theorem 6.8.1 is obtained. If, in particular, f belongs to STA, then it follows that

for 1 E Zn, u E (Rn)f and x E Rn. By another property of 8, we have that a - v neighborhood of W ( x ; u) in Rn contains h(W (x; u)) for s E Rn. It may be shown that &(Zn v) c Zn h(v) (v E Rn) if f belongs to TAN U PEM U SSTA, and then 8 induces a conjugacy map on the torus between f and A, from which Theorem 6.8.2 is obtained. The proof of Theorems 6.8.1 and 6.8.2 will be precisely described in Chapter 8.

+

+

CHAPTER 7 Solenoidal Groups and Self-covering Maps

In the previous chapter we have investigated topological structures of the inverse limit systems of self-covering maps (Theorems 6.5.1 and 6.5.3). In this chapter we shall discuss the structure of solenoidal groups, and provide another approach to the study of dynamics of self-covering maps on tori. The results of this chapter axe useful to prove Theorem 6.8.1 mentioned in $6.8 of Chapter 6. $7.1 Geometrical structures of solenoidal groups In this section, some fundamental properties of topological groups and the geometrical structure of solenoidal groups are discussed. We shall deal with abelian groups except special cases. In the case the group operation will be written by addition. Let X be an abelian topological group. Let T be the one-dimensional circle group. A continuous homomorphism of X into T is called a character of X. The set of all characters of X is called the character group of X and is denoted as G. The addition in G is defined as follows: for g,gl E G, g+gl(x) = g(x) g'(x) for each x E X . The inverse -g of g is given by (-g)(x) = -g(x) for each z E X. It is elementary to show that G is an (additive) abelian group. The homomorphism which maps each element of X into the identity of T is called the identity of G and is denoted as 0. 0 is also called a zero-character. Let X be a compact abelian group and let G be its character group. Then G is discrete. If G is discrete then X is compact (Pontrjagin [Po]). Let X be a compact abelian group and G the character group of X. Let XI be the character group of G. For x E X define a map

+

Then a natural homomorphism w : X

+ XI

is defined by

and w(X) = XI. Under this map w we identify X with XI. Then x E X is a character of G defined by x(g) = g(x),g E G. This is a duality theorem due to Pontrjagin([Po]). For a subgroup GI of the character group G we can define a subgroup H = {x E X : g(x) = 0 for all g E GI} of X. Then H is called the annihilator of GI in X. We sometimes write H = ann(X, GI).

57.1 Geometrical structures of solenoidal groups

223

.

Let G be a countable discrete abelian group. A set of r points gl , -.,g, in G is said to be linearly independent if algl a,g, = 0 (al, ,a, E Z) implies a1 = . = a, = 0. An infinite set of points in G is linearly independent if all finite subsets are linearly independent. The maximal cardinal number of a linearly independent set of G is called the rank of G. If, in particular, G is a torsion group (i.e. for any x E G there is n > 0 such that nx = 0), then the rank of G is zero. If X is a compact metric abelian group, then its character group G is countable and the topological dimension of X is equal to the rank of G ([Po]). We say that S is a solenoidal group if S is compact, connected, finitedimensional and abelian. Thus, every finite-dimensional torus is clearly solenoidal. A compact metric space which is homeomorphic to a solenoidal group is called a classical solenoid. Denote as G the dual group (or the character group) of S. If S is n-dimensional then the rank of G is n. Thus there is 8 c G such that 13 = {el,.. ,en) is linearly independent, and so the factor group G/gp8 is a torsion group (the notation gpE means the subgroup generated by E). Hence every 0 # g E G is expressed as

+ -+

--

.

.

.

for some a # 0 and some a l l . . . ,a, with (al,.. ,a,) # (0,. . ,0). Since the existence of ( a l l a y . . ,a n l a ) is unique, we can define an injection cp : G + Qn by cp(g) = (allat .. ,anla). To simplify the notations, we identify g with (al/a,. . ,anla) under cp. Then we can consider 8 is the canonical base of Zn (i.e. el = (1,0,. ,O), . . . ,en = (0,. . ,0, I)), and so gp8 = Zn c G c Qn c Rn. For t = (tl , ,t,) E Rn define

.

-

.

.

...

..

+ . .+

(addition mod 1) $(t)g = t l a l / a t,a,/a for all g = ( a l l a y . . ,a n l a ) E G, i.e. $(t)g = (t, g) where ( , ) denotes the inner product. Then we obtain +(t) E S. In fact : Rn + S is a continuous homomorphism (Pontrjagin [Po]). The following lemmas will be used subsequently. L e m m a 7.1.1. +(Rn) is dense in S. If S is the torus then +(Rn) = S. Proof. From the definition of $ it follows that if g E G and $(t)g = 0 for all t E Rn then g = 0. This shows the first statement of the lemma. If S is an n-torus, then G/gp8 is finite (since gp8 = Zn). Let F = ann(S, gp8). Then F is finite since so is GIgp8. Thus

.

for some m

> 0.

+

Therefore, by the following Lemma 7.1.2 (2) we have

CHAPTER 7

224

L e m m a 7.1.2. Let F be the annihilator of gp6 in S. Then (1) F is totally disconnected and $-'{+(Rn) n F} = Zn, (2) S = $(Rn) F, (3) thew is a small open neighborhood V of 0 in Rn such that $(V)nF = (0) and the direct product V x F is homeomorphic to $(V) F, (4) Let V be as in (3). Then $(V) + F is an open neighborhood of 0 in S, (5) $(Rn) is the path connected component of 0 in S.

+

+

Proof. The dual group G/gp6 of F is a torsion group (i.e. for any x E G/gp6 there exists n > 0 such that nx = 0). Hence F is totally disconnected. The equality $-'{$(Itn) n F } = Zn follows from the definition of $. (1) was proved. Let ? r :~S --+ S/F be the natural projection. From (1) we have TF

o $(Rn) = ?

r o~ ${(tl,.

.., t n ) E Rn :0 5 ti 5 1,l 5 i 5 n ) ,

which implies that ?rFo$(Wn) is compact. Since $(Rn) is dense in S by Lemma 7.1.1, ? r o~$(Rn) = S/F and so

S = $(Rn)

+ F.

(2) was shown. To show (3), we take a bounded closed set

and define a map n:UxF+$(U)+F

+

by n(x, y) = +(x) y (x E U and y E F). Obviously n is a continuous surjection. Let n(u, x) = n(ul, 2') for some u, u' E U and some x , x' E F. Then $(u - u') E +(Rn) n F and so u - U' E Zn by (1). Since the distance from each component of u - U' to 0 is less than 213, we have u - u' = 0 and hence x = x', i.e. n is bijective. Since ? r o~ $ : Rn + S/F is an open map, $(U) F is a closed neighborhood of 0 in S. (4) follows from the fact that ? r o~ : Rn -+ S/F is an open map. To show (5), let u : [O, 11 + S be a path in S starting at 0 and put J = {t E [0,1] : u([O,t]) C $(Rn)). Let V be as in (3). Since F is totally disconnected, by (3) and (4) there is to > 0 such that u([O,tO])C $(V). Hence [O, to] C J. Let x E S and denote as cp, : S + S the translation defined by cp,(t.) = x z. Then cp,($(V) F ) is homeomorphic to V x F, which implies that $(Rn) f l cp,($(V) F ) is expressed as cp,($(V) C) for some C C F. From this fact it follows that J is open and closed in [0, 11. Therefore J = [O, 11. (5) was proved.

+

+

+

+

+

+

s7.1 Geometrical structures of solenoidal groups

Lemma 7.1.3. $(Zn) is a closed subgroup in $(Rn). Proof. By Lemma 7.1.2 (1) it is clear that $(Zn) C F. By Lemma 7.1.2 (2)

and so cl($(Zn))

n $(Rn) = $(Zn). Here cl(E) denotes the closure of E.

Lemma 7.1.4. $(Zn) is dense in F.

+

+

Proof. Write B = cl($(Zn)). Then SIB = {($(Rn) B)/B) {FIB). Since $(Rn)/$(Zn) is a factor group of Rn/Zn, it is a torus and so ($(Rn) B ) / B is also a torus because of

+

by the following lemma. On the other hand, SIB is connected, from which we have F = B.

Lemma 7.1.5. Let H be a normal subgroup of a topological group G and let M be any subgroup of G . Let cp(mH) = m(M n H ) for m E M . Then cp : M H / H -t M / ( M n H ) is an open map. Proof. Let A H be an open subset of M H I H . That means A H = cp(AH),A c M , and A H is relatively open in M H c G. But (p(AH) = {x(M n H ) : x E A). Since A(M n H ) = A H n M,A(M n H ) is relatively open in M. Therefore, by the definition of the quotient topology of M / ( M nH), (p(AH) = {x(M n H ) : x E A) is open in M / ( M n H).

Lemma 7.1.6. If S contains no torus subgroups, then 11, : Rn bijective.

+

$(Rn) is

Proof. If K is the kernel of $, then K is discrete and $ induces the continuous bijective homomorphism $' : Rn/ K -,$(Rn). If K # (0) then Rn/ K contains a torus T and so (0) # $'(T) c S. Since S has no torus, we arrived at a contradiction.

Lemma 7.1.7. The family of all open subgroups of F is a base of open neighborhoods of the identity 0 of F. Proof. Since F is totally disconnected, obviously F has a base of open closed neighborhoods of 0 in F. To obtain the conclusion it suffices to see that if U is an arbitrary open neighborhood of the identity 0 of F, then there is an open closed subgroup H such that H c U. Since 0 is the connected component of F, there is an open closed subset P such that 0 6 P c U. Define Q = {q E F : P q C P). Then H = Q n (-Q) is an open closed subgroup satisfying H C U. To see that Q is open in F take and fix q E Q. Let x E P be arbitrary. Since x + p E P and P is open, there are open neighborhoods U, and V, such that

+

CHAPTER 7

226

+

s E U,,q E V, and U, V, C P.Since {U,: x E P)is an open cover of P,we can find a finite cover {U,,,. . . ,U,,)in the cover. Set V = V,,n . n V,,. Then P V C P and q E V c P.This means that Q is open in F. To prove that Q is closed in F we show that F \ Q is open. Take r E F \ Q. Since P r $! P,there is p E P such that p r E F \ P.Since F \ P is open, we can find an open neighborhood W of r such that p W c F \ P. This implies that W C F \ Q.Thus F \ Q is open and so Q is closed. Since 0 E P,for y E Q we have y = y 0 E P and thus Q c P. Since P 0 = P C P,clearly 0 E Q. Therefore H = Q r l (-Q)is an open closed subset containing the identity 0. It remains to see that H is a subgroup. Take hl,h2 E H,then hl E Q and -h2 E 4! and so P (hl- h2) = (P hl) - h2 c P - h2 c P. This shows that hl - h2 E Q. Similarly we have -(hl- h2)= h2 - hl E Q. Therefore, hl - h2 E H and H is a subgroup of F.

.

+ +

+

+

+

+

+

+

+

Lemma 7.1.8. S = $(Rn) Fofor every open subgroup Foof F.

Proof.There is a subgroup Zt with finite index such that Fon$(Rn)= $(Z,"). Thus we have Rn= C Z t for some compact subset C of Rn,and so

+

is dense in S, we have S = +(W) Since $(Rn)

+ Fo.

Let d denote a translation invariant metric for S and d p denote the euclidean metric for Rn.For 6 > 0 put

Since F is totally disconnected and Fa(0)is symmetric (i.e. if x E F6(0) then -a: belongs to F6(0)), F6(0) contains an open subgroup of F. To avoid complication we may assume that F6(0) itself is an open subgroup of F. Let CYO > 0 be small enough. By Lemmas 7.1.2 (3) and 7.1.7

is an open neighborhood of 0 in S. Throughout this chapter, a0 is fixed. Define a function

+

K

+

by

=W, and put for s,y E S for x = $(v) y E $(Uao(0))Fa,(0)

do(.,

y)=

{ ir

- ')

forx-YEW for x - y @ W.

$7.1 Geometrical structures of solenoidal groups

227

Then do is equivalent to the original metric d for S and a translation invariant metric for S. If, in particular, v, v' E Rn and dRn (v, vt) < a0 then we have We sometimes adapt the metric do in stead of the metric d . For 0 the open neighborhood W6(0) with respect to do is expressed as

< 6 < cro

where U6(0) and F6(0) are open neighborhoods of Rn and F respectively. Clearly W6(0) is symmetric, and W6(0)n F is a symmetric open neighborhood of 0 in F. Remark that $(U6(0)) is not open in $(Rn). If Fl = F6(0) is a subgroup of F, by Lemma 7.1.2 (1) there is a subgroup Z? C Zn such that Fl

n

= $(Z;)

and then If Fl is open in F, then ZT( is of finite index. Let G ( c o n ) be a discrete countable abelian group and let 2 : G + G if G be an automorphism. Then G is said t o be finitely generated under contains a finite set (91,. ,gn) such that G = g P { ~ i ( g j ): i E Z, 1 j n). Especially G is said to be finitely generated if there is a finite set E such that G = gp(E). Let Sn be an n-solenoidal group and A : Sn + Sn an automorphism. As before, let (G,Z)be the dual of (S,, A) (i.e. G is the character group of Sn and (xg)(z) = g(Az) for all g E G and z E S,). Since Sn is an n-solenoidal group, there is a continuous homomorphism : Rn + Sn such that

x

< <

..

+

where F is a totally disconnected subgroup of S,. Obviously A o +(Rn) = $(Rn). As seen above, since $(s)g = (s,g) for s E Rn and g E G, we have (A 0 $(s))g = $(s)(&7) = (s,&) = (%, g ) (where denotes the transposed matrix of 2 )

tx

= ($ o tx(s))g Here

(S E Rn,g E G).

< , > denotes the inner product.

This shows that the diagram

I+

commutes.

CHAPTER 7

228

Theorem 7.1.9. A : Sn -t Sn is ezpansive if and only if all eigenvalues of

tx are 08the unit circle and G is finitely generated under x.

-

Proof. First we prove that A : S, -+ Sn is expansive when G = gp{A (g) : -00 < n < oo) and all the eigenvalues of 2 are off the unit circle. Since Sn is n-dimensional, we have rank(G) = n. Thus {g,?l(g), , --n-1 A (g)) is linearly independent in G. By Lemma 7.1.2 (3) there is a small open neighborhood V of 0 in Wn such that $(V) n F = (0) and the direct tx-n(~) product V x F is homeomorphic to $(V) F. Define Vl = and Fl = A-"(F). Then $(Vl) Fl is an open neighborhood of the identity in S,. Indeed, Vl is an open neighborhood of 0 in Wn. Since the character group of F/Fl is a finite group

...

n;=-,

+

n:=-l

+

+

we have that F/Fl is finite (see [Po]). Thus Fl is open in F. Since $(V) F is open in Sn (by Lemma 7.1.2 (4)), we have that $(Vl) Fl is open in Sn. Since all the eigenvalues of are off the unit circle and 2 is similar to the linear map : Rn + Wn is expansive by Lemma 2.2.33, from which we have

+

x

tx

m

Since Xz-,A gp{g,X(g),

i

---

...

--1

(g)) = G and F is the annihilator of gp{g,x(g), ,A ---n-1 ,A (g)) in Sn, we have

Since $(V) r3 F = (0) by Lemma 7.1.2 (3), it is easy to see that

r)rm

for all j 2 0. Thus A j ( $ ( ~ l ) +~1 ) = {0), which implies that A : Sn -+ Sn is expansive. From now on we prove the implication e).Since G is finitely generated ,gk) such that G = under A, by definition G contains a finite set {gl, E!=~G~where Gi

= g~{Aj(gi): -00

< j < oo),

1

< i < k.

$7.1 Geometrical structures of solenoidal groups

229

Let Ki denote the annihilator of Gi in Sn for 1 5 i 5 k. Then each of Gi is the character group of Sn/Ki. Thus the factor automorphism A : Sn/Ki -t Sn/Ki is expansive. Denote as pi the projection from Sn onto Sn/Ki for 1 5 i 5 k, and define u =p,'(~~)n...n~;~(u~) where each of Ui is an open neighborhood of the identity of Sn/Ki. Then we have m

n

A.(u)

= {O).

-00

c:=~

Ki = (0) because of Gi = G. This follows from the fact that Therefore A : Sn + Sn is expansive. +) : To see that all the eigenvalues of are off the unit circle choose a small neighborhood U of the identity in Sn. Then there is a neighborhood V of 0 in Rn such that +(V) c U and : V + U is injective (by Lemma 7.1.2). Thus m 00 m

'x

+

tx

and so,:)( 'Xn(V) = (0). This implies that : Rn + Rn is expansive. By Lemma 2.2.33 all the eigenvalues of are off the unit circle. To prove that G is finitely generated under it suffices to see that G/gp( U, 7i'(Zn)) is finite. Since F = ann(X, Zn) by Lemma 7.1.1, we have

'x

n>

Aj(F) = ann(X, g p ( U z xj(Zn))). If H = nYW A ~ ( F ) is finite, then G / g p ( U r m 7 i ' ( ~ n ) )is finite since it is the character group of H. To prove that H is finite take a small open neighborhood V of 0 in Rn. As above define V, = ()11 'x'(v) and Fl = ()11 Aj(F). Then +(Vl)+ Fl is open in Sn. Since A : Sn + Sn is expansive, we have

nrrnAj(F1) = nrmA ~ ( F )= H. Therefore G is finitely gener-

and so (0) = ated under A.

In the remaining part of this section, we give a simple example of a solenoid which admits a TA-homeomorphism (due to Smale).

Remark 7.1.10. A 1-solenoidal group S1 is constructed by the map a (a : z H z2) of the circle S1 = {z E @ : Izl = 1). Obviously a : S1 + S1 is expanding. Let D2 = {W E (C : IwI 5 1) and define a map f of the solid torus N = S1 x D2 into N by

CHAPTER 7

230

nr=,

Set A = fn(N), then A is 1-dimensional. Define the natural projection p : N -, S1 by p(a, w ) = a and a homeomorphism h: A -+ (S1)Nby h(x) = ( p ( f - i x ) ) ~ o . Then we have h(A) = S1 where S1 = {(a;) E ( s ' ) ~ : ,(,;+I) = ail, and the following diagram

A - Af

1.

commutes.

It is easily checked that ho f oh-' = a. Therefore, f is a TA- homeomorphism.

Figure 29

Remark 7.1.11. Solenoids introduced in Williams [Wi11,2,3] is defined as follows. We proceed in defining a 1-manifold K such that the following three types of coordinate neighborhoods are allowed: R, H = {x E R : x 2 0) and Y = {(x, y) E R : y = 0 or y = cp(x)) where cp : R -, R is a fixed Coo function such that cp(x) = 0 for x 5 0 and cp(x) > 0 for x > 0. The boundary aK is defined as usual to be points of K corresponding to 0 E H. The branch set B of K is the set of all points of K corresponding to (0,O) E Y. Obviously aK and B are finite if K is compact. A CT-structure for a branched 1-manifold is defind as usual. Since two the submanifolds containing a branch point b have the same tangent at b, the branched 1-manifold K has a tangent bundle TK. A differentiable map f: K -, K induces a map D f : TK TK. Then f is an immersion if D f is a monomorphism on the tangent space a t each point. Branched manifolds occur naturally in topological dynamics and they can be defined for all dimensions.

57.2 Inverse limit systems of self-covering maps on tori

231

If g: K + K is an immersion of a branched manifold, then g is an eqansion relative to a Riemannian metric on a tangent bundle TK if there are c > 0 and X > 1 such that JIDgnvlJ2 cAnlJvllfor v E TK and n 0. Let K be a compact branched 1-manifold with branch set B and let g: K + K be an immersion such that g is an expansion, all points of K are nonwandering under g and each point of K has a neighborhood whose image under g is an arc. Let Kgbe the inverse limit system of (K,g). Then Kgis called a solenoid. For a point a = (ao, a l , ) E Kfdefine h-'(a) = (al, az, al, ) as before. Then h: K -t K is a homeomorphism. The dynamics of h: K -+ K is discussed in [Will, 21 and the dynamical systems on solenoids for all dimensions were developed in [Wi13].

>

...

...

$7.2 Inverse limit systems of self-covering m a p s o n t o r i

Let Tn be an n-torus and let .~r: Rn + Tn be the natural projection. Let dRn denote the euclidean metric for Rn. Then dRn induces a metric d for Iln by d(x Zn, y Zn) = inf{dp(x d, y dl) : d, d' E Zn)

+

+

+

+

for x, y E Rn. We denote as G(T) the covering transformation group for ?r. Then G(T) is isomorphic t o Zn (see Remark 6.3.8). Throughout this section, we suppose that f : Tn -t 11'" is a self-covering map. Let 7 : Rn + Rn be a lift off by T . Then 7 : Rn -t Wn is a homeomorphism by Theorem 6.3.12 (1). Since G(T) Z Zn, the homomorphism 7* : G(T) -+ G(T) defined as (6.1) can be considered as a group homomorphism of Zn, in which case 7, : Zn -+ Zn can be written as an n by n matrix 2, all the entries of which are integers. - The matrix A can be considered as a linear map of Rn such that Zlzn = f,. Then by Lemma 6.3.10 we have

(7.4)

f(v

+ e) = A(e) + T(v)

for L E Z" and v E Rn.

Note that f: Iln -+ Iln a homeomorphism if and only if ?I(Zn) = Zn. Since f is a self-covering map in our case, we have 2 ( Z n ) c Zn and 2 ( Z n ) is of finite index (see Theorem 6.3.12 (2)). = ZF and every element h of ZF Put ZF = n r $ ( Z n ) . Then satisfies the property that there is a non-zero polynomial p(x) E Z[x], where p(x) is monic and its constant term is 1 or -1, such that p(2)k = 0. In this case there is a subgroup Zlf c Zn such that

-

A@;) CZ;,

~;n~;=o

and Z; $ Z t is of finite index. Indeed, let Qn denote the n-vector space over Q and let QF be the smallest subspace containing Then Qn is 2-invariant

q.

CHAPTER 7

232

and since z(Z,") = Z,", Q," is 2-invariant. If p(x) denotes the characteristic then p(x) is divided by the characteristic polynomial PI(%) polynomial of of and so p(x) is expressed as p(x) = pl(x)p2(x) for some p2(x) E Q[x]. Obviously Q; = {v E Qn :pl(2)v = 0).

x,

zlQ;,

We write

Qr = {VE Qn : p 2 ( X ) =~ 0).

Then Qn = QF $ Qlf and z ( ~ l f )= Qlf. Therefore x(Zlf) Q t n Zn, and Z," $ Zlf is of finite index.

c Zlf where Zlf =

nf

Remark that zJ(Z:) = (0). Let 'b' and 9, denote the tori with Z," and Z,: respectively, as the groups of all covering transformations. Then 9" = 9~x @ is a finite covering space of Tn. We denote as T' the finite covering map from 'k" onto P. Since 2($) = Z," and x(Zlf) c Zlf, 2 induces an automorphiim A, : @' + @' and an endomorphism A, : ?q + 9,. The following cases are considered: (i) Zn = Z," (Zlf = (0)) (ii) Zn = Zlf (Z> (7.5)

(iii) Zn

> Z:

(0))

$ Zlf

J

11." = @,

=$

Tn = 99,

(Z," # (0) and Zlf # (0))

* 9" is a finite

covering space of P with covering map T' : 'k" -,P. In the case (i) we have that f : P + P is a homeomorphism. Hence, for the discussion about the inverse limit system of ( P , f ) it suffices to consider the cases (ii) and (iii). From now on we deal with the cases (ii) and (iii) of (7.5), i.e. Zlf # (0). - Since 'k" has Z; x Z t as the group of all covering transformations and Alz; x is the group homomorphism with

xIzG

f: Tn -,T" induces a self-covering map f' : 'k" -t ?" such that the diagram

1 T"

f

j, T"

commutes.

$7.2 Inverse limit systems of self-covering maps on tori

tx

233

As above denote as the transposed matrix of 2. Then det(3) = det(tx). Since x(Z:) Z:, we have t x ( ~ : ) Z t and write

5

5

Obviously :SI(G) = G and G is a subgroup in Qq. Impose the discrete topology in G and denote as (S,, A) the dual of (G, :7i) (g(Ax) = (txg)(z), g E G and x E S,). Then S, is a q-solenoidal group with metric do defined as in (7.2) and A : S, -+ S, is an automorphiim. Thus we can find a continuous homomorphism $1 :Rq + Sqsuch that S, is expressed as

where Fq is the annihilator of Z t in S, (by Lemma 7.1.2 (2)). Then F, is the closure of $1 (Zt) by Lemma 7.1.4. It is clear that A o $1 (Rq) = $1 (Rq), and SO

for s E Rq and g E G. Thus we have

That S, contains no tori is checked as follows: Suppose that S, contains a torus and let T be the maximal torus of S,. Then A(T) = T and the character group of T is the factor group GIG' where G' is the annihilator of T (G' = ann(G,T)). Thus GIG' is finitely generated. Then there exist gl,. . ,gk E G such that G = K $ G' (direct sum) where K = gp{gl,. ,gk), from which Sq splits into the direct sum S, = T $ Si of the maximal torus T and a solenoidal group Si where T = ann(S,,G1) and S i = ann(S,, K). Obviously K is the character group of T and G' is that of S;. Since F, is totally disconnected, T n F, is finite and so T n LF, = (0) for some L > 0. Thus we have

..

.

If Z1 = ann(G,LFq), then Z t C Z1 and Z1/Zt is algebraically isomorphic to F,/CF,, from which mZ1 c Z i for some m > 0. Remark that Z1 is the character group of Sq/CF,. Let Zl,1 and Z1,2 denote the character groups of (T $ CF,)/LF, and (Si LF,)/LFq respectively (i.e. Zl,1 is the annihilator of

+

CHAPTER 7

234

(Si+LF,)/LF, and Zl,z is that of (T$LF,)/LF,). From the splitting of Sq/LFq it follows that Z1 = Zl,l 63 Z1,2. Since the automorphism 2 : S, + Sq induces an automorphism from (T $ LF,)/LF, onto itself, we have 2(ZlV2)= Zl,2 and so Z(rnZl,z) = rnZl,Z C Z i . oo -i This shows mZl,z c A (Z;) = (0)) thus contradicting. Therefore the homomorphism '$1 : Wq + S, is injective by Lemma 7.1.6. We define the homomorphism '$ : BP $ Bq + WP $41 (BQ)by

no

for up E

IWP and v,

E

Rq,

and put

-I

(7.9)

f ='$ofo'$-l

and x ' = ' $ o ? l o $ - ' ,

i.e. the diagrams

$1

1

commute.

Since ?l : WP $ WQ + BP $ WQ is expressed as the direct sum 2 = of the linear maps : Rp + WP and : Bq + Bq, we have

xlRp

xlwq

zlwP $21wq

x'

which shows that : +(Wn) -+ $(Rn) is continuous, because Al+l(Rq) = $1 0 Alwq 0 $1' by (7.7). Let dWPand dWqbe the euclidean metrics for RP and WQ respectively and define for (up, v,), (vk, vi) E BP x WQ (7.10)

-

d((vp, v,), (vi, vi)) = rnax{dwp(vp, vk),daq(~n,vb))-

Then, 2 is uniformly equivalent to dnn. A metric 2' for (7.11)

IWP $ S, is defined by

-I

d ((up,y), (vk, y')) = max{dwp(vp7vk),do(~, Y'))

for (up, y), (vk, y') E IWP x S, where do is the adapted metric defined as in (7.2) ) we have for S,. Since $ J ~ ( Bc~S,,

$7.2 Inverse limit systems of self-covering maps on tori

235

if dWq(vq,vL)< a0 (aOis a number chosen for S , as in (7.2)), and so (7.11) equals maxidm (v,, v;), dwq(v,, vb)) whenever d ~ ~ ( vv,t,) < 1 ~ 0 .Thus we have -I

d ( $ ( v ) ,$ ( w ) ) = Z(v, W )

if Z(V,W ) < a0

( v ,w E Rn).

Note that a base of neighborhoods of the identity of S, is

$l(U)

+ F,,x

:

U is an open neighborhood of 0 in Rg and FqVxis an open subgroup of F,

(see Lemma 7.1.2 (4)). From the existence of a subgroup Zyvxof Z: with finite index such that

the relative topology of $l(Rq) is given by the following base of neighborhoods of 0 in $1 (R'J):

$1

(U

+ Zlf,-+):

U is an open neighborhood of 0 in RQ and Z;,x is a subgroup such that $ I ( Z ; , ~ )= F,,x f7$(RQ) for some open subgroup F,,x of F

For Z;J a subgroup of Z t with finite index let Fq,x = ~ l ( $ ~ ( Z i , , ) )Then . it follows that F,,x n $1 ( R 9 )= $1 (Z;,,).

Lemma 7.2.1. ( 1 ) f l ( $ ( v + e)) = xl($(k!)) + f l ( $ ( v ) ) for v E Rn and P E Zn, ( 2 ) and 2' are -d -biunifomly continuous.

fl

Proof. (1) follows from (7.4). Since & , l ( ~ q ) = $1 0 Alw- 0 $1' by (7.7) and d : S, + S, is an autornorphism, we have 2'-biuniform continuity of X I . Since f : Rn -,Rn is 2-uniformly continuous, for every open sets U c RP and U 1 C IWq there are open sets V c IWP and V' c RQsuch that ?(V@V')c U@U1, and so

fl(v@ $1 (vl)) C u @ $1 (ul)-

Since 2;+,(p,) : $l(Rg) + d l ( R q ) is continuous, there is a subgroup Z:,2 of

Z: with finite index such that

and

x1

0 $1

@;,2

c $1 (Z;,l).

CHAPTER 7

236

Therefore we have -I

f

( V CB $ l ( V 1

+ ZY,,)) C U $

$1

(U'

+ Zlttl)

from which f' is d-uniformly continuous. The same result holds for the inverse of 7'.

x'

7

From Lemma 7.2.1 it follows that the maps and induce homeomorphisms of RP@Sq onto itself respectively. We denote them as the same symbols. Let p : RP $ S, + %"P $ S, be the natural projection defined by

Then we can define a homeomorphism j : %"P $ S, + TP $ Sq and a group automorphism A : %"P $ Sq + @' $ S, by

respectively. The following diagram illustrates the relationship among maps described above.

Lemma 7.2.2. For z E %"p $ S, and y E F,, j(z

+ y ) = A(y)+ f ( z ) ,

Proof. This follows from Lemma 7.2.1 (1) and continuity. 0 Since

j'

o $l(Zi) c

$1

( Z y ) , we have

A(F,) c F,. Thus a continuous map

: (%P $ Sq)/Fq+ (%"p $ S,)/F, is defined by

$7.2 Inverse limit systems of self-covering maps on tori

Since S, = $(Rq)

Thus a map

<:

237

+ Fq, it follows that

( f p @

Sq)/Fq+ 'b'@ f q is defined by

Then ( is continuous and bijective.

Proposition 7.2.3. The following diagram

commutes.

Proof. The commutativity is calculated as follows: for x E 'b and v E Rq, 0

j' 0 I-'(.

+v +Z t ) = t o f ( x +

+ = r ( j ( x + $ I ( v ) )+ F ~ ) = t ( j0 P(.

$1

( v ) F,)

+ 9 l ( ~+) )Fq)

(where a:

+ 9 1 ( v )= P(U + 9 1 ( v ) ) )

= t ( p 0 7' 0 $J(U

+ v ) + Fq)

=t(~O$Of(u+v)+Fq) = t ( 90 f ( u

=t((w

+

+ + Z: + Fq) V)

+ Z;) + 41(wt)+ Fq)

(if f (u v ) = w+wt where w E Rp and w' E IWP)

=(w+Z;)+w1+Z; -

= f(u+v)+iZpn@Z;

+

= f ' ( ( u Z;)

= fl(x

+ ( v + ZY))

+v +Zt).

Theorem 7.2.4. Let f : T n + T n be a self-covering map of an n-torus and A : W -+ Tn denote the endomorphism homotopic to f . Then the following properties hold :

CHAPTER 7

238

(1) There are an n-dimensional solenoidal group S and an automorphism + S such that (S,A) is topologically (algebraically) conjugate to the inverse limit system of (Tn, A). (2) There is a homeomorphism f of the solenoidal group S such that (S, j ) is topologically conjugate to the inverse limit system of (Tn, f ) .

2 :S

Proof. We define the inverse limit system (X, j")of ((Tp $ S,)/F,,

and .f"((f-"(~)

j')by

+ Fq)n20) = ( J - n + l ( ~ )+ Fg)n>o.

Then X is a closed subset of an infinite product topological space ((@' $ s,/F,)~, and : X -, X is a homeomorphism. To see the relation between the systems ('ifp $ S,, j ) and (X, j"),we let ((x) = (fn(x)

+ F,),?O

for x E

Pp@ S,.

Obviously ( : 'ifp$ S, + X is continuous and surjective. The injectivity of C is checked as follows. If ((x) = ( ( y ) then j-"(x) F, = j-n(y) F, for n 2 0, and hence jn(j-,(x) F,) = Fq), from which x - y E A"(F,) for n 2 0 (by using Lemma 7.2.2). , -i Since ~ ( Z Yc) Z; and A (ZF) = (01, we have xl($l(Zy)) c

+

+

no

+

p(j-n(y) +

no

oo -13

and A ($l(iL;)) = (0). Since F, is the closure of $l(Z;), cl(xl($l(Zy))) = A(F,) c F, and the following is easily checked:

obviously

Therefore, x = y. The following commutative diagram was obtained :

X

T fp @ f9

f"

f'

1

(: homeomorphism

X

1 1

inverse limit

.$: homeomorphism

VP $ q9

57.2 Inverse limit systems of self-covering maps on tori

239

From the diagram it follows that (@ @ S,, f ) is the inverse limit system of (Pn,f'). Therefore, ( 2 ) is concluded by the following Lemma 7.2.5. Letting f = A, we obtain (1) as a special case of (2).

Lemma 7.2.5. Let f : T n + T n be a self-covering map which is not a homeomorphism. Let n' : Pn t T n be a finite covering map and denote as G ( n l ) the covering tmnsformation group for n'. Then a self-covering map f' : @' t @' satisfying f on' = n' o f' induces a homomorphism f: : G(?rl)4 G ( n l ) (see j ' : ( ~ ( n ' )= ) { i d ) , then the inverse limit system of Remark 6.3.11). If ( T n ,f ) is topologically conjugate to the inverse limit system of (p, f'), i.e. letting

nj2,

X f = {(xi)i20 : xi E T n and f ( ~ i + l=) xi (i 2 O ) ) , X f ' = { ( f i)i>, : f i E 9" and f l ( f i+l) = f ; (i 1 O)),

P 1 ( f i )H( ~ ' ( f i ) ) , one has that @ : X j l

4

X f is a homeomorphism and the diagmm

01

18

commutes.

Here a' and u denote the shift maps induced by f' and f respectively. Proof. Since f o n' = n' o f', by definition we have the above commutative diagram. Thus it suffices to show only that /3 is a homeomorphism. The continuity of n' implies that of P. To show injectivity of P, for ( i i ) ,($;) E X I let ( n t ( f i )= ) (TI(&)). Then for i 2 0 there is ai E G ( n l )such that f i = a i ( f i ) (see Remark 6.3.6). Thus we have

njZo

f ' : ( ~ ( r '=) ){id), i.e. ai = id. and so ai = f$(ai+l),which implies ai E Therefore, (&) = (gi). This shows that P is injective. It is easy to see that /3(X;) is dense in X f . Since X ; is compact, it follows that /3 is surjective. Therefore P : X f , t X f is a homeomorphism.

Let C(Tn) be the set of all self-covering maps of 'IP'. Then C(Tn) becomes a metric space with respect to d defined by d ( f ,g) = m a x { d ( f( x ) , g ( x ) ): x E T n ) for f,g E C(Tn) where d(x,y) denotes the metric for 'IP' induced by the euclidean metric 2 for 8".

CHAPTER 7

240

As above, let f be a self-covering map in C(W) and denote as 7 : Bn + Rn a homeomorphism which is a lift of f by ?r. Let {fi) be a sequence of self-covering maps in C(Tn) and suppose fi converges uniformly to f under d. If i is sufficiently large, then each of fi is : Rn + Rn is a lift of fi by homotopic to f (see Remark 6.7.10). Hence, if ?r, then the following property holds:

Ti

Ti

converges uniformly Since d(fi, f ) + 0 as i 4 oo, we may suppose that to f under 2. Indeed, let F : Tn x [O, 11 + W be a homotopy from f to fi. By applying the homotopy lifting property (Theorem 6.2.1) we can find : Rn x [0,1] + Rn starting at f . Define : In+ Rn a homotopy as the terminal map of F. Obviously Ti is a lift of fi by ?r. Since ?r is a locally isometric covering map, it is easy to check that ~(f(x),fi(x)) = d(f-( ~-( x ) )fi(?r(x))) , for all x E Rn if i is large (see Remark 6.7.10). Thus d(fi, f ) =sup{~(Ti(x),f(x)) : x e R n ) + O asi-, oo. As in (7.9) define : W$+I (Rq) + RP$+1(Rq) by = +of i0rlt-l. Then Therefore each 7: induces a homeomorphism Lemma 7.2.1 holds for - - - from BP $ Sqonto RP$ Sq, which is denoted as the same symbol. Since d( f ,, f ) -+ 0 as i + 00,it follows that the sequence (7;) - - - of homeomorphisms of RP $ S, --I converges uniformly to f under b, where f 1denotes the homeomorphism of RP $ Sqinduced by f and d is a metric for RP $ Sq as in (7.11). Thus we have the following lemma.

Ti

3

7;.

Lemma 7.2.6. Let f :W + W be a self-covering map and suppose { fi) is a sequence of self-covering maps in C(Tn). If d(fi, f ) -+ 0 as i + oo, then there exists a sequence (7:) of homeomorphisms of IWP $ Sq such that 7; converges uniformly to f under 2'. Theorem 7.2.7. Let f and ifi) be as in Lemma 7.2.6. Then there exist a homeomorphism j of 'b $ Sq and a sequence {fi) of homeomorphisms of ?b$ Sq such that fi converges uniformly to j. Proof. As in (7.13) define homeomorphisms j and ji of Lemma 7.2.6 the conclusion is obtained.

'b $ Sq. Then

by

CHAPTER 8 TA-Covering Maps of Tori

In this chapter we shall restrict ourselves to tori and deal with the class ?A of all TA-covering maps. We recall that each map f in ?A belongs to the subclass 'TAX if it is injective, and ortherwise f is a member in one of the following three subclasses

PEM, SS'TA,

?A

\ (TAX U P E M U SSTA).

The purpose of this chapter is to show that for each subclass, a map in it is characterized, in a topological sense, as an algebraic endomorphism which indicates a characteristic of the subclass.

$8.1 Toral endomorphisms homotopic to TA-covering maps In this section we investigate some properties of TA-covering maps which are invariant under homotopy. To show the theorems stated below we need some results on fixed point indices of TA-covering maps, which will be established in Chapters 11 and 12.

Theorem 8.1.1. Let f : Tn + 'll"' be a self-covering map of an n-torus and A : Tn + 'I"' denote the tom1 endomorphism homotopic to f . I f f is a TAcovering map, then A is hyperbolic. Proof. When f is not expanding (i.e. f is a map in 'TA \ PEM), from Theorem 6.7.3 it follows that the family F; of all stable sets in strong sense is a generalized foliation on Tn.In this case we can say whether or not the family F; is orientable (this will be discussed in Chapter 10). If 3;is non-orientable, we use Proposition 10.7.2 and take a double covering space of 'll"'. Then a lift of f by the double covering map belongs to ?A \ P E M (see Lemma 2.2.34) and the lift of 3;is orientable. Hence we may suppose that 3;is orientable. Since Tnis orientable, we can use Theorem 10.9.1. When f belongs to P E M , we apply Theorem 10.8.1. In any case the following holds; there is P > 0 such that for each m L all fixed points of f m have the same fixed point index 1 or -1. Choose a positive integer mo with mo P such that fmo is topologically mixing on each elementary set, and write g = f m o . Obviously g: T n + 1P' is a TA-covering map and g is homotopic to AmO. It is enough to show that AmO is hyperbolic.

>

>

CHAPTER 8

242

By the result stated above together with the property of fixed point index (see Property 10.4.4 of Chapter 10) we have for m 2 1

where N ( g m ) is the number of fixed points of gm, I ( g m , z ) is the fixed point index of gm at 2; and I ( g m ) is the fixed point index of gm. Let Xi ( 1 i n ) denote the eigenvalues of the linear map of Bn which is a lift of AmO. Then from Lefschetz fixed point formula it follows that

< <

For the details, see 510.5 of Chapter 10. If e is an expansive constant for g, then there is E > 0 such that any &-pseudo orbit of g, ( x i ) , is e/3-traced by some point in (ll'"),. Since g is topologically mixing on an elementary set B , there is k > 0 such that g k ( ~ n) K t # 0 for any K , K' of a finite cover consisting of €12-balls in B. Let x E B be a fixed point of gm and choose y E B such that d ( x , y ) < E and d ( ~ , ~ " < ~ E. ) )Then we construct a one sided ( m k)-periodic &-pseudoorbit

+

+

which coincides with the one sided sequence ( z i ) r of a two sided ( m k)periodic €-pseudo orbit ( z ; ) in (Tn)'. Hence there is ( y i ) E ( T n ) , such that d(yi,~ i<)e/3 for all i E Z. By c-expansivity we have gm+"yo) = yo. Therefore N ( g m ) N(gm+k) for m 1. Note that each X i is not a root of unity. Indeed, this follows from the fact that Per(g) # 0 and N ( g m ) = 11 - XTI. To see 1A;I # 1 for 1 5 i 5 n, suppose [Ail = l ( 1 i 5 s ) , I X i ( < 1 (s+l 5 i 5 t) and 1X;I > 1 (t+l i n). Since N ( g m ) N(gm+" for m 1, we have

<

<

<

> nL1 >

< <

ny+l

Then the left hand side of (8.1) tends to IXf" ae~m m CQ. Since [ X i [ = 1 and X i is not a root of unity ( 1 i s ) , we can find a subsequence { m j ) such that X? + X f k as j 4 CQ. Therefore the right hand side of (8.1) tends to 0, thus contradicting.

< <

From Theorem 8.1.1 we have the following corollary.

$8.1 Toral endomorphisms homotopic to TA-covering maps

Corollary 8.1.2. If f:Tn

+

243

'ITn is a TA-covering map, then Fix(f) # 0.

Proof. This follows from the fact that the Lefschetz number L(f) is non-zero (by Theorem 8.1.1). For details, see $10.5 of Chapter 10.

Theorem 8.1.3 (Franks [Fl]). Let f : Tn + Tn be a self-covering map and let f : Rn + Rn be a lift off by the natural projection .rr : Rn + Tn. I f f is a TA-covering map, then 7 has exactly one jixed point. Proof. For the proof we need Theorems 10.8.1 and 10.9.1 which will be proved in Chapter 10. The propositions ensure the existence oft! > 0 such that for m 1 e each fixed point of f m : Tn + Tn has the same fixed point index 1 or - 1. Fix the natural number rn here. Let 3 : Rn + Rn denote the linear map which covers the toral endomor: Rn + Rn is hyperbolic by Theorem phism A homotopic to f . Since 8.1.1, it follows that -7irn has the single fixed point 0 and the fixed point index, I ( ~ , o ) equals , to f1 (see Remark 10.4.8 of Chapter 10). Since f m and Am are homotopic, the relation between and is expressed as

7

xm

r

from which we have M = s u p { ~ l r ( x-7 ) m(x)l(: x E Rn} < m. since -id is an isomorphism of Rn by hyperbolicity of we let p = inf{ll(?lm id)(x)ll : 11x11 = 1) > 0. Here id is the identity map and 11 11 denotes the euclidean norm of Rn. Choose r such that p r > M. If 11x11 > r, then

zm,

which implies that all fixed points of 7 are contained in the ball of radius T . Since is expansive, the fixed points must be isolated. Therefore ~ i x ( 7 ) is finite. Let us define kt : Rn + Rn,O t 1, by

7

< <

Then kt is a homotopy from f m to

Am and if

llxll

> T then

Hence, for any t, kt has no fixed points outside the ball of radius T . This shows I ( 7 , x ) (see Properties 10.4.4 and 10.4.5 that I(Zm) = I ( 7 ) = of Chapter 10).

xrEFilf

CHAPTER 8

244

7,

By the fact that f m o .lr = .lr o it is clear that n ( F i x ( 7 ) ) c Fix(fm). Let x E F i x ( 7 ) and choose an open neighborhood U, of x such that the restrictions TI(I=and T l ~ m ( U = )are both injective. Since the diagram

IT

commutes,

we have I ( 7 , x ) = I(fm,.lr(x)) (see Property 10.4.7 of Chapter lo), from which each a: E F i x ( 7 ) has the same index by Theorems 10.8.1 and 10.9.1 of Chapter 10. Hence

Therefore,

7 :Rn + Rn has exactly one fixed point and so does 7.

$8.2 Construction of semi-coqjugacy maps

In the previous section we have seen that toral endomorphisms homotopic to TA-covering maps are hyperbolic. In this section we discuss dynamics of hyperbolic linear automorphisms of the euclidean space Rn, and show the following Theorem 8.2.1. Let 2 be a metric for Rn induced from a norm of Rn. For continuous maps f and g of Rn we write

Notice that

a(f , g) is not necessarily finite.

Theorem 8.2.1. Let f: 'ITn -t 11'" be a self-covering map and A : lr" + W denote the tom1 endomorphism homotopic to f . Let 7 : Rn -, Rn be a lift of f by the natuml projection n : Rn + 'ITn and let 2 : Rn -, Rn be the linear automorphism which is a lift of A by 7 ~ . If A is hyperbolic, then there is a Rn such that the following properties unique continuous surjection : Rn hold: (1) 2 o h = E o 7 , (2) d(h, id) is finite, ( 3 ) E is uniformly continuous under 2, where id denotes the identity map of Rn. First we establish the following proposition to obtain Theorem 8.2.1.

$8.2 Construction of semi-conjugacy maps

245

Proposition 8.2.2. Let L : Rn + Rn be a hyperbolic linear automorphism and let T : Rn + Rn be a homoemorphism. ~f 2 ( ~T) , is finite, then there is a unique map 4 : Rn + Rn such that (1) L o q 5 = 4 o T , (2) a(4, id) is finite. Furthermore, for K > 0 there is a constant SK > 0 such that if Z(L, T) < K , then the above map 4 has the following properties : (3) a(+, id) < S K , (4) 4 is a continuous su jection, ( 5 ) 4 is uniformly continuous under 2 if so is T . For the proof we need the following Lemmas 8.2.3 and 8.2.4.

Lemma 8.2.3. Let L :Rn + Rn be a hyperbolic linear automorphism. Then the following properties hold : (1) for K > 0 and E > 0 there exists N > 0 such that if ~ ( L ~ ( xLi(y)) ), 5 K for all i with )i)5 N , then Z(x, y) 5 E , ( 2 ) for given K > 0, if ~ ( L ~ ( x~)' (, y ) )5 K for all i E Z , then x = y. Proof. Since L : Rn -t Rn is hyperbolic, there are L-invariant subspaces Em, EUand constants 0 < X < 1, c > 1 such that Rn = EU$ E d and (8.2)

((LYE.(1 5 cXn

( n2 0 )

I(LYEU(1 2

(Vn 5 0).

For simplicity, let L, = LIEU and L, = LIE.. Obviously L is linearly conjugate to L, x L,. - (1): Let K > 0 and E > 0. Choose N > 0 satisfying cKXN < E. If d ( ~ r ( x )~, $ ( y ) )5 K, then we have

Similarly, if Z(L;"(X), ~ ; ~ ( y )I ) K then ~ ( X , Y<) E. Hence, Lu x L, possesses the property of (1) with respect to a metric induced from a norm of Eu x Em.Since all norms of Rn are mutually equivalent, from the fact that L is linearly conjugate to L, x L, we obtain the conclusion of (1). (2) is clear from (1).

Lemma 8.2.4. Under the assumptions of Lemma 8.2.3, for K > 0 there exists bK > 0 such that for any K-pseudo orbit {xi : i E Z) of L there is a ) , 2 bK for i E Z. unique x E Rn so that ~ ( L ~ ( xXi) Proof. Let L, and L, be as in the proof of Lemma 8.2.3. Since L is linearly conjugate to L, x L,, it is enough to show the lemma for Lu because the analogous result holds for L,.

246

CHAPTER 8

Let c and X be numbers satisfying (8.2), and put 6~ = cKl(1- A). Let {xi : i > 0} be a K-pseudo orbit of L,. Then ((a&(( < K where ark = xk+l - Lu(xk) for k 2 0, and we have

Write g = L i l for simplicity. Then

(Jgkl(

( cXk and xk = L;(XO

+ vk) where

and so {ak} converges to some p in Eu. If y = xo + p then

for k 2 0. Applying this fact, we can easily prove the existence of a SK-tracing point for any K-pseudo orbit {xi : i E Z}.By Lemma 8.2.3 (2) its uniqueness is clear.

Proof of Proposition 8.2.2. Since ( L , T ) is finite, we can take K > 0 such , < K for all x E Rn.Let bK > 0 be as in Lemma 8.2.4. For that ~ ( L ( x )T(x)) any x E Rn the sequence { T ~ ( x ): j E Z) is a K-pseudo orbit of L. Hence there is a unique y E Rn such that

We define a map q5 : Rn + Rn by +(x) = y. Since 2(x, y) I tiK, obviously sup{2(q5(x), x) : x E Rn) 5 SK. Hence (2) holds. Since {Tj(T(x)) : j E Z} is SK-traced by a point L(y), we have L($(x)) = L(y) = q5(T(x)), from which (1) is obtained.

$8.2 Construction of semi-conjugacy maps

The uniqueness of 4 is easily checked as follows. If a map satisfies ( 1 ) and (2), then for v E Rn and i E Z

247

4' : Rn + Rn

Thus +(v) = #(v) by Lemma 8.2.3 (2). Next, we show the second statement. For K > 0 let 6~ be as in Lemma 8.2.4. If ~ ( L , T< ) K, then Z(4,id) < 6~ by the construction of 4. Hence ( 3 ) holds. To see ( 4 ) , suppose 4 is not continuous at some xo E Rn. Then there is a sequence xn + xo ( n + oo) such that yn = 4 ( x n ) does not tend to yo = +(xo) as n + oo. Since {x,) is &bounded and J ( 4 ( x n ) , x n ) 5 bK for n 2. 0 , the set {+(xn)) is also &bounded. Hence we can suppose, by taking a subsequence if necessary, yn + y; # yo as n + m. By Lemma 8.2.3 ( 2 ) we can find k E Z satisfying

Fix k and choose no > 0 such that J ( L k ( y n ) ,~ ~ ( ( y< )K)/ 4 for n 2 no. Since xn + x0 as n + m, we have ; ~ ( T ~ ( x ~ ) , T < ~ (SKx ~for ) ) n 2 no if no is sufficiently large. Then we have

and hence

which is a contradiction. By ( 2 ) the map 4 is extended to a continuous map on S n = Rn U {oo) by $ ( x ) = 4 ( x ) for 2 E Rn and &oo) = oo, and a homotopy ht between and the identity map is defined by

4

h t ( x ) = t+(x) Hence 4 :Rn

6 : Sn -t

+(1- t)x

( x E Rn)

and

4

h t ( m ) = m.

+ S n is surjective (see Remark 10.6.4 of Chapter 10) and so Rn is surjective. Therefore (4) was proved.

CHAPTER 8

248

To show (5), let 6~ > 0 be as above. For given E > 0, by Lemma 8.2.3 (1) ~ i with (il 5 N, then - there is N > 0 such that if 2(li(x), ~ ~ ( y<) 3) 6 for d(x, y) < E . Since T is uniformly continuous, we can take y > 0 satisfying the property that 2(Ti(x),Ti(y)) < SK (-N 5 i 5 N ) whenever Z(x, y) < y. If d(x, y) < y, then we have for i with Jil 5 N

which implies 2(4(x), +(y))

< E.

Therefore (5) holds.

Proof of Theorem 8.2.1. Since f and A are homotopic, we have

-

f(.

+ e) = X(e) + T(x)

for t' E Zn and a: E Rn

Hence, there is K > 0 such that ~ ( ~ ( x ) , ~ (<x K ) )for all x E Rn. Since 7 is uniformly continuous under 2 (see Lemma 6.5.4), the conclusion is obtained from Proposition 8.2.2. In the remainder of this section we discuss the case when a TA-covering map is expanding, i.e. it is a member in PEM.

Proposition 8.2.5 (Coven and Reddy [C-R]).Iff:Tn + 'IP is expanding, then there exist a constant X > 1 and a compatible metric D for Rn such that (1) is complete, (2) all covering tmnsfonnations for .~rare isometries under D, Y) for 2, Y E Itn. (3) D(f 7(Y)) 1

(XI,

Proof. Since f is positively expansive, by Theorem 2.2.10 there exists a compatible metric D for Tn and constants 6' > 0 and X > 1such that D(x, y) 6' implies D(f (x), f(y)) _> AD(x, y). Since .~r: Rn -+ Tn is a covering map, there exist a metric D for Rn and a constant 60 > 0 satisfying the properties in Theorem 6.4.1. For 6 = min{6',So), Lemma 6.5.4 ensures the existence of 0 < 61 < 6 such that ~ ( f ( x ) , T ( y ) ) < 61 implies D(X,y) < 6. Note that D(X,y) = D(x, y) since 6 5 60. Fkom these facts we have that b ( x , y) < / A if ~ ( (X),?(Y)) f < 61. For x, y E Rn let {xi : 0 i 5 t' 1) be a bl-chain from x to y (i.e. D(xi, x;+l) < S1 for 0 i 5 t ) and define by

<

<

<

+

$8.3 Nonwandering sets

249

where the infimum is taken over all finite 61-chains from x to y. By the triangle inequality of D we have D ( x , y) 2 D ( X , y), from which D is a metric for Rn. Clearly D ( X , y) = B ( x , y ) if D(x, y ) 5 61. Thus 'Zr is compatible and by Theorem 6.4.1 ( 3 ) , ( 1 ) holds. ( 2 ) is clear from the construction of IS together with Theorem 6.4.1 (2). It remains to show only (3). Let { x i : 0 i 5 m ) be a finite sequence from f ( x ) to f ( y ) with D ( X ~ , X ~ + ~ ) -- 1 < 61 for 0 5 i m - 1. Then {7-'(zo),.. ,f (x,)) is a finite sequence from x to y such that

<

for 0

<

.

< i 5 m - 1 and thus the sequence is a &-chain. Thus we have --

>

and therefore C ~ ( x i , x i + l ) XD(x, y), from which ~

(( ~f )

, f ( ~>)XD(x, ) y).

Lemma 8.2.6. Under the assumptions and notations of Proposition 8.2.5, given K > 0 there exists bK > 0 such that for any K-pseudo orbit { x i : i 0 ) of 7 there is a unique x E Rn so that ~ ( f . ( x x) i,) bK for i 0.

<

>

>

Proof. This is proved in the same technique as the proof of Lemma 8.2.4. In fact, put x: = fWi(xi) for i 0. By Proposition 8.2.5 we have

>

Thus (24) is a Cauchy sequence and so there is a point x in Rn such that x: -, x as i -+ oo. Fix i > 0 and let 0 5 j < i , then we have

x )6n ) for j 2 0. where bK = K/(X - 1). Therefore ~ ( x ~ , f j ( <

Proposition 8.2.7. Under the assumptions and notations of Proposition 8.2.5, there ezists a unique continuous surjection : Rn -,Rn such that (1) T o E = ItoX, ( 2 ) sup{D(k(x),x ) : x E R") is finite, ( 3 ) It is unifomly continuous under

n.

CHAPTER 8

250

--

Proof. The finiteness of s u p { ~ ( f(x),z(x)) : x E Rn)is followed from the fact t!) = X ( t ) 7(x) for L E Zn and x E Rn. The existence of that ?(x satisfying (1) and (2) is easily checked from Lemma 8.2.6. The proof of (3) and the uniqueness of K is similar to that of Proposition 8.2.2.

+

+

$8.3 Nonwandering sets

In Chapter 7 we have seen that the inverse limit system of a self-covering map of 'P is topologically conjugate to a homeomorphism of an n-dimensional solenoidal group S (see Theorem 7.2.4). In this section we shall establish the following Theorem 8.3.1, and show that for every map belonging to ?A the nonwandering set coincides with the entire space Tn. Throughout this section, let f : 'P + 'P be a self-covering map of an ntorus and A : 'P + Tn denote the toral endomorphism homotopic to f . Let f be a lift of f by the natural projection n : Rn -, Tn and A denote the linear map which is a lift of A by T. As seen in $7.2, these lifts induce a homeomorphism j : 'b@ Sq -, 'b' @ Sq of an n-solenoidal group and an automorphism 2 : f p $ Sq + f p $ Sq respectively (see (7.13) of $7.2).

Theorem 8.3.1. If A is hyperbolic, then there is a continuous surjection

i : 'b'$ Sq -+ f p @ Sq such that (1) A L o k = L o j, (2) each path connected component in 'b $ Sq is k-invariant. To show the above theorem, suppose A is hyperbolic. By Theorem 8.2.1 there is a semi-conjugacy map 'il : Rn + Rn between and 7.

Lemma 8.3.2. Let Z,"and Zt be as in (7.5) of 57.2. Then (1) for v E Rn and t! E Z,"

(2) for any X > 0 there exists a subgroup Zt,lc Zt,which is of finite index, and a continuous map i@: Rn x Z?,,+ Bx(0), where Bx(0) is a closed ball at 0 of mdius A, such that

-

h(v

+ 1) = P + 'il(v) + q(v, t )

for v E

Rn and P E ZiV1

Proof. (1) : Since f and A are homotopic, there is K > 0 such that 2(7(v), A(v)) < K for v E Rn. Hence, for t E Z," the sequence { ~ ' ( v t!) : j E Z) is a K-pseudo orbit of 51. Let bK be a number as in Lemma 8.2.4. Then there

-

+

$8.3 Nonwandering sets

251

+

is a unique w E Rn such that a y ' ( v i ! ) , ~ j ( w )5) bK for j E Z , and by the construction of X we have x ( v 1 ) = w. Also { ~ ' ( v :) j E Z } is a K-pseudo - -j orbit - of Thus d ( f ( v ) , ~ j ( w ' )5) hK ( j E Z ) for some w' E Pn, and hence h ( v ) = w'. Let d = w - 1. Since Z(Zb)) = Zb) by definition, we have 7 ( v + Zb)) =

+

x.

Zb)

+ 7( v )for j E Z. Thus z @ ( ~ )~, ' ( 6=) ;i(7(V ) + r),7 ( a+ 1 ) ) < bK

for all j E Z and so w' = 65 by Lemma 8.2.3 (2). Therefore

( 2 ) : Let X be a positive number. Since 2 is hyperbolic, by Lemma 8.2.3 (1) we can find N > 0 such that for v , w E Rn

--

d(~j(v),~j(w 5 )2) 6 (Vj ~ 2 -N)

(8.3)

Z(v, w ) < A.

Choose then a subgroup Z;,, of finite index such that j 2 - N . Obviously

x ~ ( z ;c~ Z) i

for

for j 2 - N . Let v E Rn and I E Zi,L Then { f ' ( v ) : j E Z ) and {Y'(v + I ) : j E Z } are both K-pseudo orbits of A. Thus, there are w', w E Rn such that -d ( f j ( v ) , ~ j ( w ' )5) bK and ~ ( f j ( v ! ) , 2 ( w ) ) 5 bK for j E Z , and X(v) = w' - -j and Z(v 1) = w hold. Put d = w - l. Then we have d ( A ( d ) x, J ( w ' ) )5 2bK for j 2 - N , and hence Z(d,w') < A by (8.3). Since d - w' depends on v and 1, letting ~ ( v , ! = ) d - w', a map TTj : Rn x Z;,, -+ B x ( 0 ) is defined, and so

+

+

from which the continuity of

is obtained.

Let 11, : RP $ Rq -,RP $ $1(Rg) be as in (7.8) of $7.2 and define

( ( u ) (, 1 ) ) = $ o ( u , )

for u E Rn and 1 E Z;,,

where : Rn x Z;,l + B x ( 0 ) is a continuous map of Lemma 8.3.2 (2). Since ~ o z = h o onRn,wehave f

where 2' and

7' denote maps defined as in (7.9).

CHAPTER 8

252

-

-

L e m m a 8.3.3. Let 2' be a metric for W $ Sq as in (7.11). Then XI : Rp $ Q1(Rg) RP d $1 ( R q ) and q1: $ ( R n ) x $1 ( q l ) $(BA(o))are both d uniformly continuous.

-

Proof. Let hK > 0 be a number such that z(E(v),v) < bK for v E Rn. Since -

A :RP $ Rq -+ RP $ Rq is hyperbolic, by Lemma 8.2.3 (1) for any sufficiently small q > 0 there is N 2 0 such that

-4

-I

By Lemma 7.2.1 ( 2 ) we have that f is d -biuniforrnly continuous. Let 0 q1 < q be a number to be determined later and take a > 0 such that

<

If F,,(O) denotes a closed neighborhood with radius 7' of 0 in F,, then there is $l (%,I) such that F,I ( 0 )n+(Rn) = $1 (Z,I) and $ J ( B (~0,) Z,,) is a closed neighborhood of 0 in RP $ $ J ~ ( R = ~+(RP ) $ R q ) (see Lemma 7.1.2). Since -d ( f j ( v ) ,2' 0 x ( v ) ) < bK for all v E Rn and j E Z , we have

+

and so for x, y E +(Rn)

1 { 1 ( ~ x) ) } If x(z, y)

+{

)- ( y ) }E ( B O ) )

< a , by (8.5) we have

and hence

from which -N

Let V be a bounded closed subset of Rn such that

( j E Z).

$8.3 Nonwandering sets

253

and choose here 7' > 0 small enough. In the same way as the proof of Lemma 7.1.2 (3), we can prove that $(V) x F,,r and $(V) F,I are homeomorphic. Thus $(V) x $ J ~ ( Z , Iand ) $ ( V ) $J~(Z,I)are also homeomorphic, from which we have

+

+

and by induction

+

and so K1(x) - K1(y) E $(B,,(O) Z,,). Therefore is ;i)-uniformly continuous. d -uniform continuity of p1follows from the fact that

-I

for x E $(Wn) and y E $(Z;,,).

Proof of Theorem 8.3.1. Since $ J ~ ( W Q is ) dense in S, by Lemma 7.1.1, by Lemma 8.3.3 it follows that Kt is extended to a continuous map of Rp $ S, onto itself, which is denoted as the same symbol. Then $(X+Z:) = 2Z,"+K1(x) for x E IWP $ Sq by Lemma 8.3.2 (I), and hence Kt induces a map of @' $ Sq onto itself. Let us denote the induced map as k, then by (8.4) we have

Then Therefore (1) holds. To show (2), let F,' be the closure of $l(Z;,,). by Lemma 8.3.3 the map p1 is extended to a continuous map, say pl, of (WP $ S,) x F,' into $(Bx(0)) where X > 0 is the number appeared in Lemma 8.3.2. By Lemma 8.3.2 we have

CHAPTER 8

254

from which there exists a continuous map @ : ( satisfying

k(x

(8.6)

+ y) = y + k(x) + @(x,y)

9$~S,) X F,'

for x E

-t

p 0 $(Bx(O))

'kP$ S, and y E F:

where p : IWP $ S, -t TP$ S, is the natural projection. By Lemma 7.1.2 (5), +l(Rq) is the path connected component of 0 in S Q and hence TP$ $ J ~ ( Ris~that ) of 0 in @ $ S,. Let L be a path connected component in TP$ S,. Since

+

there is y E F,' such that L = 'kp $ ql(Rq) 3. By (8.6) we have k(% $l(Rq) Y) = Tp$ +l(Rq) y, and so k(L) = L. Thus (2)was proved.

+

+

@

The remainder of this section is devoted to the proof of the following theorem. Theorem 8.3.4. I f f : Tn -t Tn is a TA-covering map, then the nonwandering set R(f) coincides with the entire space S.

E o m the above theorem together with Theorem 7.2.4 (2)we have the following theorem. Theorem 8.3.5. I f f : llYZ 4 Tn is a TA-covering map, then R(f) = 'Ir".

To show Theorem 8.3.4, suppose f : ?m -+ Tn is a TA-covering map. Then by Theorem 8.1.1 it follows that A is hyperbolic. Hence, by Theorem 8.2.1 we have a semi-conjugacy map % : Rn Rn between 2 and 7. Lemma 8.3.6. is compact).

is proper (i.e. the inverse image by

of any compact subset

Proof. This is clear by Theorem 8.2.1 (2).

- --1

Since 2 o fE = il o f on Rn,we have f (h Lemma 8.3.7.

7 : TI-' (0)

4

(0))= 5~-'(0).

SE-'(0) has POTP.

Proof. Since f has POTP, for E > 0 there is 6 > 0 such that every 6-pseudo orbit of 7 is E-traced by some point of Rn. If {vi)c x-'(o) is a 6-pseudo orbit o f f , then an €-tracing point v for { v i ) exists in Rn. Since x(v;) = 0 for all i, each of o X(v) is near to 0 in Rn,and hence x(v) = 0 by expansivity of 2, i.e. v E ~ ~ ' ( 0 ) .

2

$8.3 Nonwandering sets

Lemma 8.3.8.

x-'(o)

is the set of one point.

Proof. f : SE-'(0) -+ ~ - ' ( 0 )has POTP by Lemma 8.3.7 and is expansive. Let R denote the nonwandering set of f l I - l ( o ) .Then the set of all periodic points

-

Rn has exactly of f l i ; - ~ ( ois) dense in R by Theorem 3.4.2. Since f : Rn one fixed point by Theorem 8.1.3, we have that R consists of one point. This ( 0 ) = R by Theorems 3.1.2 and 3.1.7. implies

x-~

Since A is hyperbolic, by Theorem 8.3.1 there is a continuous surjection $ Sq -+ f?p $ Sq such that A o k = A o j.

k : ?p

Lemma 8.3.9. If g: @'$s~ -+ @'$sqis a continuous surjection and satisfies g(0) = 0, then g ( q p $ $1 ( R q ) )c $ $1 (Wq). Proof. This follows from the fact that component of 0 in f ? $ ~ Sq.

?p

$

$ I ( R ~ is ) the path connected

Lemma 8.3.10. If g: ~ ? P $ +s@' , $Sq is a local homeomorphism and g(0) = 0, then g(?p $ $1 ( R q ) )= 'b $ $1 ( R q ) . Proof. Let x be a point in 'b'$ $l(Rq). Since ?p $ $1 ( R Q )is path connected, we can take a path u joining 0 and x in ?P $ Sq. Since g is a covering map and g ( 0 ) = 0, by Theorem 5.1.7 there is a lift v of u by g starting at 0. Then the image of v is contained in @P $ $l(Wq), which implies that g(@' $ $l(Wq)) > TP$ $ J ~ ( Rand ~ ) ,hence the conclusion is obtained from Lemma 8.3.9.

Proposition 8.3.11. If xo E ~ e r ( A )then , k-'(xo) is the set of one point. Proof. Without loss of generality we may suppose A ( x o ) = $0. Since A o = k o j, we have j(k-'(xo)) = ( 5 0 ) .Thus j : k-l ( x o )-t k-l ( x o )has POTP. The proof is similar to that of Lemma 8.3.7. Since f l l - l ( x o ) is expansive, clearly k-l(x0) contains a periodic point yo of j. To avoid complication, suppose f(yo) = yo. We denote as cp,, and cpgo translations of @ $ Sq -+ $ Sq defined by cp,,(z) = xo z and cpgo(z)= yo z respectively. Write

+

Then ij(0) = k ( 0 ) = 0 and A o X E ? P $ S ~ ~ ~ ~ ~ Z E F ,

+

k

=

o ij.

By Lemma 7.2.2 we have for

CHAPTER 8

256

Since k : 'b' $ S , + ?P @ Sq is a continuous surjection, by Lemma 8.3.9

and since j is a homeomorphism, by Lemma 8.3.10

Define a continuous bijection $I : ' i f p $ Rq t + $ l ( v ) for t v E ?P $ RQ,and put

+

-1

+ @' $ $ l ( R q )

-I

by $'(t

+

V)

=

--I

Then, A o k = 03'. It is easy to see that 3' and A are uniformly continuous bijections, and that %I is uniformly continuous and El(@' $ R Q )C ?p $ RQ. By (8.7) we have

+ e) = X ( o + g l ( ~ )

for v E

+

CBRq and

e E Z;.

Let 3, A and E denote lifts of 3i1, 2' and 2 by the natural projection RP$RQ ?P $ RQrespectively. Then the following diagram

+

commutes and

(8.8)

Tj(v + P ) =

x(P)+ g ( v )

for v E RP $ R4 and P E Z;

$ Z;.

Since A ( Z n ) c Zn and 3 : Rn + Rn is bijective, by (8.8) the map 3 induces a self-covering map g: w + W , and we have then the commutative diagram

nr

2 ( ~ ,= ) (0) where [ is a homeomorphism as in Proposition 7.2.3. Since by (7.14), it follows that (@ $ 3) is the inverse limit system of ('kp$ @, gl).

s~,

$8.4 Injectivity of semi-conjugacy maps

257

Thus, g' is a TA-covering map because j is a TA-homeomorphism. Since

A o h = h o 3, we have g(h-l(0)) = K-'(o) and therefore by Lemma 8.3.8, -- 1 k (0) is the set of one point.

On the other hand, by Theorem 8.3.1 (2) we have ~ - I ( L ) = L for each path component L in 'fp $ S,. Since L o q m = and k(0) = 0, it follows that k l ( f p $ (Rq)) = 'Ib $ +l(RQ), and so k1(0) = h-l (0). Since k1(0) = q;?' (i-' (xo)), by the above result we conclude that k-l (30) consists of one point.

Lemma 8.3.12. L(R(j)) = @ $ S,. Proof. The periodic points of A are dense in @$s,.~ f j L ( ~ ( j#) )@$s,then k(R(j)) is a proper compact subset of 'b $ S,. Hence we can find z E 'b' @ S, such that p ( z ) = z for some r and e 4 L(R(j)). Then L-l(2(z)) is a non-empty compact j-invariant subset of @ $ S, that disjoints from ~ ( j ) , which is impossible.

ULl

Lemma 8.3.13. Them exists a basic set Rl such that L(R1) =

$ S,.

Proof. Let z be a point in @' $ S, such that the orbit {&(z) : i E Z) is dense in @' $ S,. By Lemma 8.3.12 there is x E a(?) such that i ( x ) = z. If R1 is the basic set in which x belongs, then i ( R l ) contains the closure of {$(z) : i E Z}, which is @' $ S,. Proof of Theorem 8.3.4. By Proposition 8.3.11, h : k7l(per(A)) -+ ~ e r ( 2 is) bijective and so ~ ( jitself ) is a basic set. Thus j,n,n is topologically transitive, in which case we have ~ ( j =) ?P $ Sq by Theorems 3.1.2 and 3.1.7 because jlnt )! is a TA-homeomorphism.

58.4 Injectivity of semi-conjugacy maps

x

The purpose of this section is to show that the semi-conjugacy map of Theorem 8.2.1 is actually a homeomorphism whenever f is a TA-covering map. For the case when f is expanding, we have the following Proposition 8.4.2.

Proposition 8.4.1. Let f : Tn + W be expanding and let % : Rn -+ Rn be the semi-conjugacy map obtained in Theomrn 8.2.1. Then is a homeomorphism and satisfies x(1 x ) = 1 x(x) for x E Bn and 1E Zn.

+

x

+

a

Proof. We already know that there exists a metric for Rn such that 'f has the property of Proposition 8.2.5 and, further, a tracing property in Lemma 8.2.6. Let h : Rn -+Rn be a semi-conjugacy map as in Proposition 8.2.7. In the similar way as the proof of Lemma 8.3.2 (I), we have h(x 1) = 1 h(x)

+

+

CHAPTER 8

258

for x E Rn and I E Zn. Thus, sup{2(~(x),x): x E Rn) is finite, i.e. there is B > 0 such that a(z(x), x) < B for x E Rn. From Theorem 8.2.1 (1) and Proposition 8.2.7 (1) it follows that

Since a(h(x), x)

+

< 6~ for x E Rn, we have for d l x E Rn

- -j

where c = B hK. Proposition 8.2.5 implies D(f o ( ~ o h ) ( x ) , ~ ' ( x )+ ) ao as j + co when o x(x) # x. But 2(T3 o o h)(x), y ( x ) ) < c for-j 2 0. This is impossible since and are uniformly equivalent. Therefore, k o h(x) = x, - and so k o h is the identity map. Similarly, o is the identity map.

z

(z

z

For TA-covering maps in 'Td \ P E M we have the following proposition.

Proposition 8.4.2. Let f : Tn 4 Tn be a TA-covering map and let : Rn t Rn be the semi-conjugacy map obtained in Theorem 8.2.1. Then is a homeomorphism and h-' is 2-uniformly continuous where 2 is a metric for Bn induced from a norm of Rn.

x

For the proof we need some lemmas. By Theorem 8.1.3 we have T(bo) = bo for some bo E Rn. Let b denote the two sided sequence (... ,bo,bo,b0,...) of boysand for i E z define : Bn = f( for d l i E Z. Since f is a map in Rn as in (6.4) of Chapter 6. Then I d \ P E M , from Proposition 6.6.5 we have the following Lemma.

2

Lemma 8.4.3. Let x E Rn and 0 < e < EO be as in Theorem 6.6.5. Then there is an open neighborhood R(x; b ) of x in Bn such that (1) diam(r(x; b)) < E O , (2) Eb : B6(x) x r ( x ; b ) 4 r ( x ; b ) is a homeomorphism where (i) Ds(z) = W:(x) n F ( x ; b), (ii) r ( x ; b ) = E ( x ; b ) n W(x; b ) (iii) {Eb(g, t)) = W;(y; b ) n W",(z )or (y, z) E Dd(x) x Z ( x ; b) (3) there is a constant p > 0 such that T ( x ; b ) > &(x), (4) Bs(x) # {x) and r ( x ; b) # {x). For x E Rn define the stable and unstable sets w d ( x ) and v ( x ; b ) as in 56.6. Then by Theorem 6.7.4 the families 7' = {ws(x) : x E Rn) and 7; = { r ( x ; b ) : x E Rn) are generalized foliations on Rn and they are transverse. Notice that Eb : Dp(x) x E U ( x ;b) is a canonical coordinate at x for 7' and 7;.

$8.4 Injectivity of semi-conjugacy maps

L e m m a 8.4.4. For a , y E Rn, w n ( x )r l W ( y ; b ) is at most one point. Proof. Let a, b E Wd(x) n T ( y ; b ) and suppose a # b. Then there is m > 0 - -such that d(f m(a),T-m(b)) < p where p is as in Lemma 8.4.3 (3). Put m a' = f- (a) and b' = j-m(b), and let e > 0 be as in Lemma 8.4.3. For sufficiently large m we have

It is clear that b' E Ws(a') since a, b E Ws(x), and that b' E &(a1) c W(a1;b ) since 2(a1,b') < p. Hence there is (bl, b2) E bn(a') x nU(a';b) such that b' = Zib(bl, b2) E W:(b2). Then we obtain b2 # a'. For, if bz = a' then ~ : ( a ' )= W:(bz) 3 b', which is inconsistent with (8.9). Let Uat and Uba be open neighborhoods of a' and b2 in Z ( a 1 ;b), respectively, such that Uat n Ub, = 0, and put Nat = ~b ( b n(a')

X

Vat),

Nbt = ?jib(bn (a') x Ub,).

Obviously Na, and Nbt are open neighborhoods of a' and b' in Rn respectively. Since Na, n Nbr = 0, by Lemma 6.6.2 we have (8.10)

-

w:(v) n W:(w) = 0

for v E Nat and w E Nbt.

Figure 30 If V, = {r E Rn : w S ( r ) n Nbt # 01, then V, is open in Rn since 7' is a generalized foliation on Rn (see Remark 6.7.2), and a' E V, since b' E Wn(a'). Since Per(f) is dense in Tn by Theorem 8.3.5, there is p E V, n Na, such ) Per(f). Let k be a period of ~ ( p and ) let u = (ui) E (EXn)? that ~ ( p E be a k-periodic sequence with p = uo. Define

7:

: Rn

4

Rn as in (6.4)

CHAPTER 8

260

-k

and write ij = f, for simplicity. Then g(p) = p. Since p E V,, we can choose w E Wd(p) n Nw . Since p, w E z(a'; u), by Theorem 6.6.5 (1) we have R ( p ; u) n w:(w) = {q) for some q E z ( a 1 ;u). Hence lim #(q) = p r+m

since wd(w) = Wd(p), and . lim $(q) = p. Using (8.10), we have p

-

#

q

14-00

because p E N,, , q E W;(W) and w E Nb,. Let E = min{';i(p, q), ef)/4 where e' is an expansive constant for ij. Then there is 0 < 6 < 2~ such that every 6-pseudo orbit of ij is &-tracedby some point of Rn. Choose t > 0 such that d(ge+'(q),p) < 612 and 2(g-'(q),p) < 612. Then the sequence

+

is a (2t 1)-periodic 6-pseudo orbit of ij. By using POTP and expansivity we ~ )qo and 2(q, go) < E. It is checked that can find qo E Rn such that ~ 2 e + ' ( q = ge+'(qo) # go. Indeed, if ge+'(qo) = go then

Thus we have Z(p, q) < 3.5 which is impossible since 4~ 5 d(p, q). Therefore ij2'+' has at least two distinct fixed points, which contradicts Theorem 8.1.3. Since ?I is a hyperbolic linear map, Rn splits into the direct sum Rn = zd(0) @ r ( 0 ) of 3-invariant subspaces Z4(0) and c ( 0 ) where Zs(0) is the subspaces corresponding to eigenvalues with absolute value smaller than one and r ( 0 ) is that of eigenvalues with absolute value greater than one. For x E Rn let zu(x) (a = s, u) denote the translation of r ( 0 ) to x. For a = s, u the family 7; = {ZU(2): x E Rn) is a (linear) foliation on Rn and the following holds: (8.11)

h(w4(x)) c Zd(h(x)), h ( v ( x ; b))

--

--

since Aoh = ho f holds and (see Theorem 8.2.1).

c ZU(h(z))

for all x E Rn

: Rn -t Rn is a 2-uniformly continuous surjection

Lemma 8.4.5 (Franks [Fl]). Let x, y E Rn. Then wU(x;b) 0 W'(y) i s the set of one point. Proof. Let yo E Rn and put s = Wd(yo).It is enough to show that r ( x ; b) n s # 0 for all x E Rn.Let us put

88.4 Injectivity of semi-conjugacy maps

then we have --U

Q={xER":W

(x;b)r)U(s)#0)

-

where U(s) = UzEsN(z; b). Indeed, choose x from the right hand set of the above equality. Then z E V W (x; b ) n U(s) and hence z E r ( x ; b ) n W(zf;b ) for some z' E s. Since r ( z f ; b) = %(DS(z') x r ( z ' ; b)), there is (yl, y2) E Bs(z') x 3 ( z 1 ;b) such that z = Eb(y1,ya) E c ( y l ; b ) . Hence yl E R ( z ; b ) C r ( z ; b ) and on the other hand, yl E Ds(z') c s. Therefore, T ( z ; b) n s # 0 which implies x E Q. Hence Q is open in Rn (see Remark 6.7.2). If Q = Rn then the lemma holds. Thus we suppose Q Rn and then derive a contradiction. Let w E Q. If %(w; b) (Z Q, then Q does not contain Ds(w). For, take x E W(w; b), then there is (x', 2'') E DS(w) x S ( W ;b )

5

such that x = E~(x',x") E C ( x f ; b). If DS(w) C Q then W ( x f ; b ) n s # since x' E Ds(w) C Q. Since T ( x ; b ) = T ( x 1 ;b), we have r ( x ; b ) ns # and therefore x E Q, i.e. %(w; b ) c Q, thus contradicting.

0 0

Tv=( w )

(w; b) Figure 31 Choose and fix a E Ds(w) \ Q. Let y : [O, 11 -+ BS(w) be a path such that $0) = w and y(1) = a, and let p : [0,1] -+ r ( w ; b ) be a path such that ~ ( 0= ) w and p(1) E r ( w ; b) f l s. If we set

then R is non-empty since ([0, 11x {O))U({O) x [O,l]) c R and by transversality of 7' and R is open in [O,1] x [0, 11. Note that R [O, 11 x [0, 11. Since

x,

5

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262

U

W ( ~ ( rb) );n W 8 ( p ( t )is ) a single point for ( ~ , tE) R, we can define a map e : R - t R n by

Then B is continuous. By (8.11) we have --U

h(W ( ~ ( rb) );)c Z'((h o ~ ( r )and ) h ( w 8 ( p ( t )c) z8(ho p(t)).

Define by pa : Rn -t z a ( 0 )the natural projection (o = s,u). Then it follows that -

h(B(R))= {z(W'(?(r): b ) )f l V s ( p ( t ) :) (r,t)E R }

c {E"(ho Y ( T ) ) n Z8(ho p(t)): (r,t ) E R )

+

= { ~ " (0 ~ h ( r ) )p U ( x 0 p(t)) : (r,t ) E R )

c p8(E 0 ~ ( [ 0 , 1 1+) )pU(K 0 P([O, 11))-

Figure 32 Notice that the last part of the above relation is compact. Since h is proper by Lemma 8.3.6, we obtain that B(R) is bounded. Let us put

to = sup{t^ : w ( 7 ( r ) b) ; fl w 8 ( p ( t )#)

0, 0 5 r 5 1 and 0 _< t 5 t^}, TO = sup{i; : w ( T ( r ) b ; ) n wS(p(to)) # 0, 0 5 r 5 i;). Then

( T O ,to)

4 R.

Since B(R) is bounded, we can choose a sequence

88.4 Injectivity of semi-conjugacy maps

263

converging to ( T O , t o ) ,such that 8(rn,t,) converges in Rn. Let lim 8(rn,t,) = v. Take a compact neighborhoods C s and Cu of v in fS8(v) and g ( v ;b) respectively, and let C = zib(Cs x Cu). Then C is a compact neighborhood of v in Rn. Since lim 8(rn,t,) = v , we may assume 8(rn,t,) E C for n 1. n+ w Then for n 1 there is (u,, v,) E C" x CU such that

>

>

and hence Y

w

( ~ ( r nb) ); n m s ( p ( t n ) )= {e(rn,tn))= { b ( u n 1vn))

c WY(U,;b)n W S ( v n ) , from which

-

U

W ( y ( ~ n ) ; b=)W u ( u n ; b ) , w 8 ( p ( t n ) )= W S ( v n ) . Thus we have { e ( ~t,))~ ,= W U ( y ( r 1 )b;) n W S ( p ( t n )= ) WY(u1;b ) n m 8 ( v n )3 E b ( ~ l , v n ) , and so 8 ( ~t,)~ =, zib(ul,v,) E C . In the similar way, 8(rn,t l ) = zib(un,vl ) E C . Since 8 is continuous, we have 8(rl,t,) -t 8(rl,to) and B(r,, t l ) -+ B(r0,t l ) ( n -t oo). Thus 8(rl,t o ) ,8(ro,t l ) E C , from which there are ( w ,z ) , ( w l ,z l ) E C" x CU such that In the same fashion we have

-

-

) ;W U ( w lb; ) ~ " ( ~ ( =t W ~ s)( z) ) , w U ( ~ ( rb~) =

and hence

w

U

( 7 ( r 0 )b; )n W s ( p ( t o )= ) W " ( w l ;b )n W 8 ( z )3 E ~ ( w 'z). , Therefore ( T O ,t o ) E R, thus contradicting. By Lemma 8.4.5 we can define 9 : Rn x Rn -+ Rn by Then

Zb

is a continuous map such that for x E Rn

-

-

2b~~y.)x~u(.ik.) = ab where Eb is a map as in Lemma 8.4.3 (2). It is easily checked that the following properties ; for x, y, z E Rn

x

ib

satisfies

Proof of Proposition 8.4.2. First we show that is a homeomorphism. Since h is surjective by Theorem 8.2.1 (3), it is enough to prove that fE-'(x) is the set of one point for all x E Rn. To do this, let x E Rn and define for y E x-'(x) I , , =( x ) ( y )

I:,, = K-'(x) n T ( y ;b).

CHAPTER 8

264

Claim 1. ib(I:,, x I:,) = i;-' (2). Indeed, for v, w E IE-' (x) -

- 3

h o ib(v, W) = h ( w (v; b ) n Wa(w))

c Zu(I(v)) n Zs(X(w ))

-

-

( since X o I = h o f )

= (21

c I-' (x) for = s,u, we have ib(I:, Conversely, let y E I-' (x). Then for any r E H-' (x)

and so ib(v, w) E I:,)

C

I-' (x).

I-' (0).

Since I;,,

(I.

x

. Therefore, r = ib(ib(r, y),

from which Zb(r, y) E I:,,. Similarly, ib(y, r ) E I,: ib(Y, z)) E ib(I:,, X I,",,).

As seen in Theorem 8.2.1, we have z(x(x),x) 5 6 (x E Rn) for some 6 > for y E K-'(0) where 0 and so diam(h-' (x)) 5 26, i.e. (x) c B6(y) = {r E In: d(z, y) 5 6). We define for y E f ~ - l ( z )

-0

Claim 2. I:,, c I,,, for (I. = S,U. Indeed, since ib(~-'(x), y) = ib(ib(C,, x I:,,), y) = ib(I:,,, have I:,, c ib(B26(~),Y) = f:,, because IE-'(x) c

Also we obtain the same result for

y) = I:,,,

(I.

we

= u.

Let us put for (y E I-'(z))

Claim 3. There is N > 0 such that R , , y E I-'(2). Indeed, by (8.12) it follows that

Since z(h(x), x) 5 6 (x E Rn) and

o

c

BN(y) for all x E Rn and

= % o 7 on Rn, we have

58.4 Injectivity of semi-conjugacy maps

and hence

from which there exists N y E I;-I ( 2 ) . Let e

> 0 such that R,,,

> 0 be as in Lemma 8.4.3

and let v E

C B N ( y )for all x E

Rn and all

Rn.We define for w E W 8 ( v )

-

8 -

d(v,w;W S ( v ) )= min{m 2 0 :F ( w ) E W e ( f ( v ) ) ) ,

and for w E W ' ( v ;b )

-

u --m

d(v,w ; W ' ( v ;b ) )= minim 2 0 : T-m(w) E W e( f

( v ) ;b)}.

Note that these are well defined by Lemma 6.6.4.

Claim 4. There exists KO > 0 such that for x E Wn and y E f;-'(2) ( 1 ) if v E R,,, and w E R,,, n W a ( v ) then , a(v,w;W 8 ( v ) )5 KO, ( 2 ) if v E R,,, and w E Rz,, n r ( v ;b),then Z(v,w ; r ( v ;b ) )5 KO. Indeed, let p be as in Lemma 8.4.3 and N as in Claim 3. Then there are P > 0 and a sequence { x l , . ..,x L ) c Rn such that B N ( y )C U: &(xi). Hence R,,, C fl(xi; b ) by Claim 3. Let v E R,,, and define

uf

Then we have D = ib(f:,,,v ) and hence D is connected. Indeed, if z E ?b(-:,,,v) C W 8 ( v )then z = i b ( z l ,V ) for some z~ E f:,,. Since v E R,,,, there is (vl ,v 2 ) E f x such that v = ib(vl,v2). Hence

:,

z,,

and so z E D. Conversely, let z E D. Then z = ib(wl,W Z )for some ( w l ,w,) E I=,, x I:,,. Since z = Zb(z,v), we have

-8

and therefore D c ib(l:,,, v). Since R,,, C U: x ( x i ;b),we have D = U: x ( x i ;b )n D. To avoid complication, we may suppose that each x ( x i ;b ) n D is non-empty. Choose yi E D n x ( x i ;b ) for 1 i 5 e. Then

<

CHAPTER 8

266

This is checked as follows. Since yi E r ( x i ; b), there is zi E z ( z i ; b ) such that yi E W:(zi). If y' E D n x ( x i ; b ) then we have also Y' E mb,(z) for some z E BU(xi;b). Since yi, y' E D c Ws(v), clearly zi, z E Ws(v) and so

which shows zi = z by Lemma 8.4.4. Therefore y' E W;,(yi), from which

By Lemma 6.6.3 there is KO > 0 such that

for z E Rn. Hence we have

Since D is connected, for il, i2 with 1 5 il, i2 5 t? we can find a sequence jl = il,jl,j2,. ,jm= i2 such that

..

By using this fact we have

<

and therefore z(v, w;ws(v)) KOfor any w E D. The analogous result holds for WU(v;b). Claim 4 was proved. Let v,w E I:,,. -s

--n

We(f

If v

# w then there is no > 0 such that j-n(v) $

(w)) for n 2 no (since f is expansive) and hence

Let KO be as in Claim 4 and write

+

--n1

n l = no + K O 1, v' = f Then we have z(vl, w';Wa(v'))

--nl

(v) and w' = f

> KO+ 1.

(w).

$8.4 Injectivity of semi-conjugacy maps

-

-

Since A o h = h o 7 on Rn, it follows that

--n1

n1

where x' = 2- (x) and y' = f (y). Therefore, v', w' E C ~ ~ l , c y l Rx,,,tn w d ( y r ) . Using Claim 4, we have 2(vr,w';Wd(v')) 5 KO,thus contradicting. This shows that I:,, is a set of one point. In the same fashion we have that I:,, is a single point set. By Claim 1 we obtain that K-'(x) is the set of one point. Next, we show that fl-' : Rn -+ Rn is 2-uniformly continuous. o ;i:= f: o 51-' and Since h : Rn + Rn is bijective, using the fact that TE-' - --1 d(h (x),x) < 6~ (x E Rn), we have that for j E Z

Since f : Rn (8.14)

4

Rn is expansive (under

for E

z), if we establish the following: ---N

> 0 there is N > 0 such that if d(f - -N

--N

(x), f

(y))

5 3bKand d(f ( ~ ) , f : ~ ( y5) )3 6 ~ then , Z(x, y) < E .

Then by uniform continuity of 2,we can find 6 > 0 such that z(x, y) < 6 implies z(z'(x),z'(~)) < 6~ for j = N and j = -N. From (8.13) we o h-'(y) E & 6 K ( ~ )for j = N and j = - N and so have 7' o K-'(x) - --I (y)) < E . Therefore h-I is &uniformly continuous. d(h (x), IE-' To prove (8.14) write

f:'

for x E Rn. Then we can find K > 0 such that B, C BK(x) for x E Rn (see the proof of Claim 3). Let e > 0 be as in Lemma 8.4.3. As above, define d(v, w;Ws(v)) for w E Ws(v) and ;i(v,w;Wu(v;b)) for w E w U ( v ; b ) . To obtain that there exists KO > 0 such that for all x E Rn w; Ws(v)) 5 KO

if v E Bz and w E B, r l Wd(v),

d(v, w; W ( V ;b)) 5 KO if v E Bx and w E Bxr l Wu(v; b) we can use the proof of Claim 4.

CHAPTER 8

268

7

Since is expansive (under there exists m > 0 such that

z), by Lemma 6.6.3 it follows that for

+

E

>0

- -j

To see that N = m KOis our requirement, suppose d ( f ( x ) , 7 ' ( y ) ) 5 3 6 ~ for j = N and j = - N . For the case j = - N we have

and thus

This implies that

from which

Therefore z ( i b ( x , y), x ) in the same argument.

< ~ / 2 For . the case j = N we have z ( i b ( x , y ) , y ) < ~ / 2

38.5 Proof of Theorem 6.8.1

By Theorem 7.2.4 it suffices to show that the semi-conjugacy map h: ?p $ S q -t ?P @ Sq obtained in Theorem 8.3.1 is a homeomorphism if f is a TAcovering map. To do this, let X : R" -t R n be the semi-conjugacy map obtained : RP $ +l(Rq) -+ RP $ 7)1(R9) be defined as in in Theorem 8.2.1 and let (8.4). Then is bijective by Proposition 8.4.2. We already know that K' is -I is d -uniformly continuous (see Lemma 8.3.3). The uniform continuity of easily checked by the following facts: ( 1 ) 7 : Rn + Rn satisfies the property of (8.14), ( 2 ) the relative topology of +(Rp $ $ 9 ) = I[$P $ $ J ~ ( R is Q )of the form mentioned in (7.12).

$8.6 Proof of Theorem 6.8.2

269

x'

Hence %' is b-biuniformly continuous and so induces a homeomorphism of RP $ S, onto Rp $ Sqwhich is denoted as the same symbol. On the other hand, by Lemma 7.2.1 (2) the maps and induce homeomorphisms of RP $ S, onto Rp $ S, respectively. We denote them by the same o = o 7' on RP $ S,. Since symbols respectively. Then we have

7

2

x' z'

for up E RP and x, E S,, a homeomorphism i: 'b' $ S, + 'b$ S, is induced : RP $ S, -, RP @ S,. Also home~m~rphisms and f of 'b' 8 S , onto by 'b' $ Sq are induced by and 7' respectively, and A o i= k o f holds.

z'

x'

$8.6 Proof of Theorem 6.8.2 Let f : Tn + Tn be a special TA-covering map, i.e. f is a map in S'TA. By Propositions 8.2.1 and 8.4.2 we have a constant 6~ > 0 and a homeomorphism h : Rn Rn such that x o x = K O ? and z(x(x),x) < bK (x E Rn). If f belongs to 'TAX, then : Zn -, Zn is an automorphism. In this case, we have 9," = Zn (see $7.2) and

xlzn

x

for v E Rn and L E Zn by Lemma-8.3.2 - (I), - -which shows that induces a homeomorphism h: Tn -t 'P. Since Ao h = ho f on Rn, we obtain Ao h = ho f on Y. Therefore, Theorem 6.8.2 (1) holds. - - - If f : 'IP -, 'IP is expanding (i.e. f is a map in PEN),then A o h = h o f by Theorem 8.2.1 and further by Proposition 8.4.1, is a homeomorphism and h(L x) = 1 z(x) for x E Bn and 1E Zn. Thus, induces a homeomorphism h: Tn -t 'P and A o h = h o f holds on Tn. Therefore, Theorem 6.8.1 (2) was concluded. Next, let f be a map in SS'TA. By Theorem 8.1.3 we have T(bo) = for some 4 E Rn. For simplicity we can suppose T(0) = 0. Indeed, choose a homeomorphism 4 of 'P homotopic to the identity map such that +(?r(bo)) = 0. Then 4 o f o 4-' belongs to SS'TA and satisfies 4 o f o +-l(O) = 0. We recall that the families 7"= {w8(x) : x E Rn} and = { V ( X ; O ): x E Rn} are generalized foliations on Rn. Here 0 = ( - . ,0,0,0,. . ). Lemma 8.6.1.(1) UeEz,, w d ( t ) is dense in Rn. (2) UtEz" ?(c; 0 ) is dense in Rn

+

+

-

.

Proof. (1) follows - from the facts that ~ " ( 0 is) dense in Tn and that n-'(wd (0) = UeEznwd(e) by Lemma 6.6.10. By Lemma 6.6.11 (2) we have

CHAPTER 8

270

Since n ( f ) = Tn, from Remark 5.3.2 (2) it is clear that WU(rO(0))is dense in Tn. Therefore (2) holds.

+

+

The end goal of this section is to show the equation E(v L) = C h(v) for all v E Rn and all C E Zn. To do this, we first investigate the relationship between Faand F:, and that between 7,"and Fx.Here F: and F: denotes transverse linear foliations associated with 2 (see 58.4). By Theorems 8.2.1 (1) and 8.4.2 we have that h i s d-biuniformly continuous. - -a - -u =F z (see (8.11)). Since ;i(h(z),z) < 6~ for ~ h u s h, ( 3 ) = F$ and x E Rn, we have

Lemma 8.6.2. For E > 0 there is 6 > 0 such that if d(z, y) u , ( ~ " ( ~ )and ) w ( x ; 0) c U, ( F ( y ; 0)).

Proof. This is clear from Z-biuniform continuity of that Fc (a= s, u) are linear foliations. A We write Fa = T ( 0 ) n Zn and

< 6 then w 8 ( x ) C

h together with the fact

= F ( 0 ; 0) n Zn.

Lemma 8.6.3. (1) rx and I?? are subgroups of Zn, (2) r j c rx.

is a subgroup of

Proof. Since ZU(0) is the linear subspace of Rn, obviously Zn. Let C E rj, then we have

L + wu(O; 0) = wu(C; 0)

(by Lemma 6.6.11 (2.b))

= F ( o ; 0) and hence C - C' E I?? when L, 1' E

(since L E WU(0;0))

r7. This shows that I?? is a subgroup of

v n. u

Next we prove (2). Take L E I?7 and suppose L have nC 4 ZU(0) for all n E Z with n # 0, and so nC E wU(O;0)

c UbK( T ( 0 ) ) which is impossible.

4

Since C

4 T ( o ) , we

( since nL E rf by (1)) (by (8.15) and the fact that h(0) = 0)

$8.6 Proof of Theorem 6.8.2

271

Since T ( x ; 0) n w s ( y ) is the set of one point for z,y E Bn by Lemma 8.4.5, we can define maps p! : Bn + ws(0) and p! : Rn + W"(0; 0) by f

f

p;(~) E F ( x ; 0) n Wa(o)

and

p;(~) E W ( O ;0) n W a ( t )

for x E Rn, respectively. For u = s, u let us put Q_O= p'l(Zn). f

f

Lemma 8.6.4. (1) Q3 is dense in ws(0), ( 2 ) Q; is dense in ?(o;

-

0).

Proof. This is clear from Lemma 8.6.1. Since F: and 3~are transverse linear foliations, we denote as pd : Bn + A -a L (0) and p?A : Rn + r ( 0 ) the natural projections. For u = s,u write Q: = p:(Zn). Lemma 8.6.5. p: : Zn + QU and pit : Zn + Qlf are injective. f

f

Proof. Since a :w s ( l ) Wa(a(l)) is injective by Lemma 6.6.8 (2), we have w a ( l )I I Zn = {C) for C E Zn. Therefore pElzn is injective. In the same way, f we have that pEIZnis injective. A

By the above lemma a map

-

: Q; + Q : is defined by

Then we have the following

Lemma 84.6. (1) X o h = h o f on ~

j ,

(2) 2 is 2-uniformly continuous. Proof. (1) is obtained by the following calculation : for 1 E Zn A )o)e(!p!F( =; io P;(O = Si(Z"(o) n Za(t))

and

k

f(p;(e)) = i? o f ( W ( o ; 0) n W8(e)) 11-

(since f (0) = 0)

11-

(since f Izn = A p )

= h (W (0; 0) nWa(f(l))) = h (W (0; 0) n Ws (Si(.!!))) = F 0 p;(Si(e)) = p;(X(l)).

-

-

CHAPTER 8

272

If ( 2 ) is false, then there is e x , y E Qlf satisfying

> 0 such that for

any 6

> 0 there exist

f

-

d(x,y) < 6

(8.16)

and

-d ( h

--U

( x ) , T ( y ) ) > e.

- -0

Choose E' > 0 such that for v , w E z U ( 0 ) ,if Z(v, w ) > e then d(L ( v ) ,z o ( w)) > e'. Take n > 0 with ne' 2hK and let 6' > 0 be a number such that n6' _< 6 ~ By . Lemma 8.6.2 we can find 6 > 0 such that a ( x , y ) < 6 implies -8 ( m 8 ( x ) ) . Now, suppose that x , y E Qg satisfy (8.16), and put W (9) C

>

-6

I. = ( P ; , ~ , . ) - ~ ( X ) and I , = ( P ; , ~ ~ ) - ~ ( Y Then ). 1, E W ( x ) and PY E w o ( y ) by the definition of p!. Thus f

Since p!!(Cx)

E %'(ex) and p;(P,)

-d Letting C = L,

E zo(P,), it

follows that

-8

( (eX),Z8(e,)) ~ > E'.

- ex, we have

and by induction

e, + ne 4 ~ , , t ( E ~ ( e ~2) )u2aK(Z0(es>).

Figure 33 On the other hand, since P, E r a ( x ) and I , E -

it is clear that

wS(e,)= W S ( y )c u 6 ' ( m 8 ( z ) )= u 6 , ( W 8 ( e X ) )

$8.6 Proof of Theorem 6.8.2 and hence L, = ex

from which

+ 1 E U ~(W8(e,)). I This implies that

r8(.!) c U a ~ ( W d ( 0 )Then ) . we have

and by induction

-

w 8 ( n 1 )c ~ , y ( W " ( (n 1)t)).

Thus Wd(ne) c U ~ ~ ~ ( WB~y (8.15) ( O ) it) .follows that W 8 ( n 1 )C C u Z ~ ~ ( Z ' ( OTherefore )). we have

U,~~(W~(O))

and arrived at a contradiction. Let x E Qb. Then p f ( t ) = x for some e E Zn and hence f

f

(P;,~.)-~(X) =T

( e ;0 ) n zn

+ g ( 0 ;0 ) ) n Zn =e+rj = {e

ce+rX = ZU(e)n

( b y Lemma 6.6.11 ( 2 b ) )

(by Lemma 8.6.3 (2))

zn,

from which pL o ( P ~ , ~ . ) - ~ ( X = ) p:(l). A f -, by sending x to e(t!).

Therefore, we can define a map

Qq Qi

Lemma 8.6.7. ( 1 ) 710%' = T (2)

o7

2

:

on Q:, f

is 2-uniformly continuous.

Proof. This is similar to that of Lemma 8.6.6. By Lemmas 8.6.6 ( 2 ) and 8.6.7 ( 2 ) the maps continuous maps

iu: W'"(0; 0 ) + z U ( 0 )

and

2 and Ed are extended to

in: W S ( 0 )+ Z 8 ( 0 )

respectively. From Lemmas 8.6.6 ( 1 ) and 8.6.7 ( 1 ) we have

CHAPTER 8

274

Let a, b E Rn, then zd(a)nzu(b)is exactly one point, say P(a, b). It follows that

p : Rn x Rn -t Rn is z-uniformly continuous. Thus a &uniformly continuous map

i :Rn

-t

Rn is defined by

h(z) = P 0 ( i u 0 p;(z),

k' o p;(z))

for z E Rn.

Then by (8.17) we have

A

onRn

and by definition for L E Zn i(e) = P(P;(~), P;(e))

= e.

- It is easy to check that sup{~(h(x),z ) : z E Rn) is bounded. For, since i is d-uniformly continuous, for m

-

d(x, y)

> 0 there is K > 0 such that

< m (z, y E Rn) * Z(i(z) - z , i(y) - y) 5 K.

Let- z E Rn, then we can find L E Zn such that a(z, 1) < m, and so z(i(z), z) = d ( h ( ~) X,O)= ;i(i(~) - 3, i(e) - e) < K. Therefore, h = i by Proposition 8.2.2 and h(e) = 1for all P E Zn.

Figure 34 From the following Proposition 8.6.8 we obtain (3) of Theorem 6.8.2. Proposition 8.6.8. E(L+ v) = !. + E(v) for all L E Zn and all v E Rn. Proof. Let v E UeEznW (L;O). Then there exists e, E Zn such that v E - -u W (L,; 0). Since h(FO) = and E(e,) = t!,, we have h(v) E ?;"(P,; 0) and for L E Zn h(v) e E T ( e v e; 0). -U

U

Fx

+

+

$8.7 Remarks

275

+

Since v +1 E r ( 1 , l ; 0) by Lemma 6.6.11 (2), clearly x(v +e) E r ( 1 , +1; 0) and therefore there exists c 7 0 such that

-- --

On the other hand, since Ao h = ho f and d(x(x), x) < bK for x E Rn, we have 2(xi OE(X), (x)) < bK for i E Z and x E Rn. Note that f(x l ) = f(z) + x ( l ) for x E Rn. Then, for i 2 0 we have

7

and so

+

- i -

+

+

d ( ( ~h ( ~ ) t),$ OZ(V e))

< 26.

(i t

+

01.

+

Therefore, by Lemma 8.2.3 (2) it follows that &(v e) = h(v) 1. Since UtEZ,,V ( t ; 0 ) is dense in Rn by Lemma 8.6.1 (2), the conclusion is obtained.

58.7 Remarks

In this section we mention some remarks about TA-covering maps of tori.

Remark 8.7.1. In Theorem 6.8.2 we required Tn to be an n-torus. However the result is true for expanding maps of an n-solenoidal group. An outline of the proof is stated as follows. Let S be solenoidal, then S is abelian. Thus S splits into the direct sum S = Sn $ Tm of the maximum torus Tm and a solenoidal group Sn without tori. If Sn = {0), then S is a torus and Theorem 6.8.1 holds. For the case Sn # (0) let n = dim(&). As saw in Chapter 7, Sn is expressed as Sn = $(Rn) + F where F is a totally disconnected compact subgroup. Let f : S + S be an expanding map. Then f is a local homeomorphism; hence, f has a fixed point. We may assume that the identity 0 is a fixed point. Then we obtain f ($(Rn) $ Tm) = $(Rn) $ Tm and a homeomorphism f : IW" x Rm + Rn x Rm which is a lift of f: $(Rn)$ Tm + $(Rn) $ Tm. It is checked that for X > 0 small there are a subgroup Z' (C Zn x Zm) of finite index and a continuous map F : Z' x (Rn x Rm) -+ BA(o) and an injective homomorphism : Z' -, Zn x Zm such that

for L E Z' and v E Rn x Rm (here Bx(0) is a closed ball of radius A). Moreover, we can prove that 5 : Rn x Rn -, Rn x Rm is expanding. This fact leads us to the end goal (see Aoki [A021 for details).

CHAPTER 8

276

For general TA-covering maps it remains a problem of whether Theorem 6.8.2 is true in the case of solenoidals. In 57.2 of Chapter 7 we have seen that a toral endomorphism splits into two subsystems throughout a method of lifting by some finite covering map (see (7.5)). In the case when a toral endomorphism A : 'IP + I[n is hyperbolic, if we have the case (i) of (7.5) then Rp = Rn and 71 : Rn + Rn is hyperbolic. For the case (ii) of (7.5) we have that WQ = Rn and 71 : Rn + Rn is hyperbolic (or expanding). The case (iii) of (7.5) implies that Rn splits into the direct sum In= RP $ WQ of z-invariant non-trivial subspaces RP and RQ,and 2 : Rn + Rn is hyperbolic. Since TilR' is hyperbolic (or expanding), RQ splits into the direct sum RQ = E d $ EU of subspaces En and Eu where Ed is a subspace corresponding to the eigenvalues of with modulus < 1 and Eu is that with modulus > 1 (or WQ = Eu). Put Z' = EU n Z t and define a subspace K = spanRZ1. Since x(Z1) c Z', we have x(v1) = Vl and Tilvl is expanding. Obviously Tilvl induces a toral endomorphism of K/Z1 onto Vl /Z1. Then Vl is the maximum subspace of RQsuch that the induced endomorphism is expanding. Thus, it is easily checked that there is an Ti-invariant subspace V2 such that

x

Rq=vl$v2,

Z1'=vznZ;

xlv,

satisfies X(Z") c Z"; and spanRZ1' = Va. induces a hyperbolic toral endomorphism of type (111). Since x(Zp") = Zp", it is clear that ZJRpinduces a hyperbolic toral automorphism. Thus the following remark is obtained as a result.

Remark 8.7.2.

If A : Tn + 'P is a hyperbolic endomorphism, then there exist a finite covering space 'k" (n-torus) of 'P and a hyperbolic toral endomorphism A' : 'k" + 2" such that the diagram

commutes, and such that there are the maximum torus subgroups 1,2), the torus subgroup 'k"3 and

el,

(a) a hyperbolic automorphism A1 : qnl + (b) an expanding endomorphism A2 : Pa+ qna, (c) a hyperbolic endomorphism Aj : 'k"3 + 'k"3,

'k"i

(i =

$8.7 Remarks

so that the diagram

commutes homeomorphically and algebraically.

( 1 ) If f: Tn -t W is a TA-covering map and if w 6 ( x o )is dense in 7"' for some xo E P, then so is w ( x ) for all x E Tn. ( 2 ) Let f : T" + 'IP be a special TA-covering map which is not injective nor expanding and let A : W + 11"" be the endomorphism homotopic to f . Let be a torus subgroup as in Remark 8.7.2. Then, f is strongly special if and only if the space p a is trivial, i.e. p a = ( 0 ) . These are checked as follows. As before let 7 : Rn -t Rn be a lift of f by the natural projection .~r:Rn + Tn and suppose 7(0)= 0 for simplicity. If 2 : Rn -t Rn denotes the linear map which covers A, then 2 is hyperbolic (by Theorem 8.1.1). Thus there exists a map E : Rn -, Rn such that 2 o = E o 7 and z ( x ( x ) , x ) < bK for all x E Rn by Theorem 8.2.1. From Theorem 8.2.1 and Proposition 8.4.2 it follows that fE is a homeomorphism and it is 2-biuniformly continuous. To show ( 1 )let e > 0 be as in Lemma 8.4.3 and choose ~1 > 0 and e > E Z > 0 satisfying

Remark 8.7.3.

x

d(v,w ) 5 E I =+ z(E-' ( v ) ,K-' ( w ) )5 e, d(v,w ) 5 E Z 3 z ( x ( v ) ,X(W)) 5 ~

1 .

For v E Rn and E > 0 denote W ; ( v ) and W : ( V ;as ~ the ) local stable sets of f and respectively. Then we have ---1

h

c W;,(fE(v);21, (w,,( v ;2))c We(H-'(v)).

Choose E such that e 2 E

> 0 and for w E W a ( v )write

h(w:,(v))

and for m

-a

>0 Dm(v,&)= { w E W a ( v ): d(v,w ; W 6 ( v ) , &5 ) m).

CHAPTER 8

Then we have

Since n(W8(v)) = wS(?r(v)), clearly ?r(Dm(v,E)) c wS(?r(v)). TOobtain the conclusion it is enough to see that for every ,d > 0 there exists P 2 0 such that if ~ ( v E) 'IP then n(De(v,e)) is P-dense in 'IP (i.e. T ( D ~ ( v , E ~n) Up(.) ) #0 for any x E 'P and any P-neighborhood Up(z)). Since is 2-biuniformly continuous, we can find 7 > 0 and 6 > 0 such that d(v, w) < 27 and d(v, w) < 6 imply ~ ( l ~ - ' ( v )K-'(w)) , < P and Z(x(v), E(w)) < -y respectively. Take and fix vo E n-'(~0). Since wa(xo) is dense in Tn, there exists k 2 0 such that a(Dk(vo;e2))is 6-dense in 'Il"'. Thus, for a 6neighborhood U = Ua(Dk(vo;€2)) of Dk(vO,E ~ we ) have T(U) = 'ITn and

x

-

h(U) C U ~ ( ~ ( D ~ ( V O ; E Z )(by ) ) the choice of 6) (by the choice of €1) C U,(~k(E(vo); €1))

where Bk(v; €1) = {WE z6(v);Z(v, W;z8(v),

< k}.

Here z6(v) denotes the leaf of the linear foliation

through s and as above , E ~ ) 2 o ; ~ ( w E) w:,($(v);x)}. Then it is define ~ ( V , W ; ~ ~ ( V=) min{i easily checked that there exists P > 0 such that for all u E u ~ ( B ~ ( K ( v ~ ) ; E ~ ) )

and then for all u E U ~ ( B ~ ( ~ (€1) V~);

u c ~-'(U,(B~(~E(V~);E~))) c E - ' ( ~ z r ( ~ e (€1~))) ; C ~p(K-l(~e(u €1))) ; C

u~(D~(x-'(,); e))

(by the choice of -y) (by the choice of €1).

Therefore, for v E U we have Up(De(v; e)) > U and therefore n(De(v; e)) is P-dense in Tn. Next, we show (2). Since f(0) = 0, we have f lzn = AIZ=and thus for P E Zn

58.7 Remarks -i

279

-

Since ;d(L(x),x ) < iiK for x E Rn, clearly d ( A o h ( I ) , 2 ( l ) )< dx for i 2 0 and --I -s -s --1 so -q I ) E Ed((). Since I E h ( L ( I ) )= W ( % - ' ( I ) ) ,we have W ( h ( I ) )= we(!)and

-

If w s ( 0 )is dense in ll"',then UeEZnwS(I) is dense in Rn and so is ULEZn Ed = ( I ) . This implies that 'k'" is trivial ( p a = (0)). Conversely, suppose (0). Then UeEZnL'(I) is dense in R" and so is UeEZ,,w d ( I ) .Therefore w s ( x ) is dense in ll"' for all x E Tn.

Remark 8.7.4. Let fi : Iln 4 Tn converge to a self-covering map f : Tn -+ Tn as i -+ oo, and let A denote the toral endomorphism homotopic to f . If each fi is a strongly special TA-covering map, then there exists a continuous surjection h: Tn + Iln such that the diagram

commutes. This is easily checked a s follows. - As above, let A : 'll'" + 'IP be the toral endomorphism homotopic to f , and A : Rn -+ Rn the linear map induced by A. Then A is hyperbolic since fi is a TA-covering map. Use Theorem 8.2.1 for f;, then there is a continuous surjection Ei : Rn -+ Rn such that Z(K;(v),v)< iiK for v E Rn. Hence, for fixedjEZ

and there is N ( j ) > 0 such that

-

f ( v ) - T ~ ( vE) BaK( 0 )

(v E Rn and i, 9 2 N l i ) ) .

Thus, we have

-

h i ( ~-) E,(v) E Z - ' ( B ~ ~ , ( O ) ) ( v E Rn and i,q 2 N ( j ) )

xi

x

which implies that converges uniformly to some by Lemma 8.2.3 (1). Obviously 6 : In+ Rn satisfies all the properties of Theorem 8.2.1. If each fi is a strongly special TA-covering map, then satisfies the condition that xi(I+x) = I+xi(x) for x E Rn and I E Zn, and so does 51. Therefore, the conclusion is obtained.

xi

CHAPTER 9 P e r t u r b a t i o n s of Hyperbolic Toral Endomorphisms

In the previous chapter we have shown that every TA-covering map of an ntorus is characterized by a hyperbolic toral endomorphism which is homotopic to the map. To advance our discussion of the dynamics, we shall investigate some of characteristics of hyperbolic toral endomorphisms. The first section of this chapter is devoted to discuss Co-perturbations of hyperbolic toral endomorphisms in the class of self-covering maps, which is a result mentioned in 51.2 of Chapter 1. 59.1 TA-C* regular m a p s t h a t are not Anosov In this section we restrict ourselves a 2-torus and deal with hyperbolic toral endomorphisms of a 2-torus which are not expanding. We notice that such a toral endomorphism belongs to the subclass T A X U S S T A . The purpose of this section is to show the following statement mentioned in Theorem 1.2.2 of 51.2. T h e o r e m 9.1.1. Let A : T2 + T2 be a hyperbolic tom1 endomorphism but not expanding and let a > 0 be an arbitmry number. ( 1 ) If A is injective (i.e. A belongs to TAX), then there exists a TAdiffeomorphism of class CO",g: T2 --t T2, with d(g, A) < a that fails to be an Anosov diffeomorphism. (2) If A is not injective (i.e. A belongs to SSTA), then there is a TAregular map of class C*, g: T2 --t T2, with d(g, A) < a that fails to be a special TA-map or an Anosov differentiable map. where d(g, A) = max{d(g(x), A(%)) : x E T2). R e m a r k 9.1.2. It seems likely that Theorem 9.1.1 can be carried over to tori with dimension greater than 2. However, we do not settle here the case. Let A : T2 -+ T2 be as in Theorem 9.1.1 and 2 : R2 + R2 denote the linear automorphism which is a lift of A by the natural projection .rr : R2 4 T2. Let p1 and p2 be the eigenvalues of 2. Since A is not expanding, we may suppose that 0 < lpl( < 1 < Ip21. Denote as Zd(0) the eigenspace of 2 for pl and write z8(x) = z8(0) x for x E R2. Then 7; = {Z8(x) : x E R2) is a linear foliation on R2. It follows that 31 = ~ ( 7 %is) the family of stable sets in -0 strong sense and for each x E Rn, ?r : L (x) + LS(.rr(x)) is bijective. Here L"(xl) denotes the leaf of 3; through x' E T2. We already know that each

+

$9.1 TA-C" regular maps that are not Anosov

leaf of 32 is dense in T2.Let x' = n ( x ) E by ds(7r(y),7r(z))= ;i(Y,z )

281

T2and define a metric ds for Ls(x') for y, z E z d ( x )

where is a metric for R2 induced by a norm of R2. For the proof of Theorem 9.1.1 we need the following lemma.

Lemma 9.1.3. Let cp : T2-4 T2 be a homeomorphism satisfying the following conditions ( i ) , (ii) and (iii); ( i ) cp is homotopic to the identity map id, (ii) each leaf of 32 are cp-invariant, i.e. cp(Ls(x'))= Ls(x') for x' E T2, (iii) for y', z' E L d ( z ' ) there esists t ( y 1 ,z') > 0 such that (a> dS(cp(y'),cp(z1))l ( ( ~ ' 9z')dd(y',z'), (b) 0 < ((Y',z') I P-' where P = [ P I [ , (c) for e > 0 there is a constant 6 > 0 such that if dS(y',z') > E then ((y',

2')

5 ,u-' - 6.

Then the following properties hold ; ( 1 ) if A is a hyperbolic automorphism, then g = A o cp is a TA-homeomorphism, ( 2 ) if A is a hyperbolic endomorphism but not injective nor expanding, then g = A o cp is a TA-covering map. Proof. Since cp(Ld(0))= L s ( 0 ) by (ii), we can choose a lift by 7r satisfying p(i;d(0)) = z s ( 0 ) . Then, for L E Z2

and hence for all x E

p : R2 -+ R2 of cp

R2

because there is always a sequence {ei) c Z2 such that Zs(ei) + Z s ( x ) as + oo. Let jj = 2 o p. Then 3 is a lift of g by n, and by (9.1) we have for x E R2

i

To obtain the conclusion of the lemma we prepare the following three claims.

Claim 1. As before let w s ( x )denote the stable set of 3 at x. Then Z s ( x ) = w S ( x )for x E R2. -3 i Indeed, by (9.2) we have 7ji(x) E L ( A ( x ) ) for i 2 0. If y E W S ( x ) ,then - -3 -i -s i d ( L ( A ( x ) ) ,L ( A ( y ) ) )converges to 0 as i + oo because ;i(?(x),$(y)) -+ 0

CHAPTER 9

282

as i + m. Since -2 is hyperbolic, it follows that Zs(x) = Z3(y), and so y E zs(x), i.e. w 3 ( x ) c Z3(x). Conversely, let y E z3(x). Then $(x), -3 -i $(y) E L (A (x)) for i 2 0 by (9.2), from which we have

Hence, to show that w 3 ( x ) > ZS(x), it is enough to see that ds (gi o n(x), gi o n(y)) -+ 0 as i + m. If this is false, then we can find E > 0 satisfying the property that for j > 0 there is ij > j such that dS(gij o n(x),g2j o n(y)) > E, and then d3(gij o n(x), gij o n(y)) = ds(A o cp o 9'j-l o n(x), A o cp o gij-' o n(y)) = pdS(cp o gij-' o n(x), cp o gij-' o n(y))

5 p<(gil--' o n(x),gij-' < d3(gij-' o n(x), 9G-l -

o n(y))dS(gij-' o n(x), gij-' o n(y))

(by (a))

o n(y))

(by (b))

...

which shows that d3(gijon(x), gij on(^)) + 0 as j 1, thus contradicting.

+ oo because

of p(p-' -6) <

Since g is homotopic t o A, for some K > 0 we have z(3, -A) < K and hence there-exist - a constant hK > 0 and a continuous surjection ?E : B2 + R2 such that Ao h = Eo7j and id) < 6K (see Theorem 8.2.1). - -Since h is-3&uniformly continuous by Theorem 8.2.1 (3), it follows that h(w3(x)) = L (h(x)) for - -3 -3 x E R2, and hence by Claim 1 we have h(L (x)) = L (h(x)) for all x E B2.

z(%,

-s

-

Claim 2. L (h(x)) = zs(x) for x E B2. for p2 and define a map Indeed, let r ( 0 ) be the eigenspace of : -u L (0) -t ZU(0) by assigning to each x E Zu(0) the point in r ( 0 ) n Z3(h(x)).

$9.1 TA-CaO regular maps that are not Anosov

Then we have

From the first part of Proposition 8.2.2 together with the fact that &distance between r' and the identity map of I U ( 0 ) is bounded, it follows that hU -8 coincides with the identity map, and therefore L (h(x)) = za(x). If y, z E Za(x) and h(y) = h(r), then we have for i E Z

and so j($(y),@(z)) < 2bK for all i E Z. Combining this fact with the following Claim 3, we obtain that h is injective.

Claim 3. If y # z for y , r E Za(x), then ;i(@(y),3i(r)) + oo as i + -oo. Indeed, let E > 0 be a number such that Z(y, r) > E and let 6 > 0 be as in (c) of the condition (iii). Then we have

-

d(3-' (y), g-' (z)) = J(v-' o

from which ((n o 3-' (y), n o 3-'(z)) and so

x-' (y), v-'

o2-I

(z))

5 ,u-l - 6 by (c) of the condition (iii),

CHAPTER 9

and by induction

which shows Claim 3 because p-l/(p-l

- 6) > 1.

By (9.3) we can show (8.14) of 88.4 (replace 7 by 9). Hence, K-' is 2uniformly continuous. This is proved by the same way as the proof of Proposition 8.4.2. Since o = 51 o 3, it follows that 9 : W2 + R2 is a TAhomeomorphism under From this fact together with Lemma 2.2.34 and Theorem 2.3.14 we conclude that g is expansive (or c-expansive) and has POTP (notice that ;iisatisfies the condition (C) of Lemma 2.2.34 (3) by (8.14)). Therefore, (1) and (2) hold.

a.

Proof of Theorem 9.1.1. Since 31 and 3 (= { T ( ~ ( x ) ): x E Rn)) are transverse linear foliations on T2, we can find a coordinate domain D at 0 such that D=D"$Du where D d= {x E Ld(0) : dd(O,x) 5 a) and DU = ?r{y E r ( 0 ) : a(0, y) l a). Here a > 0 is a small number. To avoid complication we may consider D" and DU as Do = [-a, a] and DU = [-a, a] respectively.

Figure 35 Let c > 0 be a small number to be determined later and choose CODfunctions I C ~: D" -t D" and h1 : DU + W, as illustrated in Figure 35, satisfying

and hl(0) = c, 0 5 hl(x)

5 c. We define a function gl : D" $ Du -t R by

59.1 TA-Cw regular maps that are not Anosov

for ( x , , xu) E D'

$ Du.

Then we have

Since gl( , x u ) is the identity on a neighborhood of the end points of D o , we have g l ( D d $ D u ) = D d and so write (PI

( x 6 ,xu) = ( g l ( x s ,xu), xu)

for ( x , , xu) E D d $ D ~ .

Obviously cpl : D -+ D is a diffeomorphiim which is the identity map on a neighborhood of the boundary OD, and d(cpl, i d ) is small if so is c. Let us put

and let p = J p l J< 1 as in (b) of the condition (iii). If c is small enough, then for fixed xu E Du we have

for all x,, x: E D o , and

It is easy to see that for given E > 0 , if d d ( ( x , ,xu), ( x : , x u ) ) > E then there is 6 > 0 satisfying € ( ( x d xu), , ( & x u ) ) 5 p - l - 6. In the case when A is an automorphism, we define a diffeomorphism cp : T2+ T2by

Then cp satisfies all the conditions of Lemma 9.1.3. Therefore g = A o cp is a TA-diffeomorphism. Moreover we have

CHAPTER 9

288

and hence g is not Anosov. (1) was proved. To see (2)) choose a sequence

satisfying p-1 # 0. Here ( T 2 ) Adenotes the space of the inverse limit system for ( T 2 , A ) . Then there is X > 0 such that B ~ ( P -f ~l B) x ( 0 ) = 0 and Bx(pi) n ( B ~ ( P -U~Bx(po)) ) = 0 for i 5 -2 where Bx(pi) is the closed disc with radius A. We may suppose that D c B x ( 0 ) (take a > 0 small if necessary).

Figure 36 Construct Cm functions n2 : Ds -+ DS and 62 : Du -+ R, as Figure 36, such that 1 g ( x ) 1 5 1and -c 5 6 2 ( ~5) C. And define a function g2 : Ds$DU -+ R by g2(xs, xu) = 2 s ~ 2 ( ~ , ) ~ 2 ( ~ s ) .

+

Since g2( , x u ) is the identity on a neighborhood of the end points of D", it follows that g2(Ds$ D u ) = D s , and so put

Then cp2 : D + D is a diffeomorphism which is the identity map on a neighborhood of 8 D , and d(cp2,i d ) is small if so is c. Moreover we have

dS(cp2(xs,xu), cp2(x:, x u ) ) = t ( ( x s , x u ) ,(4,xu))dd((xs, xu), (x:,xu)) where

E((xs,x u ) ,(2:)x u ) ) = I1

+ & ( x u )6 2

It is easily checked that for fixed xu E DU

-n2(4) I. x, - 2:

(2.3)

$9.1 TA-Coo regular maps that are not Anosov

and for that

E

287

> 0, if d8((x,, xu), (xi, xu)) > E , as above we can find 6 > 0 such

€((xa, xu), ( 4 , x u ) ) 5 p-I - 6. Now, let us define a diffeomorphism cp :T2+ T2by

Figure 37 Then cp satisfies all the conditions of Lemma 9.1.3. Since A is an endomorphism but not injective, the composite g = A o cp is a TA-covering map by Lemma 9.1.3 (2) and clearly it is a C" regular map.

Figure 38 It only remains to see that g is not special and fail to be Anosov. Since c p l ~=~c p l l ~ "= id, we have 910%= AID"and hence D u = W,"(O)where 0 denotes the two sided sequence consisting of 0's. Since

CHAPTER 9

288

we have g l ~ , ( ~= , )AIBx(Pi) for i 5 -2 and hence

+

where a : ('P)A+ ('IP)Adenotes the shift map. Thus W,"(p) = g(Du P-1) n BIZ(()). Since g ) ~ ' + p - = ~ A O (PIDU+~-~ and P(X) = ( P ~ ( x P-1) p-1 for x E DU p-1, the sets W,U(O) and W,"(p) are illustrated as Figure 38. Therefore, g is not special by Theorem 6.6.13. The derivative Dog of g at 0 is calculated as above. Hence, Dog is non-hyperbolic, i.e. g is not Anosov. (2) was proved.

+

+

$9.2 One-parameter families of homeomorphisms Let X be a compact metric space with a metric d and let C(X) denote the set of all continuous maps of X equipped with the Co-topology, i.e. the topology induced by the metric d( f, g) = max{d(f (x), g(x))). For h E C(X) the path connected component of h in C(X) is the homotopy class of h, written as Hom(h). In the case when X is a compact topological manifold, from Remark 6.7.10 it follows that C(X) is locally contractible, which implies that each homotopy class is open and closed in C(X). Let H(X) be the set of all homeomorphisms of X onto itself. Then H(X) is a topological group. Given f E H(X), the isotopy class of f is defined as the path connected component of f in H(X). We denote it as Iso(f). By the following Theorem 9.2.1 we have that each isotopy class is also open and closed in H(X) if X is a compact topological manifold. Theorem 9.2.1 ( ~ e r n a v s k i i[Ce], Edwards-Kirby [E-K],). The homeomorphism group H(M) of a compact topological manifold M is locally contractible. In particular, H(M) is locally path connected. We omit the proof. If it is of interest, the reader should see ~ e r n a v s k [Ce] i and Edwards-Kirby [EK]. In this section we first discuss one-parameter families of homeomorphisms in a general setting, and show the following theorem. Theorem 9.2.2. Let A : 'P + Tn be a hyperbolic tom1 automorphism of an n-torus. Then there is a unique continuous map H : Iso(A) + Hom(id) with H(A) = id such that A o H ( f ) = H ( f ) o f for all f 6 Iso(A), where id denotes the identity map. A path w in H(X) is called an isotopy or a path of homeomorphisms. It follows that the map F : X x I + X defined by F(x,t) = w(t)(x) is continuous, i.e. F is homotopy such that for each t € I, F ( ,t) : X + X is a homeomorphism. Here I = [0, I]. Conversely, if F : X x I + X is a homotopy

59.2 One-parameter families of homeomorphisms

289

and each of F ( , t ) is a homeomorphism, then the map t t-+ F ( ,t) is a path of homeomorphisms of X. As before, let 2X denote the family of non-empty closed subsets of X. Then 2X is a compact metric space under the Hausdo& metric (see Remark 3.1.5). Let Y be a metric space with a metric dl. A map f: Y + 2X is said to be upper semi-continuous (resp. lower semi-continuous) if for y E Y and E > 0 there is 6 > 0 such that f (z) c U,(f (y)) (resp. f (y) C U,(f ( z ) ) ) whenever dl(y, z) < 6 (compare with the definition of semi-continuity in 53.3 of Chapter 3). Here U,(f (y)) denotes the &-openneighborhood of f (y) in X. It is easy to see that f: Y + 2X is upper semi-continuous if and only if {(x, y) : x E f (y), y E Y) is closed in X x Y.

Remark 9.2.3. If f: Y 4 2X is upper or lower semi-continuous, then the inverse images of open sets are F,-sets, i.e. a countable union of closed subsets. Indeed, if G is open in X , then B(G) = {F E 2X : F c G) and C(G) = {F E 2X : F n G # 0) are elements of a subbase of 2X (see 53.1 of Chapter 3). Thus it sufficies to see that f-'(B(G)) and f-'(C(G)) are F,-sets. For the case when f is upper semi-continuous, f-'(B(G)) = {y E Y : f (y) c G) = {y E Y : f(y) n Gc = 0) is open in Y. Since Y is a metric space, we see that any open subset 0 is expressed as 0 = Ai for an increasing sequence of closed subsets A;. Applying this fact, then we have that f-'(B(G)) is a F,-set. Since X is a metric space, we have also that G = U="=,; for closed subsets Ki. Thus

Uzl

where Fi = {y E Y : f(y) n Ki # 0) for i 2 1. Since each F; is closed, fF1(C(G)) is a F,-set. For the case when f is lower semi-continuous, the open set G is expressed as a union of closed sets K; such that Ki C int(Ki+') for i 2 1. If y E f (B(G)), then we have f (y) C G C U:"=,nt(K;). Since f (y) is compact in X, there is n > 0 such that f(y) C K,, and therefore f-l(B(G)) = U,",,{y E Y : f (y) C K,). Since f is lower semi-continuous, {y E Y : f (y) C K,) = {y E Y : f(y) n KE = 0) is closed in Y. Therefore, f-'(B(G)) is a F,-set. Since f -'(C(G)) = {y E Y : f (y) n G # 0) is open in Y, f-'(C(G)) is also a Fe-set.

-'

Let X and Y be metric spaces and let f : X + Y be a map. Denote as D(f) the set of discontinuous points of f .

CHAPTER 9

290

Remark 9.2.4.

This is checked as follows. (1): Take x $ D(f). Then, for a subset A C X we have x $! cl(A) \ f-'(cl(f (A))) when x $ cl(A). If x E cl(A), for {z,) C A satisfying x, + x we have that f(x,) + f(x) since f is continuous at x. Thus f (s,) E f (A) c cl(f (A)), and so x E f -'(cl(f (A))). Therefore, x $ cl(A) \ f-'(cl(f (A))). In any case we have

If x E D , then there exists a neighborhood Vf ),( of f (3) such that f (U,) P Vf(,) for any neighborhood U, of x. Put A = {y E X : f(y) $ Vf(,)}, then z E cl(A). Since x $ A, we have x $ f -'(cl(f (A))). Thus x E cl(A)\ f -'(cl(f (A))) and therefore D c UAcx[cl(A) \ f -'(cl(f (A)))]. (1) was proved. (2): Let x E D and take a neighborhood B of f (x) such that f(U,) (t B for any neighborhood U, of x. It is clear that a: E f-'(int(B)) and x $ int( f (B)). Thus

-'

Conversely, take a point x from the set of the above right side. Then x E f-'(int(B1) \ int(f-'(B)) for some B C Y. Thus f(U,) C int(B) for any neighborhood U, of x. This shows that f is not continuous at x, i.e. x E D. (2) was proved. It remains to show (3). Since int(Y \ B ) = Y \ cl(B) and f-'(Y \ B) = f-'(Y) \ f-'(B) = X \ f-'(B), by (2) we have the conclusion.

Remark 9.2.5. (4) G: open

=

u

F: closed

[cl(f -'(F))\ f -'(F)I

59.2 One-parameter families of homeomorphisms

By the above Remark 9.2.4 we have

(J [f-'(G) \ int(f -'(G))I

c

U lf -'( i n t ( ~ )\) int(f - ' ( ~ ) l l . BCY

G: open

Since int(B) is open and int(f -'(B)) the equality (5) is concluded. Remark 9.2.6. Then

> int(f -'(int(B))),

we have (4). By (3)

Suppose Y is separable and let {On) be an open base of Y.

D ( f ) = U[f

-'(0.)

\ int(f -'(On))] = U[cl(f - ' ( ~ n ) ) \ f -'(Sn)]

n

n

where Sn = Y \ 0, for n. For the proof it sufficies to see that

IJ

G:open

[f-'(G) \ int(f -'(G))] = U[f-'(on) \ int(f -'(On))]. n

To do so take a point x from the set of the left side. Then we have x E f-'(G) and x $ int(f-'(G)) for some open set G. Since G = IJOj for Oj E {On), there is no such that x E f-'(On,). But, x $! int(f-'(On,)) since x 6 int(f (G)) = int(Un f (0,)). Therefore, x E f (On,) \ int(f -'(On,)) c Un[f -'(on) \ int(f -'(On))]-

-'

-'

-'

Remark 9.2.7. Let Y be a metric space and X a compact metric space. If f: Y -t 2X is upper or lower semi-continuous, then Y \ D ( f ) is a Baire set. This may be shown by using the above remarks. Indeed, by Remark 9.2.6 we have D(f) = Un[cl(f-'(0,)) \ f-'(On)]. Denote as Pn the complement for n. Then each Pn is open and dense. Thus of cl(f-'(0,)) \ f-'(0,) Y \ D(f) = Pn is a Baire set.

nn

Let {At : t E I) be an upper semi-continuous family of points in 2X, i.e. the map from I to 2X defined by t H At is upper semi-continuous. For t E I let f t : At -t At be a homeomorphism. If the map (x,t) H ft(x) is a continuous map from {(x,t) : x E At,t E I) to X , then we say that the family {ft : t E I) is a path of homeomorphisms on {At). Remark 9.2.8. For a homeomorphism f : X -t X let CR(f) denote the chain recurrent set of f . If {ft : t E I) is a path of homeomorphisms of X, then

(1) { C ~ ( f t :) t E I) is an upper semi-continuous family, (2) { f t p ~ ( :~t E~ I) ) is a path of homeomorphisms on {CR(ft)).

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Indeed, suppose (1) is false. Then we can find to E I and EO > 0 such that for every n > 0 there is t , E I with Ito - tnl < l l n so that C R ( f t n )(Z Uco(CR(fto)).Take xn E C R ( f t , ) such that xn $ Uc,(CR(fto)).We may suppose that x, converges to some xo E X. Then xo $ C R ( f t o ) . Since x, x,, for 6 > 0 there is a periodic 6-pseudo orbit { x , = yo, yl, ,yk = x,) of f t n . Since d ( f t o ,ft,) < 6 for large n, it follows that the sequence is a xo for fto, thus periodic 26-pseudo orbit of f t o , from which we have xo contradicting. Therefore, ( 1 ) holds. (2) is clear from the fact that ift)is a path of homeomorphisms of X.

-

-

. ..

Let A be a closed set of X and let f : A -+ A be a homeomorphiim. As above, let {At : t E I ) be an upper semi-continuous family and suppose { f t : t E I ) is a path of homeomorphisms on { A t ) . A family {ht : t E I ) of continuous surjections ht : At -t A is said to be a continuous family of semi-conjugacy maps between f and { f t ) if the following properties hold; ( 1 ) the map ( x ,t ) H h t ( x )is a continuous map from { ( x , t ): x E At, t E I ) to A, (2) f o ht = ht o f t for all t E I.

Lemma 9.2.9. Let f and { f t : t E I ) be as above. If f is expansive and h: A0 -t A is a continuous surjection satisfying f o h = h o fo, then there exists at most one continuous family of semi-conjugacy maps {ht : t E I ) between f and { f t ) such that ho = h. Proof. Let { h t ) and { h i ) be continuous families of semi-conjugacy maps between f and i f t ) satisfying ho = hb = h. If { h t ) # { h i ) , then for E > 0 there is t E I such that ht # hi and d(ht,hi) < E , which contradicts the following Lemma 9.2.10.

Lemma 9.2.10. Let (2,d z ) and (W,d w ) be metric spaces and let f : Z -t Z and g: W -+ W be homeomorphisms. Let h, h' : W -+ Z are continuous maps satisfying f o h = h o g and f o h' = h' o g respectively. Suppose f is expansive and e > 0 is an expansive constant for f . If

then h = h'. Proof. If h # h', then we have

Therefore, the conclusion

in

obtained.

$9.2 One-parameter families of homeomorphisms

293

Remark 9.2.11. In Lemma 9.2.10 the assumption that f is expansive can be changed for the weaker assumption that f has sensitive dependence on initial conditions. Let {ft : t E I) and {fi : t E I) be paths of homeomorphisms on upper semi-continuous families {At) and {A:) respectively. An inverse path of {ft} is defined by {ft) = {fl-t : t E I) which is a path of homoeomorphisms on {A1-t : t E I). If A1 = Ah and fl = f;, then we can define aproduct of {ft) and {f:) by {ft).{fi) = {ft. fi : t E I) where 0 5 t 1 112 f t f; = f;t-l 112 5 t 5 1. Then {ft) {fi) is a path of homeomorphisms on an upper semi-continuous family {At A: : t E I}where

.

{

.

Let Ao = Ah, A1 = A: and fo = fi, fl = fi. We say that Ift) and {fi) is homotopic if there are an upper semi-continuous family {A,,, : t , s E I) and a family {ft,, : t , s E I) of homeomorphisms ft,, : At,, -+ At,,, with the property that the map from {(x,t,s) : x E At,,,t,s E I } to X defined by (x, t, s ) I+ ft,,(x) is continuous, such that A t , = At, ft,o = f t (vt E I ) A O ,= ~ Ao, fa,, = fo (Va E I ) All, =A,, fl,, = f i (Vs E I). At,l = A:, ft,l = fl (W E I), Here ({At,,), {ft,,)) is called a homotopy from {ft) to {f:). Remark 9.2.12. Let {ft) and {fi) be homotopic and let ({At,s),{ft,8)) denote a homotopy from {ft} to {fl). Let f : A -,A be a homeomorphiim and suppose there is a continuous family {ht,,} of semi-conjugacy maps between f and {ft,,). If f is expansive, then ho,, and hl,, do not depend on s. This is easily checked by Lemma 9.2.10. Let f : X + X be a homeomorphism and denote as C(f) the set of homeomorphisms h of X commuting with f , i.e. f o h = h o f . It is clear that C ( f ) is a subgroup of X(X). We call this subgroup the centralizer of f . If f is expansive, then by Lemma 9.2.10 it follows that C(f ) is discrete in X(X).

Remark 9.2.13. Let M be a compact topological manifold. Let {At : t E I) be an upper semi-continuous family of points in 2M and let {ft : t E I) be a path of homeomorphisms on {At). Suppose A. = A1 and fo = fi. Then, it seems likely that the following (MC) is true whenever fo is expansive. (MC) If {hi : t E I}is a continuous family of semi-conjugacy maps between fo and {ft) and if ho is the identity map of Ao, then hl is a homeomorphism, i.e. hl belongs to C(fo

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294

It is easy to see that (MC) is equivalent to the following assertion. If for some ho E C ( f o l A othere ) is a continuous family of semi-conjugacy maps {ht :t E I ) between fo and { f t ) , then hl also belongs to C ( f o l A o ) . A sequence of groups and homomorphiims

is called a short exact sequence if h is injective, Im(h) = Ker(hl) and h' is

surjective. If, in addition, h has a left inverse, the short exact sequence is said to split. In this case Gzis isomorphic to the direct sum of G1and G3.

Lemma 9.2.14. Let X be a pamcompact Hausdorff space and let p : X + S 1 be a fiber bundle over a circle with connected fiber. Let bo E X and denote as Xo the fiber over p(b0). Then the following sequence of fundamental groups and induced homomorphisms 0

+

xi ( X Obo) ,

-

( X ,bo) JL7 r l ( ~ ',p(bo))

is a short ezact sequence where i : Xo

-, X

O

denotes the inclwion map.

Proof. This is easily checked by the fact that p : X -+ S 1 is a fibration (see Remark 6.5.2) and x z ( S 1 ) = 0 (see Remark 6.7.9). The detail is left to the readers.

Theorem 9.2.15. Let S 1 be a circle R/Z and e : R -t S 1 denote the natuml projection. Let X be a compact connected locally connected metric space and let p : X + S 1 be a fiber bundle with connected fiber. Suppose X is semilocally 1-connected. Let f: X -+ X be a homeomorphism which preserves each fiber for p, and for t E R let Xe(t) = p-'(e(t)) and f e ( t ) = flxe,,,. Suppose A : T n -t T n is a hyperbolic tom1 automorphism and for some to E R there is + )x l ( T n , C O ) , where bo and co are base a homomorphism 4 : T ~ ( X ~ ( ~ , ) ,+ points, such that the following dtagmm commutes:

where u , (resp. v,) denotes the induced isomorphism by a path u (resp. v) in X (resp. T n ) from bo (resp. A(co)) to f ( b o ) (resp. co) as in Lemma 6.1.4. Then the following properties holds: (1) there exists a unique continuous family ht : Xe(,) -t T n , t E R, of continuous maps with hto(bo)= Q such that hto* = q5 and A o ht = ht 0 f e ( t ) for all t E R,

59.2 One-parameter families of homeomorphisms

295

(2) there exists a homeomorphism T : Iln -, Iln homotopic to the identity map such that T E C(A) and To ht = ht+l holds for all t E R, (3) if the following short exact sequence splits:

then the map T is the identity map. From the above theorem we have the following corollary.

Corollary 9.2.16. Let A : Iln + Iln be a hyperbolic tom1 automorphism and let f t : Iln + P, t E I, be an isotopy starting at A. Then there is a unique continuous family {ht : t E I) of semi-conjugacy maps between A and {ft) such that ho is the identity map. Furthermore, if fo = fl then hl is also the identity map. Proof. Let {gt : t E I) = {ft).(ft) and define g: 'IP' x S1 -r 'IP' x S1by g(x, t) = (gt(x), t). Here S1 = [O, 1]/{0,1). Since go = gl = A, by Theorem 9.2.15 there is a unique continuous map H : Tn x S1 + Iln such that HITnx(Ol(x, 0) = z and A o H = H o g, which shows the first part of the corollary. When fo = fl, let us define g: Iln x S1 -, Tn x S1 by g(x, t) = (ft(x), t). Then we have a semi-conjugacy map H' from g to A such that HiTnx(olis the identity map. Since n l ( P x S1) E ?rl(Tn) $ ?rl(S1), by Theorem 9.2.15 (3) it follows that ho = hl. Proof of Theorem 9.2.2. For f E Iso(A) let {ft) be an isotopy from A to f . Then, by the first part of Corollary 9.2.16 there is a continuous family {ht) of semi-conjugacy maps between A and {ft), and so we define H : Iso(A) -r Hom(id) by H ( f ) = hl. This is well-defined by the last part of Corollary 9.2.16. By this fact together with Theorem 9.2.1 we have that H is continuous.

Remark 9.2.17. The statement of Theorem 9.2.15 can be extended to the more general case of hyperbolic infra-nil-automorphisms. This implies that Theorem 9.2.2 and Corollary 9.2.16 also hold in such a general setting. For the proof the readers may refer to Hiraide [Hig]. For the proof of Theorem 9.2.15 we prepare the following lemma.

x x

Lemma 9.2.18. Let X be a compact metric space with metric d and let be a connected topological space. Let p : X X be a covering map. If X is connected and locally connected, then there is a compatible metric p for such that (1) all covering tmnsfonnations for p are isometries under p, p) is a complete metric space, (2) (3) for all x E X and T > 0 the closed ball {y E : p(x, y) T ) is compact.

(x,

x

<

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286

x

Proof. Since p : --, X is a covering map, there exist a metric ;ifor X and a constant So > 0 satisfying the properties in Theorem 6.4.1. Fix 0 < 6 < h0/2. For x, y E X let {xi : 0 5 i 5 !+I) be a 6-chain from x to y, i.e. a(xi, xi+l) 5 6 for 0 5 i 5 I , and define p by

where the infimum is taken over all finite 6-chains from x to y. By the triangle z(x, y), from which p is a metric for inequality of we have p(x, y) Clearly a(%,y) = p(x, y) if z(x, y) 5 6. Thus p is a compatible metric and by Theorem 6.4.1 (3) we obtain (2). From the construction of p together with Theorem 6.4.1 (2) it is easy to see (1). To show (3), let K be a compact subset of X and write Na(K) = {y E X : p(y, K) 5 6) where p(y, K) = min{p(y, x) : x E K). If p(y, x) 5 6, then by the definition of p we x) = p(y, x), from which - have z(y, x) 5 6, and hence N6(K) = {y E X : d(y, K ) 5 6). Let {yi) be a sequence of points in Na(K). Then we can find a sequence {xi) of points in K such that yi E B6(xi) for all i. Here B6(xi) = {z E : z(xi,z) 5 6). Since K is compact, clearly ) to some x, E K. Hence yj E Bz6(xrn) there is a subsequence { ~ jconverging if j is sufficiently large. Since 26 < 60, by Theorem 6.4.1 (1) we have that BZa(x,) is compact, and so {yj) contains a subsequence {yk) converging to some y,. Since a(Yk,xk) 5 6, it follows that y, E N6(K). Therefore, N6(K) is compact. From the fact that z(x, y) = p(x, y) if p(x, y) 5 6, it is easy to see that Nza(K) = Na(Na(K)), which shows that NZa(K) is also compact. By and T > 0, then we can induction, so is Nna(K) for every n 1. Let x E find n 1 such that {y E X : p(x, y) 5 r ) c Nna(K). Therefore, (3) holds.

a

x.

>

x

>

>

x

Proof of Theorem 9.2.15. Since X is semilocally 1-connected, by Theorem 6.2.11 there is the universal covering .~r: X + X. By Proposition 6.2.7 we can find a continuous map p :X -+ R such that the diagram

-

commutes. Since R is contractible, it follows that j5 : M + R is a trivial fiber bundle, i.e. there is a homeomorphiim cp : p-'(0) x P -, X such that j j o cp(x, t) = t for all ( x , t). For t E R denote as Xt the fiber over t. For simplicity let to = 0. Then for i E Z we have n(Xi) = Xe(o) and each restriction ?r : Xi + Xe(o)is the universal covering.

59.2 One-parameter families of homeomorphisms

297

Choose $0 E Eo such that ~ ( $ 0 )= bo. Since u is a path in X starting at bo, we can find a lift Ti of u by n satisfying Ti(0) = Zoo. Then Ti(1) E Xo such that -and - n(B(1)) = f(bo). Hence there is a lift f : X + X of f by -.~rf (bo) = B(1). Since f preserves each fiber for p, it follows that f (Xt) = X t for all t E R. Put fo = f lXo. Let n' : Rn + Tn be the natural projection and choose 4 E Rn such that 7r1(EO)= co. Then there is a lift 71 : Rn + En of A such that x(&) = g(0) where 5 is a lift of v by T' with E(1) = 6. Since A is a hyperbolic toral automorphism, it is clear that is conjugate by some translation to the linear map which covers A. Hence by Lemmas 8.2.3 and 8.2.4, has the following properties:

-4

( P l ) for K > 0 and od > 0 there exists J > 0 such that if d(A (x),T(y)) 5 K 5 J, then J(x, y) 5 E , for all i with JiJ (P2) for given K > 0, if 97i'(x), z ( y ) ) 5 K for all i E Z, then x = y. (P3) for K > 0 there exists bK > 0 such that for any K-pseudo orbit -i {xi : i E Z) of there is a unique x E Rn so that d(A ( x ) , xi) bK for i E Z. where 2 denotes the euclidean metric. Let G(T) denote the covering transformation group for n. Then G(n) is isomorphic to nl(X, bo) by Theorem 6.3.4. Denote as F the subgroup of G(n) which corresponds to T ~ ( X , ( ~bo). ) , Since X,(o) is the fiber over e(0) for the fiber bundle p : X + S1,G(n)/F is isomorphic to Z. Sincep,(nl(Xo, bo)) = 0, it follows that a ( x t ) = Xt for all a E F and t E R.

x

<

- Let f, : G(n) -, G(n) denote the induced homomorphism as f. Then fop = f ,(p) o 7 holds for all P E G(n) (see Lemma 6.3.10). Since f , ( ~ ~ ( x , (bo)) ~), = .~rl(X,(~), f(bo)), we have f,(F) = F and the restriction f, : F + F is

To,.

consistent with the induced homomorphism Denote as G the covering transformation group for n' and as A, : G -r G the induced homomorphism. As usual, we consider 4 : T(X,(~),bo) + nl('lP, co) as a homomorphism from F to G. Since A, o 4 = (v, o 4 o u,) o fe(0)+by the assumption, from the choice of 7 and we have 2, o 4 = 4 o 7, on F. Hence, letting F(b) = {a(b) : a F) (b E W) and G(c) = {L(c) : L E G ) (c E Rn),we have the following commutative diagram :

x

-

-

where 4' and 4" are defined by 4'(a(bo)) = 4 ( a ) ( ~ 0 and ) 4"((afo(bo))) = 4(a)(x(&,)) for all a E F, respectively.

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298

x (xo)

Let d be a metric for X and choose a metric p for as in Lemma 9.2.18. Then the following property holds: (P4) for t E R there is Kt > 0 such that C UK, where UK,( X O = ) { x E : p(x,xo) < Kt). For, let D be a compact covering domain for ?r : X t + Xe(t) and choose K > 0 such that D c UK(&). Here ~ ~ (= {8y E~ ): P ( Y , 8 ~<) K ) . Then UaEFa ( D ) = Hence, by Lemma 9.2.18 ( 1 )we obtain (P4). -Let x E and choose t E R such that x E Since f ( X t )= the orbit { 7 ( x ): i E Z ) is a subset of Sit. By (P4) there is Kt > 0 such that for i E Z there is ai E F such that ai(80)E UK,(?(X)). Notice that ai(go)E for all i E Z. Since 7 is uniformly continuous under p, we can find Ki > Kt such that p(7(~),7(~)) < K; whenever p(x, y) < Kt. Put Bt = { z E X : p($, t) I Kt Ki). Then Bt is compact by Lemma 9.2.18 (3). Hence

xt

xt.

xt.

zt, xo

+

Z ~ ) .by (P3) there exists is finite. Let L = Lt = max,~x p ( + ( a ) ( ? l ( ~ ~ ) ) ,Then bL > 0 such that any L-pseudo orbit of 2 is GL-traced by a unique point in Rn.Notice that L depends on t.

Claim 1. For z E X let {ai : i E Z ) be as above. Then (4' o ai(80)): i E Z) is an L-pseudo orbit of A. i+l Indeed, since J,(ai)(7(&o))= T(ai(80))E UK:( f (z)),we have ( a i ) ( T ( 8 0 ) ) , ~ i + l ( 8<0 )Kt ) Ki, and hence a z l o ? , ( a i ) E X, from which there is -yi E E such that J,(ai) = ai+l o -yi. Then we have

+

and so

for all i E Z. Therefore, Claim 1 holds. Let

t,

be a bL-tracing point of (4' o ai(Z0): i E Z).

$9.2 One-parameter families of homeomorphisms

Claim 2. z, is independent of the choice of { a i ) . Indeed, for i E Z let a:(&)E uK, ( 7 ( x ) )for some a: E F. Then {qYoa;(&)] is an L-pseudo orbit of 2 and there is a unique z : E Rn which is bL-tracing {4'0a:(io)). since p ( ~ i ( 6 0 )a:(&)) l < Kt Kt < Kt +Ki, we have a: = ai o-yi for some -yi E E, and hence for i E Z

+

Therefore

t,

= 2: by ( P 2 ) .

-

Define a map H : X -+ Rn by x w z,. This is well-defined by Claim 2. Let x = $0. Then, from the choice of f and 3 it is easily checked that D(4' 0 c ~ i ( c o ) , ; l ' ( ~ o<) )L = Lo for all i. This shows IT(i0) = 4. For x E X let { a i ) be as above. Then we have for i E Z

-

L

and hence A o H = 1? o f holds on 5?. Let x E X and { a i ) be as above. Since ui(so)E U K ~ ( ? ( X ) )for all i E Z , for given J > 0 there is a neighborhood U ( x ) of x in such that ai($o) E y E U ( s ) and all i with liJ 5 3. Then D($ o ~ ( y ) , oZ U K ~ ( ? ( ~ for )) H ( x ) ) 5 2 6 for ~ i with lil 5 J. Choose J large enough, then D ( Z ( x ) , g ( y ) ) is small by ( P I ) , which shows continuity of p. -

Claim 3. H o a = $ ( a ) o W for a E F. Indeed, let x E and { a i ) be as above. Since a,(&) E u K t ( 7 ( x ) ) ,it Ifollows that ?:(a10 E ~ ~ . ( f ) ( a ( z and ) ) , so D ( A o H o a ( x ) , $ ' o f i . ( a ) o ai(bo))< bL for all i E Z. Since

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we have

from which

z o a(x) = d ( a ) o z ( x ) . Therefore, the conclusion is obtained. - -

By Claim 3 for each t E R the restriction H-: Xt -+ Rn can be projected to a continuous map ht : Xe(t) + Tn. Since A o H = o 7, we have that A o ht = ht o fe(,) for all t E R. Since a& (), = 50, clearly ho(bo) = co, and by Claim 3, hoe = 4 holds. Since H is continuous, it follows that {ht : t E R) is a continuous family. - hh(b0) To show uniqueness, let {hi) be another continuous family such that = Q, hb, = 4, and A o hi = hi o f,(,) for all t E R. Since n : X -+ X is the universal covering, there is a lift ffl : X + Rn of the family {hi) -I such that H (bo) = 4. Then o 3 = 3 o 7. Since hh, = 4, it follows for all a E F. By this and Claim 3 we have that that H o a = 4(a) o D(H,R1) = s u p { ~ ( ~ ( x ) , ~ ( :xx) )E X) is finite, and hence = R1by Lemma 9.2.10 and (P2). (1) of Theorem 9.2.15 was proved. Next we show (3). Let P denote the subgroup of F corresponding to Ker(4). Since the sequence of (3) splits, we can choose p E G(n) with @(To)= XI such t h a t f o p = r e o ~ o ~ f o r s o m P. e n ~~ a k e a l i f t ~ ' : x + R ofthe " continuous family {ht) such that z1(p(60)) = Z0. Then HI o ,f?= Hence

z

z

x

z.

z1

Let a E F. Then 0 o a = K' o a o /3 for some n1 E P. This shows oa = 4 ( a ) o 3.Hence = 1? by Lemma 9.2.10 and (P2). Since /3(xo) = XI, it is clear that p ( X t ) = Xt+1for all t E R, and therefore ht = ht+l for all t E R. (3) of Theorem 9.2.15 was proved.

89.2 One-parameter families of homeomorphisms

301

To show (2), take p E G(n) such that p(Xo) = 15'. Then there is a lift H : X + Rn of the family {ht} such that z1(p(6o)) = 6 , i.e. 3 o p = Z. Let a = 7 '; (p) o p-' . Then a E F and 2 o = 3 o ( 7 o a). Hence we have

-I

where y = 4of,(a) E G. Notice that y is a translation on Rn by some e E Zn. We choose c E Rn satisfying the equation 2(c) - c = e, and define 'i: Rn -t Rn by T(x) = x c. Then

+

Combining this and (9.4) we have

Since (F o P')o a = 4 ( a ) o ( po z l ) for CY E F, by Lemma 9.2.10 and (P2) we = (2) was proved. obtain that T o

z.

Most of the above discussion will be applied for one-parameter families of self-covering maps (more generally, also for those of continuous maps). In the remainder of this section we discuss the case of hyperbolic toral endomorphisms of types (11) and (111). Let M be a compact topological manifold and denote as C(M) the set of self-covering maps of M equipped with the C0 topology. For given f E C(M) we denote as Iso(f) the path connected component of f in C(M). &om the following theorem we have that Iso(f) is open and closed in C(M).

Theorem 9.2.19. C(M) is locally contractible. Proof. Let f E C(M) and let d be a metric for M. By Theorem 9.2.1 together with the fact that the map from X(M) to C(M) defined by h w f o h is continuous, it suffices to show that for g E C(M), if d( f,g) = max{d( f (x), g(x)) : x E M} is small enough, then there is h E X ( M ) with d(h, id) small such that f oh=g. a lift of f by Let n : M + M be the universal covering and 7 : M + n. Let g E C(M) and suppose d( f , g) is sufficiently small. By Remark 6.7.10 there is a homotopy f t : M + M (0 5 t 5 1) from f to g. Choose a homotopy f t : M + a s u c h that 7o= f and f t o 7 r = n o T t for allt. put g = f l . Then -d( f , 3) = d(f , g). Here z(f,g) = sup{z(f (x), g(z)) : z E ?i?) and 2 is a metric for as in Theorem 6.4.1. Let G(n) denote the covering transformation group for n. By Lemma 6.3.10 : G(n) + G(7r) such that there are homomorphisms

Tt,

-

f t 0 a = ft,(a)

o

7t

for all a E G(n),

CHAPTER 9

302

Tt+

7,

and by continuity it follows that does not depend on t. Hence, = 3,. - --1 Let a E 1m(7,), then f o f , (a)(x) = a o f (x) for x E from which we have -- 1 f o a(y) = ?;'(a) o jhl(y) where 7(x) = y. Hence

z,

a.

for all a E G ( r ) and x E Therefore, 7-l o 3 can be projected to a homeomorphism of M , say h: M -t M . Then f o h = g. Since d( f , g) = a(7,I ) , we obtain that d(h, id) is small if so is d(f,g). For expanding toral endomorphisms the following theorems are easily checked by the same way as the proof of Theorem 9.2.2. The proofs are left to the readers

Theorem 9.2.20. Let A : W -+ "IF be an expanding endomorphism of an n-torus. Then there is a unique continuous map H : Iso(A) -t Hom(id) with H(A) = id such that A o H ( f ) = H(f ) o f for all f E Iso(A). By Theorem 6.7.11 together with Theorem 9.2.19 we have that a hyperbolic toral endomorphism of type (111) does not possess the property as in Theorems 9.2.2 and 9.2.20. However, the following theorem holds.

Theorem 9.2.21. Let A : W -t Tn be a hyperbolic endomorphism of an n-torus. Suppose {ft : t E I) is a path of self-covering maps ft : "IF + Tn starting at A. Here I = [O, 11. Then there ezists a unique homotopy kt : S -r S, t E I, such that (1) ho =_ id, (2) d o ht = ito itfor all t E I where it: S + S (resp. A : S -t S) is the homeomorphism of the solenoidal group (resp. the automorphism) associated with f t (resp. A) as in Theorem 7.2.4. Proof. Let r : Rn -t Tn be the natural projection. Denote as 71 : Rn + Rn the linear map which covers A, and as 7, : Rn + Rn, t E I, a unique path of homeomorphisms satisfying

From the proof of Theorem 9.2.2 we see that there exists a unique path of continuous surjections Kt, t E I, such that (1) =_id, (2) A 0 ht = 0 (3) a(x,(~),.) < ,6 for a l l v E R ~ .

5

xt Tt,

59.2 One-parameter families of homeomorphisms

303

By (7.8) of 57.2 we can find a homomorphiim 11, : Rn -, Rn such that ll,(Rn) is dense in a certain solenoidal group S. Thus, as in (7.9) define

and then we have a commutative diagram

xi

{Ti

where = $ o Kt o 1,-' for t E I . Since : t E I ) and 2 ' are biuniformly continuous by Lemma 7.2.1, 'J: and A' induce continuous bijections ft and A of the solenoidal group S. Moreover, {hi : t E I ) induces a homotopy { i t : t E I ) of continuous surjections it : S -+ S such that io= id and A o it = ito jt for all t E I . The uniqueness of { k t ) follows from Lemma 9.2.9.

C H A P T E R 10 Fixed P o i n t Indices

In Chapter 6 we have seen that the family of stable sets in strong sense of a TA-covering map has a structure which is called a generalized foliation. The purpose of this chapter is to discuss orientability of generalized foliations on topological manifolds without boundary, and to establish the fixed point index theorem for TA-covering maps.

510.1 C h a i n complexes In this section we shall introduce some algebraic preliminaries to discuss singular homology. Let C = {an: Cn -+ be a chain complex, i.e. C is a sequence

of abelian groups Cn and homomorphisms a,, called boundary operators, such that an o an+, = 0 for all n E Z. Since an o an+l= 0, we have Im(an+l) C Ker(an). The factor group Hn(C) = Ker(Bn)/Im(an+l) is called the n-th homology group of C, and an element in Ker(an) is a cycle. We say that an element in Hn(C) is a homology class, and denote as [c] a homology class represented by a cycle c. Let C = {a, : Cn -t Cn-,) and C' = (8; : C:, -, CL-,) be chain complexes. The family cp = {cp,) of homomorphisms cp, : C, + C:, is said to be a chain map if 8; o cp, = cpn-1 o an for all n E Z, i.e. the diagrams

1vn-

1

commute.

A chain map cp = {cp,) is an isomorphism if each of cp, is an isomorphism. The family of identities id = {id,) : C -, C is a chain map. If cp = {cp,) : C -4 C' and cp' = {cp;) : C' --, C" are chain maps, then cp'ocp = {cp',ocp,) : C -+ C" is also a chain map. Let cp = {cp,) : C + C' be a chain map. Then cpn(Ker(Bn)) c Ker(8;) and cpn(Im(an+l)) C Im(8;+,). Thus, for each n E Z we can define a homomorphism V* : Hn(C) -t Hn(Ct) by P*([C])= (vn(c)]-

$10.1 Chain complexes

305

Two chain maps cp, cp' : C + C' are said to be chain homotopic if there is a family = {a,) of homomorphisms an : Cn -+ C;+l such that for n E Z In this case, the family 8 is called a chain homotopy from cp to cp' and we write cp LX 9'. It is easily checked that the relation N is an equivalence relation. Proposition 10.1.1. If chain maps cp, cp' : C + C' are chain homotopic, then cp, = cp: : H,(C) -+ Hn(C1)for all n E Z .

Proof. This follows from the fact that ~ + ( [ c=] [) ~ n ( c = ) ] [ ~ h ( c ) a;+, 0 @ n ( ~ )an-I 0 an(^)] = [cpk(c)] ( since &(c) = 0 )

+

+

= cp:([cl).n

A chain map cp : C -+ C' is said to be a chain homotopy equivalence if there exists a chain map 11, : C' -+ C such that 11, o cp N id and cp o 11, N id. Here 11, is called a chain homotopy inverse of cp. If cp : C -+ C' is a chain homotopy equivalence, then the induced homomorphism cp, : H n ( C ) -t Hn(C1)is isomorphic for all n E Z. A sequence of abelian groups and homomorphisms

-

hn .-.--+An+1hn+, An -An-l -... is called an exact sequence if Im(hn+1) = Ker(hn) for all n. If, in particular, a sequence O--tAh.A1ZA"+O is exact, then it is a short exact sequence. A sequence of chain complexes and chain maps

is called a short exact sequence if all rows of the following commutative diagram are exact :

0-

cn

- Vn

I

C:,

Pn

c::

-0

306

CHAPTER 10

Given a short exact sequence of chain complexes and chain maps

homomorphisms cp, : Hn(C) + Hn(C1) and cp: : Hn(C1) + Hn(C1') are induced as stated above. Moreover, a homomorphism a, : Hn(C1') + Hn-l(C), which is called the connecting homomorphism, is defined as follows. Let a E Hn(C1'). Then a = [c"] for some c" E Ker(8:). Since cpk : Ck + C: is surjective, we have cp',(c') = c" for some c' E CL, and

Thus ak(cl) E Ker(~:-~) = Im(cpn-1). Since cpn-l is injective, there is a unique c E Cn-1 such that ak(cl) = cpn-l(c) and then

which shows Bn-l(c) = 0 since cpn-2 is injective. Thus, c E Ker(an-l), from which [c] E H,_l(C). The element [c] is independent of the choice of c' and c". Therefore, we can define a map a, : Hn(C1') + Hn(C) by &(a) = [c]. It is easy to see that a, is a homomorphism.

Proposition 10.1.2. (1) For a short exact sequence of chain complexes and chain maps O+C3C'LC"+O, the sequence

is exact. (2) Given the diagram of chain complexes and chain maps

if the first row and the second mw are both exact, then the following diagram commutes :

510.1 Chain complexes

307

Proof. (1) is checked as follows. Im(cp) = Ker(cpl): It is clear that cp: o cp, = 0, i.e. Im(cpl) c Ker(cp). Conversely, let y E Ker(8k) such that cpL(y) E Im(8:). Then we can find z' E C:+, with 8:+l(z') = cp',(y), and y' E CL+l with cpL+l (y') = z'. Since cpL(y (y')) = 0, there is x E Cn such that cpn(x) = y - 8n+1(y'). Then x E Ker(8,). Therefore, [y] = cp,([x]) E Im(cp,). Im(cpt) = Ker(8,): If y E Ker(al), by the definition of 8, we have 8, o cp:([y]) = 0, and so Im(cp:) C Ker(8,). Let z E Ker(8:). If B,([z]) = 0, then there exist y E Ck and x E Cn such that cpL(y) = z and an(y) = 8, o cpn(x). Thus, y - cpn(x) E Ker(8,) and pL(y - cpn(x)) = z, from which [z] E Im(cp:). Im(8,) = Ker(cp,): By the definition of 8, we have cp, o 8, = 0. For x E Ker(BnW1)let cpn-l(x) E Im(8;). Then there is y E Ck such that 8L(y) = cpn-~(x),and so a:(%) = cpk-, o cpn-l(x) = 0 where z = cpL(y). Therefore, z E Ker(8:) and [XI = a,([z]). (2) is clear by definition.

The following lemma is easily checked and so the proof is left to the readers. Lemma 10.1.3 (Five lemma). Given the commutative diagmm of abelian groups and homomorphisms with exact rows

if XI, X2, Xq and X5 a= isomorphisms, then so is X3.

Let C = (8, : Cn -t Cn-l) be a chain complex. Then C' = (8, : C; + Ck-,) is said to be a subcomplex of C if for each n E Z (1) Ck is a subgroup of Cn, (2) an = anlch9 (3) an(cL) C CL-1. It is clear that a subcomplex C' is a chain complex. are subcomplexes If C' = (8, : CL + Ck-,) and C" = (6, : C: -t C:-,) C:-,) is a of the complex C, then C' + C" = (8, : CL C: --+ C;-, subcomplex of C which is called the subcomplex generated by C' and C". Given chain complexes C = (8, : Cn -, Cn-1) and C' = (8; : C; --+ Ck-,}, the direct sum of C and C',

+

+

is a chain complex. Here each 8, is defined by &(c, c') = (dn(c), 8;(c1)).

CHAPTER 10

308

Lemma 10.1.4. For chain complexes C and C'

for all n E Z. Proof. This is easily checked and so we omit it. Let C = {an : Cn + Cn-,) be a chain complex and let C' = {an : CA + CA-,) be a subcomplex of C. Then for each n E Z the boundary operator 8, induces a homomorphism an : Cn/CA + Cn-l/C~-, defined by 8,(c CA) = Bn(c) CA-l, and the sequence

+

+

is a chain complex, which is called the quotient complex. We denote it as C/Ct.

Remark 10.1.5.

Let a E Hn(CICt). Then the element a is expressed as a = [c CL] for some c 6 Cn with an(c) E CA-,. Thus, a = 0 if and only if c = an+1(d) c' for some d E Cn+1 and some c' E C;.

+

+

Let CIC' and DID' be quotient complexes and let cp = (9,) : C + D be a chain map. If cpn(CA) c DL for n E Z, then the chain map cp induces a chain map cp : C/C1 + DID' and moreover cp, : Hn(C/C1) + Hn(D/D1), n E Z, is defined. Let A and B be abelian groups. Denote as F(A, B) the free abelian group generated by the set {(a, b) : a E A, b E B), i.e. F(A, B ) is the direct sum $(o,blZ. Let F1(A,B) be the free abelian group generated by elements of the form (a1 + az,b) - (ai,b) - (az,b),

(a,bl

+ bz) - (a, b1) - (a, bz)

where a, ai E A and b, b; E B, i = 1,2. Since F1(A, B) is a subgroup of F(A, B), an abelian factor group F(A, B)/F1(A, B) is defined. We denote as A @ B the abelian factor group, which is called the tensor product of A and B. For (a, b) E F(A, B) let a @ b denote the element of A @ B represented by (a, b). It follows that every element in A @ B is expressed as a finite sum of such a @ b with integer coefficients, and that the map (a, b) H a @ b from A $ B to A @ B is bilinear. Hence, we have the following properties :

$10.2 Singular homology

309

Let cp : A -* A' and 11, : B -* B' be homomorphisms of abelian groups. Then a homomorphism YJ QD 11, : A @ B -, A' @ B' is defined by

For chain complexes C = ( 8 , : Cn -t Cn-1) and D = ( 8 , : D, -* Dn-l) we define C @ D = ( 8 , : ( C 8 D ) , -+ ( C @ D),-l}, which is called the tensor product of C and D , where

(C @ D), =

x

C p @ D,

(x

denotes direct sum),

p+q=n 8 n ( ~ @ d ) = 8 p ( c ) @ d + ( - 1 ) P ~ @ B q ( (d c) ~ C ~ , d E D , , p + q = n ) . Since 8, 08,+~ = 0 for n E Z , it follows that C @D is actually a chain complex. For n E Z we can define a homomorphism

by n([c]@ [dl) = [c @ dl. Let cp : C -+ C' and $ : D define achain map c p @ $ : C @ D -+ C ' @ D ' b y

-+

D' be chain maps and

Then (cp 8 $), o n = n o (cp, @ +,) holds. Next, let us introduce the torsion product for abelian groups. To do this we need the following lemma.

Lemma 10.1.6. Let A be an abelian group. Then there exists free abelian groups Fo and Fl such that Fo > Fl and A g Fo/Fl. Given an abelian group B , let i@id : Fl @B -* Fo@B be the tensor product of the inclusion i : Fl -+ Fo and the identity id : B -* B . Then Ker(i @ id) is independent of the choice of the groups Fo and Fl, that is, they are mutually isomorphic. For the proof we refer to Dold [D2]or Spanier [Sp]. Let A and B be abelian groups and let Ker(i @ i d )be as in Lemma 10.1.6. We call Ker(i @ i d )the torsion product of A and B , and denote it as A * B. It is checked that (1) A * B Z B * A , ( 2 ) if B is torsion free (i.e. nb # 0 for all b E B , n E Z with b # 0, n # 0 ) then A* B = 0, ( 3 ) A * B is torsion (i.e. for d E A * B there is n E Z with n # 0 such that nd = 0).

CHAPTER 10

310

Theorem 10.1.7(Kunneth formula). Let C = {an : Cn -t Cn-1) and C' = (8, : CA + C;-l} be chain complexes. Suppose each of Cn (n E Z) is a free abelian group. Then one has the following short exact sequence :

and this sequence splits. For the proof, see Dold [D2] or Spanier [Sp]. $10.2 Singular homology The purpose of this section is to recall some properties of singular homology groups of topological spaces, which will be used frequently in later sections. Let {eo, ,en) be the standard base of an euclidean space Rn+', i.e. eo = (1,0,. ,O), ,e, = (0,. ,0,1). We denote as An = [eo,. ,en] the smallest convex set containing {eo, ,en), which is called a standard n simplex. Let X be a topological space. A continuous map u : An -, X is said to be a singular n-simplex of X. Denote as Sn(X) the collection of all singular n-simplexes of X and as Sn(X) the free abelian group generated by Sn(X). For n < 0 we let Sn(X) = 0. Let 0 5 i 5 n and define an affine map EL : An-1 A, by

-. - . .. .

..

For u E Sn(X) the composites a o E:, we let n

-.

..

,u o E:

belong to Sn-l(X), and so

i=O

by which a homomorphism 8, : Sn(X) -+ Sn-1(X) is induced. This called the boundary homomorphism. From the following lemma it follows that the sequence

is a chain complex, which is denoted as S(X). Lemma 10.2.1. 8, o an+l= 0 for all n E Z. Proof. For u E Sn(X)

an is

510.2 Singular homology

Since E: o

= ~ j o, &1Ll:

for i

> j, it follows that

As above let X be a topological space. We denote as Hn(X) the n-th homology group Hn(S(X)) of the chain complex S(X), which is called the f n-th (singular) homology group of X.

Remark 10.2.2. The following properties are easily checked : (1) If X is the set of one point, then Hn(X) 2 Z (n = 0) and r 0 (n # 0). (2) If X is path connected, then Ho(X) 2 Z. For a subspace A of X the pair (X,A) is called a topological pair. It is clear that &(A) c Sn(X) and &(&(A)) c Sn-1(A) for n E Z. Thus, S(A) is a subcomplex of S(X). The n-th homology group of the quotient complex S(X)/S(A) is denoted as

and it is called the n-th (singular) homology group of the topological pair (X, A). If A is empty, then Hn(X) = Hn(X, 0). Let (X, A) and (Y, B ) be topological pairs. If a continuous map f: X + Y safisfies f (A) c B, then we say that f is a continuous map form (X, A) to (Y, B), and write f : (X, A) 4 (Y, B). For u E Sn(X) the map f o u belongs to Sn(Y). Thus, a homomorphism f n: Sn(X) + Sn(Y) is induced by the continuous map f . Since (f o o) o EL = f o (u o EL), we have 8, o fn(u) = f no &(a), which implies that the diagram

1

f*

commutes.

Therefore, ffl= {fn} : S(X) -+ S(Y) is a chain map. For a continuous map f : (X, A) -+ (Y, B), we have fn(Sn(A)) C S,(B), and so a chain map of the quotient complexes, ft : S(X)/S(A) -+ S(Y)/S(B), is induced by the homomorphism ftl : S ( X ) + S(Y). Therefore, we have a Hn(Y, B). homomorphism f, : Hn(X, A)

312

CHAPTER 10

Theorem 10.2.3. Let X , Y and Z be topological spaces. Then the identity map id : (X, A) -+ (X, A) induces the identity map id, = id : Hn(X, A) + Hn(X, A). If f : (X, A) + (Y, B) and g: (Y, B ) + (2, C) are continuous maps, then (g o f), = f, o g, holds. Proof. This is clear by definition. For (X, A) a topological pair, the inclusion i : A + X induces an inclusion it : S(A) + S(X), and the inclusion j : (X, 8) -+ (X, A) induces a quotient map jp: S ( X ) + S(X)/S(A). Thus, the following sequence is exact :

By Proposition 10.1.2 (1) we have an exact sequence

which is called the homology exact sequence of the pair (X, A).

Theorem 10.2.4. For a continuous map f: (X, A)

-+

(Y, B ) the diagram

commutes. Proof. This follows from Proposition 10.1.2 (2). Two continuous maps f , g : (X, A) + (Y, B ) are said to be homotopic, written as f z g, if there is a continuous map F : ( X x [0, 11,A X(O,l]) + (Y, B ) such that F(x, 0) = f(x) and F ( z , 1) = g(x). Here F is called a homotopy from f to g.

Theorem 10.2.5. Let f , g : (X, A) + (Y, B ) be continuous maps of topological pairs. Iff and g are homotopic, then f, = g, : H,(X, A) + Hn(Y, B ) for all n. Proof. Let F : ( X x [O,l],A x [0, 11) + (Y, B) be a homotopy from f to g and define continuous maps io,i1 : (X, A)

-+

( X x [O,l], A x [O, 11)

by io(z) = (x, 0) and il(x) = (x,l). Since F o io = f and F o il = g, by Theorem 10.2.3 it suffices to show that for all n

(10.1)

io, = il, :Hn(X, A) + Hn(X x [0, 11,A x [O,l]).

510.2 Singular homology

For 0 5 i 5 n let Oi =

: A,+l

+ A,

313

x [ O , l ] be an affine map defined

bv

Then the composite 8&, oak+l : A,

8z+1 o a!+,

= ii,

8:+,

=:;8;

o

4

A, x [0,1] have the following properties o 8"" =

n+l

.'

2~

o

(E;-~ x id) o 8;-I

(i > j )

( ~n-1 j - l x id) o 8;

(i

+ 1< j)

where ih,i: : A, -+ A, x [O, 11 and ib(x) = (x,O),i:(x) = (x,1) for x E A,. Let a E S,(X) and consider a composite

2 An x [O, 11

( a x id) o 8' : An+,

u%d

X x [O, 11.

Since (o x id) oei is an element in S,+l ( X x [0, I]), we define a homomorphism : sn(X) Sn+l(X X [O, 11) by

@n

+

n

@,(a) = x ( - l ) ' ( o x id) o 8'. i=O

From (10.2) it follows that

Since @, maps S,(A) to Sn+l(A x [0, I]), we have that ioa,iip : S(X)/S(A)

-t

S(X x [O, 1])/S(A x [O, 11)

is chain homotopic. Therefore, (10.1) is obtained.

A continuous map f: (X, A) -+ (Y, B) is said to be a homotopy equivalence if there is a continuous map g: (Y, B) -+ (X, A) such that

In this case, by Theorems 10.2.3 and 10.2.5 the induced homomorphism f, : H,(X, A) -+ H,(Y, B) is isomorphic for all n. A topological space X is contractible if there is a homotopy equivalence from X to the set of one point.

CHAPTER 10

314

Remark 10.2.6. If X is contractible, then H n ( X ) G Z ( n = 0 ) and 2 0 ( n # 0). Let Y and Z be subspaces of a topological space X such that X = Y U Z . It is clear that S ( Y ) and S ( Z ) are subcomplexes of S ( X ) . If e : S ( Y ) S ( Z ) -, S ( X ) denotes the inclusion, then e is a chain map. The couple {Y,Z ) is said to be an excisive couple of X if for any n the induced homomorphism

+

is isomorphic.

Theorem 10.2.7. Let Y and Z be subspaces of a topological space X . If X = int(Y) U int(Z), then {Y,Z ) is an excisive couple of X . We omit the proof. See Dold [D2]or Spanier [Sp] for the detail. Theorem 10.2.8(Mayer-Vietoris exact sequence). Let {Y,Z ) be an excisive couple of X and define homomorphisms

where il:YnZ-+Y, iz:YnZ+Z, j l : Y -+YUZ, j 2 : Z - , Y U Z are inclusions. Then the sequence

is exact. Here 8, is a homomorphism satisfying

a* ([.I

+ C Z ] ) = [&(.I

)] = -[&(cz)]

for ~1 E S n ( Y ) and cz E S n ( Z ) . Proof. Since the sequence

-

o --+ S(Y n z ) A S ( Y ) $ s(z)2 S ( Y ) + s(z) is exact where i ( c ) = (c, -c) and j(c1, c2) = c1 we have the following exact sequence :

o

+ c2, by Proposition 10.1.2 ( 1 )

Since {Y,Z ) is an excisive couple, from Lemma 10.1.4 the conclusion is obtained.

$10.2 Singular homology

Remark 10.2.9. Let Sn be an n-dimensional sphere. Then

Indeed, it is enough to give the proof for the unit sphere, i.e. Sn = {x E denotes the euclidian norm. Since Sn is path Rn+' : llxll = 1) where connected, clearly Ho(Sn) E Z. Define subsets

11 11

then {U+, U-) is an excisive couple. Hence, the following Mayer-Vietoris exact sequence is obtained : -+

H,(u+)e H,(u-)L H,(s~)3 H,-I(u+nu-)

H,-I(u+)e H,-~(U-) -+

.

Since U+ and U- are homeomorphic to Bn, they are contractive, and so Hp-l(U+) = HP-l(U-) = 0 for all p > 1. It is easy to see that the inclusion b : Sn-I + U+ nU- is a homotopy equivalence. Thus, the map k induces an isomorphism b, : H~-~(S"-') + HP-l(U+ n U-), and by the above exact sequence kc1 o 8, : Hp(Sn) + Hp-l(Sn-l) is an isomorphism for p > 1. Since So = {+I,-1) c R, we have Hp-1(U+ n U-) = 0 for all p > 1 when n = 1, and Ker(i,) c Ho(U+ n U-) is a free abelian group generated by [+I] - [-I] if n = 1. For n > 1, i, : Ho(U+ fl U-) + Ho(U+) $ Ho(U-) is injective . Therefore, we have H~(S') = 0 (p > I),

Hl(Sn) = 0 (n

> 1),

H~(S') E Z

from which the conclusion is obtained by induction. Theorem 10.2.10 (Excision Theorem). If {Y,Z) is an excisive couple of X , then the inclusion i : (Y,Y n Z ) + ( X , Z ) induces an isomorphism i, : Hn(Y,Y n Z ) + Hn(X,Z) for a l l n E Z.

Proof. Let us consider the following commutative diagram of inclusions and quotient maps :

It is clear that the rows of the above diagram are short exact sequences. Hence, by Proposition 10.1.2 we have the following commutative diagram :

-

Hn(Z)

+

Hn(S(Y) S(Z))

Hn((S(Y) + S(Z))/S(Z))

Hn-l(Z)

316

CHAPTER 10

+

Since {Y,Z) is an excisive couple, the inclusion e : S(Y) S(Z) + S(X) induces an isomorphism e, : Hn(S(Y) S(Z)) -,Hn(X). From this together with Lemma 10.1.3 we have

+

On the other hand, by the homomorphism theorem we have

Since S(Y) n S(Z) = S(Y n Z), we have Hn(Y, Y f l Z ) inclusion. Therefore, the conclusion is obtained.

E Hn(X, Z)

under the

Theorem 10.2.11(Excision isomorphism theorem). Let (X, A) be a topological pair and let U be an open subset of X . Suppose cl(U) c int(A). Then the inclusion i : ( X - U,A - U) + (X, A) induces an isomorphism i, : Hn(X - U,A - U) + Hn(X,A) for all n.

Proof. Put Y = X - U and Z = A. Since cl(U) c int(A), we have X = int(Y)Uint(Z), and by Theorem 10.2.7, {Y, 2)is an excisive couple. Therefore, Theorem 10.2.10 derives the conclusion. Lemma 10.2.12. (1) Hp(Rn,Rn - 0) 2 0 (p # n) and Hn(Rn,Rn - 0) Z Z. is the euclidean (2) Given T > 0 let 0: = { x E Rn : 11x11 < T ) whew n o m . Then the inclusion i : (Rn,Rn - 0:) t+ (Rn,Rn - 0) induces an isomorphism

11 11

Proof. Since i : Rn - 0: -t Rn - 0 is a homotopy equivalence, we have that i, : Hj(Rn - 0:) + Hj(Rn - 0) is an isomorphism for j. Theorem 10.2.5 yields the following comutative diagram whose rows are exact.

Since there is a homotopy equivalence from sn-'to Rn - OF, we have that Hj(Rn - 0:) Z Hj(Sn-') Z Z ( j = 0 , n - I), S 0 ( j # 0,n - 1). Note that Hj(Rn) "L O(j # 0) and Ho(Rn) 2 Z. Therefore, (1) and (2) are obtained from the above diagram together with Lemma 10.1.3. For topological pairs (X, A) and (Y, B) we write (X,A) x (Y,B) = ( X x Y , A x Y u X x B).

510.2 Singular homology

Define affine maps E; : Ap -+ An and qq" : Aq + An by

and let PI : X x Y + X and P2 : X a E Sn(X x Y) define

xY

+Y

be the natural projections. For

where

Then a homomorphism p : Sn(X x Y)

+ (S(X) 8 S(Y)),

is defined by

We call the family p = {p) : S ( X x Y) + S(X)@S(Y) the Alexander- Whitney map. For a E Sn(X) and r E Sn(Y) write ( a x r)(z) = (o(a), ~ ( 2 )where ) a E A,. Then a x r E Sn(X x Y). For 0 5 i 5 n - 1 affine maps qk : An + An-1 is defined by (0 5 12 5 i) q:(ek) = (i < k 5 n). ek-l Let I = {il,i2,. ,ip : ik < ik+1,1 5 k < p) be a subset of {O,l,. ,n - 1). A map q~ : An + An-p is defined by

.

-.

If, in particular, I = 0 then we let q~ = id. Let 8 be the collection of all subsets consisting of p elements of {O,1, ,n1) and for I E 8 we write J = {0,1, ,n - 1) - I. For I E 8 denote as €(I) the cardinal number of the set {(i, j) : i E I,j E J,i > j). Let p q = n and take a E Sp(X) and r E S,(Y). Then

-.

- -.

+

is an element of Sn(X x Y). Thus a homomorphism . call the family V = {V) : S(X) 8 S(Y) -r is defined by a 8 T H ~ V T We S ( X x Y) the Eilenberg-MacLane map, and write xVy = V(x 8 y) for a: E Sp(X) and y E Sq(Y).

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Theorem 10.2.13. The Alezander-Whitney map p = {p) : S ( X x Y) + S(X) 8 S(Y) and the Eilenberg-MacLane map V = {V) : S(X) 8 S(Y) -+ S ( X x Y) are chain maps with the following properties : (1) p and V are natural with respect t o continuous maps, i.e. i f f : X -+ X ' and g: Y -+ Y' are continuous maps, then p o (f x g)n = (fn @I gI1)o p and V 0 (fn 8 gn) = ( f x 9111 0 V. (2) p and V are chain homotopy equivalences and these are a chain homotopy inverse i n each other. Moreover, one can choose the chain homotopy between p o V (resp. V o p) and the identity (resp. the identity) to be natural with respect to continuous maps. For the proof we refer to Eilenberg-Maclane [E-MI. Let (X, A) and (Y, B ) be topological pairs. If {A x Remark 10.2.14. Y,X x B ) is an excisive couple, then the Alexander-Whitney map p = {p) : S ( X x Y) + S(X) 8 S(Y) induces an isomorphism

for all n. Indeed, since p : S ( X x Y) -+ S ( X ) 8 S(Y) maps S(A x Y) S(A) 8 S(Y) S ( X ) 8 S(B), by Theorem 10.2.13 (2)

+

+ S ( X x B ) to

is a chain homotopy equivalence. The right hand of the above relation is isomorphic to S ( X ) / S (A) 8 S(Y)/S(B). Since {A x Y, X x B) is an excisive couple, we have for all n

Therefore, the conclusion is obtained. For a = [z] E Hp(X, A) and b = [y] E Hq(Y, B) we define a homomorphism

by ~ ( [ z8] [y]) = [a:8 y] as stated in 510.1, and define a x b by

p*(a x b) = n(a 8 b), which is called the cross product of a and b. By Theorem 10.2.13 the EilenbergMacLane map V : S ( X ) 8 S(Y) -, S ( X x Y) is a chain homotopy inverse of the Alexander-Whitney map p. Thus, we have the following corollary.

$10.2 Singular homology

319

Corollary 10.2.15. Let a E Hp(X,A) and b E H,(Y, B). If a = [XI and b = [y] where x E Sp(X) and y E S,(Y), then the cross product a x b is expressed as a x b = [xVy]. R e m a r k 10.2.16. The cross product has the following properties : (1) x : Hp(X,A) @I H,(Y, B) + Hn((X,A) x (Y, B)) is bilinear, (2) (f xg),(a x b) = f,(a) xg,(b) for continuous maps f:(X, A) -+ (X', A'), 9: (Y, B) (Y', B'), (3) i,, (a) = a x [y] and i,, (b) = [XI x b for inclusions i, : (X, A) + (X, A) x Y, i, : (Y, B ) 4 X x (Y, B) defined by i,(x) = (x,y), i,(y) = (x, y), (4) ( a x b) x c = a x ( b x c), (5) if T : X x Y + Y x X is a continuous map defined by T(x, y) = (y, x), then T,(a x b) = (-1)PQb x a for a E Hp(X, A) and b E H,(Y, B). Indeed, (1) and (3) is easily checked from definition. (2) follows from Theorem 10.2.13 (1). For the proof of (4) and (5), see Dold [D2] or Spanier [Sp]. +

&om Theorem 10.1.7 we have the following theorem. T h e o r e m 10.2.17 (Kiinneth formula). Let (X, A) and (Y, B) be topological pairs. If {A x Y, X x B ) is an excisive couple, then the cross product yields the following split exact sequence

R e m a r k 10.2.18. Let Tn = Rn/Zn be an n-torus. Since 'IP is homeomorx S1 (the direct product of n times) and phic to S1 x

...

by Theorem 10.2.17 we have that Hp('IP) is a free abelian group of rank In particular, Hl(Tn) "L Zn holds. Let u1, u2, ,un : [O, 11 + Rn be paths defined by

-.

and put wi = x o u i for 1 5 i 5 n where IT : Rn + Tn is the natural projection. Then each wi is a closed path from 0 to 0 in Tn, and {[wl],... ,[w,]) is a system of generators of the fundamental group ?rl(Tn,0) 2 Zn. On the other hand, each wi can be considered as a singular 1-simplex and then they are cycles since al(wi) = 0. It is easy to see that the set { [wl], ,[w,]) of homology classes is a system of generators of Hl (Tn).

- ..

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320

Let Z : ?rl(Tn,O) + H l ( P ) denote an isomorphism defined by Z([wi]) = [wi] (1 i n). Then, for a continuous map f : Tn -t P we have the following commutative diagram

< <

where w is a path from 0 t o f (0) and w, : ? r l ( F ,f (0)) the induced isomorphism in Lemma 6.1.4.

-t

m(Tn, 0) denotes

510.3 Euclidean neighborhood retracts (ENRs)

Let X be a topological space and A a subset of X . The subset A is said to be a retract of X if there is a continuous map r : X + A such that the restriction of r to A is the identity map, i.e. r l = ~ id. In this case, r is called a retraction. We say that A is a neighborhood retract (abbreviated NR) of X if there exists an open neighborhood U of A in X such that A is a retract of

U. Remark 10.3.1. (1) If A is open in X , then A is an NR of X. (2) If A is an NR of X and B is an NR of A, then B is an NR of X . (1) is clear. (2) is easily checked as follows. Since A is an NR of X , there is an open subset U of X and a continuous map r1 : U + A such that rl is a retraction. Since B is an NR of A, an open subset V of A and a retraction rz : V + B exist. Put W = r l l ( v ) and define a continuous map r = rz o ( r l l w ) : W + B. Then W is open in X and T : W -r B is a retraction.

A topological space X is said to be locally compact if for any x E X and any neighborhood V of x there is a compact neighborhood of x containing in V. A subset A of X is called a locally closed set if there exist an open subset 0 of X and a closed subset C of X such that A = 0 n C. Lemma 10.3.2. Let X be a subset of Wn. Then (1) if X is an NR then X is locally closed set, (2) X is locally closed set if and only if X is locally compact, (3) if X is a locally closed set then X is homeomorphic to some closed set of IWn+l .

510.3 Euclidean neighborhood retracts (ENRs)

321

Proof. (1) : Since X is an NR of Rn, there exist an open set 0 of Rn and a continuous map T such that T : 0 + X is a retraction. Since r p is the identity map, X is closed in 0.Thus, X = 0 n C for some closed set C of Rn. (2) : If X is a locally closed set, there exist an open set 0 and a closed set C of Rn such that X = 0 n C. Let x E X. Then there is a compact neighborhood K of x in Rn such that K c 0. Since K n X = K n C is a compact neighborhood of x in X, X is locally compact. Conversely, suppose X is locally compact. For x E X there is a compact neighborhood K , of x in X . Thus, K, is expressed as K , = X n V, where V, is a compact neighborhood V, of x in Rn. Put Ox = int(V,). Since

we have

and therefore X is a locally closed set. (3) : Since X = 0 n C for some open set 0 and some closed set C, we can define a continuous map j : 0 + Rn+' by

where d denotes the euclidean metric. Then j ( 0 ) = {(x,t) C Rn x I[$ : td(x,Rn - 0 ) = 1) is closed in Rn+'. Obviously j : 0 + j ( 0 ) is a homeomorphism, and so j ( X ) is closed in j ( 0 ) . Therefore, j(X) is closed in Rn+'. T h e o r e m 10.3.3. Let X be an NR of Rn and let Y be a subset of Rm. If Y is homeomorphic to X , then Y is an NR of Rm. Proof. By Lemma 10.3.2, X is locally compact and so Y is locally compact. Thus Y is a locally closed set. This ensures the existence of an open set V of Rn such that Y is closed in V. Let h: Y + X be a homeomorphism. Since V is a normal space, there is a continuous map g: V -+ Rn such that g ~ y= h. Since X is a NR of Rn, an open set 0 of X in Rn and a retraction T : 0 + X exist. The set W = g-'(0) is an open set of Rn such that W > Y, and h-' o T o g : W + Y is a retraction.

A set X is said to be an euclidean neighborhood retmct (ENR) if X is homeomorphic to an NR of some euclidean space Rn. By Theorem 10.3.3, ENR is topologically invariant, i.e. if X is an ENR and X is homeomorphic to some subset Y of Rm, then Y is an NR of Rm.

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322

Lemma 10.3.4. Let X be a topological space and A a subset of X . Suppose Y is an ENR. I f f and f' are continuous maps from X to Y with f i A = then there exist an open neighborhood W of A in X and a homotopy F from f l w to fiw such that F : W x [O,1]t Y and F ( a , t ) = f ( a ) = f l ( a ) for all a E A and all t E [ O , l ] .

flA,

Proof. Since Y is an ENR, there are an open set 0 of an euclidean space Rn and continuous maps r : 0 4 Y and i : Y t 0 such that r o i = id. Define a subset of X by W = { x E X : ( 1 - t ) i o f ( x ) + t i o f'(x) E 0 for all t E [0,1]} and a homotopy F : W x [ O , l ] -+ Y by

+

F ( x , t ) = r ( ( 1- t)i o f ( x ) t i o f 1 ( z ) ) . Then we obtain the conclusion.

Theorem 10.3.5. Let X be a Hausdorfl space and suppose { X I ,- - - ,X,} is a finite open cover of X . If each of Xi is an ENR, then X is an ENR. Proof. By the assumption X is homeomorphic to some closed set of a euclidean space Rn. By Lemma 10.3.2 ( 3 )for each X i there is a homomorphism g; : Xi -, Rni such that gi(Xi) is a closed subset of Rni. As one point compactification of Rni we consider the sphere Sni = Rn' U { m ) . Define a map hi : X -t Sni by

Then hi is continuous. Indeed, let C be a closed subset of S n i . If m $2 C , then h f l ( c ) is homeomorphic to C n g ; ( X i )which is compact. Thus, h f ' ( c ) is closed. When oo E C, h f ' ( S n i - C ) = g i l ( ~ n i- C ) is an open subset of X i , which implies that h f (s"'- C ) is open in X , and so h f l (c)is closed in X. To obtain the conclusion it sufficies to show the case when { X I ,X 2 } covers X . We put N = nl nz 2 and define a continuous map h: X + Snl x Sna c R N . Then h is injective and h: X -, h ( X ) is a homeomorphism. By Lemma 10.3.2 each Xi is locally compact, and so is X = X l U Xz . Thus, h ( X ) is locally compact. Lemma 10.3.2 ( 3 ) ensures that h ( X ) is homeomorphic to some closed subset of a euclidean space R N . It remains to see that X is an ENR. We identify h ( X ) with X and then X is a closed subset of R N . Since each Xi is an ENR, by Theorem 10.3.3 an open set Oi of RN with Oi > Xi and a retraction ri : Oi 4 Xi exist. Put

+ +

310.3 Euclidean neighborhood retracts (ENRs)

323

Obviously rllo : 0 + Xl n X2 and r210 : 0 + X1 fl x2are retractions. Since Xl n X2 is an ENR, by Lemma 10.3.4 there exist an open neighborhood Wo of XI n Xz and a homotopy H from rllwo to r21w0 such that Wo c 0 and H : WO x [O, 11 + Xl n Xz with H(a, t ) = a (a E Xl n X2,t E [0, 11). Note that X - X1 and X - X2 are closed in Rn and

Choose open sets Wl and Wz such that

Then W = Wo U Wl U W2 is an open neighborhood of X in Rn. Let T : Rn + [O, 11 be a continuous map such that r(W1) = 0, and 7(W2) = 1. Define a continuous map p : W -, X by

Since p is a retraction, X is an ENR. Theorem 10.3.6. Let X be a Hausdorfl space and suppose {Xi : 1 5 i < oo) is a countable open cover of X . If (1) each Xi is an ENR, (2) for suficiently large n, each Xi is embedded in Rn, i.e. there is a homeomorphism from Xi onto a subset of Rn, then X is also an ENR. Proof. By Lemma 10.3.2 each Xi is locally compact, and so X is locally compact. Since each Xi has a countable base, so does X and thus X is paracompact. By the assumption (2) we have dim(X) 5 n. From a fact of the dimension theory we can find a refinement {Yi) of {Xi) such that {Yi)is an open cover of X and for Y,,, ,Yj, E { Y , ) ,Yj, n.. # 0 implies t 5 n + l . Take subfamilies C1, .Cn+l of { x ) such that C1 U . . . U Cn+l = { x ) , and all elements of Ce are mutually disjoint for 1 5 t! 5 n + 1. Then for 1
--

. ..

an?,

.

5, +

i, +

CHAPTER 10

324

Theorem 10.3.7. Every topological manifold is an ENR. Proof. Each point of X has a neighborhood which is homeomorphic to Rn or Rn x [O, 00). It is clear that Rn and Bn x [O,oo) are EN&. Since X has a countable base, by Theorem 10.3.6 it follows that X is an ENR.

Fix n

> 0 large enough.

Let vo, vl ,

.. ,vp be points in Rn. If

-- .. .-

--...

= 4 = 0, then v0, v1, ,up are said to be linearly implies Xo = X1 = independent. In thii case, we denote as [vo, vl, ,up] the smallest convex set containing {vo,q, ,up), which is called a p-simplex. Let UP = [vO,v1, ,up] be a psimplex and let {io, ,i,) c {O,1, ,p). A q-simplex [vi,, . ,viq] is called a face of up. We say that a collection K = {a) of simplexes is a simplicial complex if (1) any face of a simplex in K belongs to K , (2) i f u , E ~ K and a n ~ # then 0 a n 7 E K, (3) for a E K and x E a there is a neighborhood V in Rn such that E K : T n V # 0) is finite. Let K be a simplicial complex. The dimension of K , denoted as dim(K), is defined to equal n if K contains an n-simplex but no ( n 1)-simplex. We write IKI = UoEKa. It follows that IKI is a compact subset of Rn if and only if K is finite. A subset L of K is called a subcomplex of K if L is a simplicial complex. A topolological space X is a polyhedron if there are a simplicial complex K and a homoemorphiim h: JKI + X. Here K is called a simplicial decomposition of X. A sequence of abelian groups, A = {A, : n E Z), is said to be finitely generated if each A, is finitely generated and A, is trivial except abelian groups of finite numbers.

...

...

+

Theorem 10.3.8. If X is a compact ENR, then the homology group H,(X) = {H,(X)) is finitely genemted. Proof. We may suppose that X is a subset of some euclidean space Rn. Since X is an ENR, an open set 0 of Rn and a retraction T : 0 -, X exist. We here define a simplicial decomposition K of Bn as follows. Let {vl,. ,v,) be a base of Bnand let n. be a permutation of (1,2, ,n). We write v = a l v l + . a,v, for (al, ,a,) E Zn and consider n-simplexes

.

- .+

--.

--

Denote as K the set of all faces of these simplexes. Then K is a simplicial complex and IK( = Rn. If the length of vi, 1 5 i 5 n, is small, every a E K with a n X # 0 is a subset of 0 (i.e. a C 0 ) since X is compact. Let L be a collection of all faces of a E K with a n X # 0. Clearly L is a subcomplex of

$10.3 Euclidean neighborhood retracts (ENRs)

325

K. Since X is compact, L is finite and X C (LI. Thus, TO = ~ l p :l ILI + X is a retraction, from which TO, : Hn(ILI) + Hn(X), n E Z, is surjective. To obtain the conclusion it sufficies to show that for every finite simplicia1 complex L, Hn((Ll),n E Z, is finitely generated and Hn(ILI) = 0 for n > dim(L). To do this we use induction on the dimension of L. If dim(L) = 0, then ILI is a finite set, and so Ho(lLI) Z Z $ . - -$ Z (direct sum of fllLl times) Hn(ILI) = 0 (n # 0).

-

Thus, the case of dim(L) = 0 holds. Let dim(L) = n and let {al, ,on) c L be the collection of all n-simplexes. Ta.ke xi E int(oi) for 1 5 i 5 k and define

Obviously {Ul, U2) is an open cover of ILI. Hence, we have the following Mayer-Vietoris exact sequence :

Let Ln-' = {o E L : dim(o) < n). Then Ln-' is a subcomplex of L, dim(Ln-') = n - 1 and (Ln-'1 C Ul. It is easy to see that the inclusion i : (Ln-'1 + Ul is a homotopy equivalence. By the assumption of induction Hp(Ul) is finitely generated and Hp(Ul) = 0 for p 2 n. It is clear that Hp(U2) = 0 for p # 0. Notice that Ul n U2 = ~ f = , ( i n t ( u i )- xi). Since each of int(oi) - xi is homotopy equivalent to Sn-', we have that Hp(Ul n U2) E 0 for p # 0,n --1 and Hp(Ul n U2) (p = 0, n - 1) is isomorphic to Z $ $ Z (the direct sum of k-times). Therefore, by the above exact sequence it follows that Hp(JLI)is finitely generated for 0 5 p 5 n, and Hp(JLI) = 0 for p > n.

...

Let X be a topological space. If the homology group H,(X) = {Hn(X)) is finitely generated, then we define the Euler characteristic of X by

where each bn(X) is the rank of Hn(X). Theorem 10.3.8 ensures that the Euler characteristic can be defined for every compact ENR.

CHAPTER 10

510.4 Fixed point indices Let X be an ENR and let W be an open subset of X. For a continuous map f : W -+ X we will define a characteristic of f , called the fixed point index I ( f ) , in the case when Fi(f)= { x E W : f ( 3 )= x ) is compact. If X = Rn, then we have Hn(Rn, Rn -0) S Z by Lemma 10.2.12 (1). Choose a generator 0 of Hn(Rn,Rn - 0 ) and let K be a compact subset of Rn. We define a fundamental homology class O K of Hn(Rn,Rn - K ) as follows. For r > 0 put 0: = {x E Rn : 11x11 < T ) where 11 11 is the euclidean norm. Take r such that K c 0,". Then, the inclusion j : (Rn,Rn - OF) + ( R n ,Rn - K ) induces a homomorphism

j* : Hn(Bn,Rn - 0,")

+ Hn(Rn,Rn

- K).

By Lemma 10.2.12 an isomorphism

is obtained. We define

O K = j* o i ~ ' ( 0 ) . It follows that ( - O ) K = -OK holds and the definition of OK is independent of the choice of r. For an open set V with K C V C Rn we have by the excision isomorphism theorem that Hn(V, V - K ) E Hn(Rn,Rn - K ) . Let 0; (or simply O K ) denote the image of OK mapped by the isomorphism. Let L be a compact subset with L C K and let V , U be open subsets with L c K c V c U c Rn. I f i , : H n ( V , V - K ) -+ H n ( U , U - L ) i s a homomorphism induced by the inclusion i : V + U , then we have

Let g: V -+ Rn be a continuous map and suppose F = Fix(g) is compact. Let (id - g ) : ( V ,V - F ) + ( R n ,Rn - 0 ) be a continuous map defined by (id - g ) ( x ) = x - g(2). We define the fied point index I ( g ) of g by

where (id - g), : Hn(V,V - F ) + Hn(Rn,Rn - 0). Since ( - O F ) = -OF, the index I ( g ) is independent of the choice of 0. If F c K c V , then (id - ~ ) * ( O K=) I ( g ) O by (10.3). For the fixed point index I ( g ) we have the following the properties. P r o p e r t y 10.4.1. If I ( g ) # 0 then F is nonempty.

Proof. Suppose F = 0, then Hn(V,V - F ) 2 0 and thus I ( g ) = 0.

$10.4 Fixed point indices

327

Property 10.4.2. For an open subset V' with F C V' C V , the indices I ( g l v d and I ( g )

Proof. This follows from (10.3). 0 Property 10.4.3. If g ( V ) = p, then I ( g ) = 1 when p E V and I ( g ) = 0 when

P ~ V . Proof. If p V , then F = 0 and by Property 10.4.1, I ( g ) = 0. When p E V we have F = {p}. Since (id - g ) ( x ) = x - p, we define a translation T : Rn -* Rn by T ( x ) = x - p. Then it sufficies to show that T,(O,) = 0 where

Take r > 0 such that p E 0: = {x E Rn : J ) ~ E J J< T } and define a homotopy ht : (Rn,Rn - 0:) (Rn,Wn - 0),0 5 t 5 1, by h t ( x ) = x - tp. Obviously ho is the inclusion and hl = T . Thus we have a commutative diagram :

Therefore, T,(Op) = 0. Property 10.4.4. Let V be expressed as the union of two open subsets Vi and V2 ( V = Vl U V - ) . V Fix(glvl) and Fix(glv2)are disjoint and compact, then I ( g ) = I ( g l v l )f I(glva).

Proof. We put FI = Fix(glvl) and Fz = Fix(glv,). Since Fl n Fz = 0, we can take open subsets Ul and Uz such that Fi c Ui c Vi,i = 1,2, and Ul n U2 = 0. Since I ( g l u i )= I ( g l v , )by Property 10.4.2, we have I ( g l u )= I ( g ) where U = Ul U U2. On the other hand, by Lemma 10.1.4

where F = Fl U F2. The isomorphism used here maps ( O S , O F a )to O F . Thus we have

I ( g l u ) O= (id - OF) = (id - OF^) + (id - s ) + ( O F ~ )

+ I(9luz))O and therefore I ( g ) = I ( g l ) + I(g2). = (I(glv,)

CHAPTER 10

328

Property 10.4.5. Let gt : V -+ Rn(O 5 t 5 1) be a homotopy and put Ft = Fix(gt). If K = Uostsl Ft is compact, then I(go)= I(gl).

Proof. Since Ft c K C V for 0 5 t 5 1, clearly (id - gt) : (Rn,Rn - K ) -+ (Rn,Rn - 0 ) , 0 5 t 5 1, is a homotopy, and hence (id - go), = (id - gl),. Thus, I(g0) = I(g1). Property 10.4.6. Let U be an open subset of Rm and let f:U -+ Rm be a continuous map. Let V be open in Rn, and g: V + Rn continuous. If Fi(f ) is compact, then the continuous map f x g : U x V -+ Rm+" has the property I ( f x 9 ) = I ( f )I(g). Proof. Let O1 be a generator of Hm(Rm,Rm - 0 ) and let 0 2 be that of Hn(Rn, Rn -0). The cross product 0 = O1 x 0 2 is a generator of Hm+,(Rm+", Rm+" - 0 ) E Z. The inclusion i induces an isomorphism

It is easily checked that O K x L = OK x OL for compact sets K C Rm and L c Rn. Thus we have O F i X ( fx) OFix(g)= O F i x ( f ) x ~ i x ( gSince ). Fix(f) x Fix(g) = Fix(f x g) and (id - f ) x (id - g) = (id - f x g), we have

and therefore I( f x g) = I ( f )I(g). Property 10.4.7. For open sets Vl C Rnl and V2 C Rna let f l : Vl -+ Rna and f 2 : V2 -+ Rnl be continuous maps. Put Ul = and U2 = f;l(v1). Let f l 0 fa : U2 +Rna. f 2 0 f l :Ul + R n l ,

fcl(vz)

Then Fix(f2 0 f l ) and Fix(f1 o f 2 ) are homeomorphic. If two the sets are compact, then I ( f 20 f l ) = I ( f l o f 2 ) . Proof- Since

f l l F i x ( f 2 o f l ) : Fix(fi 0 f i )

F k ( f 1 0 f 2 ) is the inverse map of it follows that Fix(f2 o f l ) is homeo-

-+

f i l ~ i x ( f f ~a )o : Fix(f1 0 f i ) + F h ( f 2 0 f l ) ,

morphic to Fix(fl o f 2 ) . Define a homotopy gt : Ul x U2 + Pnl+na by

Then

Fix(gt) = {

( ~ 1 , ~ 2E )Ul X

U2 : 2 2 = f 1 ( x 1 )x1 , = f2(x2))

$10.4 Fixed point indices

329

is homeomorphic to F i ( f i o f l ) and so Fix(gt) is compact. Thus I(go) = I(gl) by Property 10.4.5. Let ht : Ul x Rna -+ Rn1+"a (0 5 t 5 1) be a homotopy defined by ht(51922) = (tfl 0 f2(21),(1 - t)fl(21)). Since gl = holulxva, by Property 10.4.2 we have I(gl) = I(ho). The image Im($) of a continuous map $ : Fix(f2 o f l ) x [O,1] + Ul x Rna, which sends (XI,t) to (31, (1 -t) f l (XI)),coincides with Uoltll Fix(ht). Thus it is compact and by Property 10.4.5, I(ho) = I(hl). Since hl(xl,x2) = (f2 0 fl(xl),O), we have I(h1) = I(f2 o f l ) by Properties 10.4.1 and 10.4.6, and consequently I(f2 o fl) = I(go). To obtain the conclusion we define gi : Ul x U2 + IWnl+"a by g:(x1, 22) = (f2(~2),t f l and hi : Rnl x U2 + Rnl+"a by

0

f2(22)

+ (1 - t ) f l ( ~ l ) )

h:(xl,x2) = ((1 - t ) f 2 ( ~ 2 ) , f 0l f2(32)). In the above way we have I(gk) = I(flo f 2 ) . Since go = g;, the conclusion is obtained. Suppose X is an ENR and W is an open subset of X. We now give the definition of fixed point index I ( f ) for a continuous map f: W -,X. Since X is an ENR, X is homeomorphic to an NR of Rn for a certain n. Thus, there exist an open subset 0 of Rn and a continuous map i : X + 0 and r : 0 + X so that T o i = id. The set Fix(i o f o r ) of fixed points of the composite is homeomorphic to Fix(f). If Fix(f) is compact, then the fixed point index I ( i o f o T ) of the composite is defined as above. We define the fixed point index I ( f ) of f by I ( f ) = I ( i o f o r). The index I ( f ) is independent of the choice of 0,i and r. Indeed, let 0', i' and r' correspond to 0,i and r respectively. Apply Property 10.4.7 for i o r' : 0' + Rn and i' o f o T : r-l(W) + Rn'. Then we have I ( i o r' o it o f o T) = I(il o f o T o i o 7') and so I ( i o f o r ) = I(i' o f o r'). It follows that the fixed point index of the continuous map f: W + X possesses also the above Properties 10.4.1-10.4.7. If a fixed point x of f is isolated, then I ( f l u ) is defined by taking a small neighborhood U of x, and it is independent of the choice of U by Property 10.4.2. Thus, we write I(f,x) = I ( f l u ) , and I ( f , x ) is called the fixed point index off at x. Let Sn be an n-dimensional sphere and let f: Sn + Sn be a continuous map. Since Hn(Sn) Z Z, there is m E Z such that the induced homomorphism f* : Hn(Sn) + Hn(Sn) satisfies the following relation We call such an m the mapping degree of f and denote it as deg(f).

CHAPTER 10

Remark 10.4.8. (1) Let V be a neighborhood of the origin 0 in Rn and let g: V 4 Rn be a continuous map with 0 E Fix(g). For E > 0 small let D r = {x E Rn : llxll 5 E ) and suppose D, f l Fix(g) = (0). Let g' : Sn-' 4 Sn-l is a continuous map defined by

Then I(g, 0) = deg(gt) holds. (2) Let g: V 4 Rn be as above and suppose g is a C1 map. Then (i) if det(id - Dog) is positive, then I(g, 0) = 1, (ii) if det(id - Dog) is negative, then I(g,O) = -1 where Dog is the derivative of g at 0 and det(id - Dog) is the determinant of id - Dog These are checked as follows. To see (I), choose r > 0 such that D, C 0: C V, and consider the following diagram

where 3 = id -g , il(x) = EX (x E Sn-') and i is the inclusion. It is easy to see that a,, i: and i, are isomorphisms and the diagram commutes. Therefore, the conclusion is obtained. To show (2), take E > 0 small enough and define a homotopy H :D: 4 Rn by H(x, t) = g(tx)/t for 0 < t 5 1 and H(x, 0) = Dog(x). By Property 10.4.5 we have I(g, 0) = I(Dog, 0). From this together with (1) the conclusion is obtained.

$10.5 Lefschetz numbers

In this section we discuss Lefschetz numbers of continuous maps on compact ENRs. If an abelian group A is finitely generated, then A is expressed as the direct sum of cyclic groups of finite number. Let {el, ,eb, eb+l, ,en) be a ,eb have infinite order and eb+l,. .. ,en system of generators of A where el, have finite order. A homomorphism cp : A 4 A has the representation

--

--

510.5 Lefschetz numbers

331

6 for some integers aij(l 5 i, j 5 n). Here we write tr(cp) = Ci=, aii, which is called the tmce of cp. Let X be a topological space and suppose the homology group H,(X) = {Hn(X)) is finitely generated. For continuous map f : X + X we define the Lefschetz number L(f) o f f by

where each f,, : Hn(X) + Hn(X) denotes the induced homomorphism. If, in particular, f is the identity id, then L(id) means the Euler characteristic of X. We have easily the following theorem. T h e o r e m 10.6.1. If continuous maps f : X topic, then L(f) = L(g).

-, X

and g:X + X am homo-

Whenever X is a compact ENR, by Theorem 10.3.8 the homology group H,(X) is finitely generated. Thus, the Lefschetz number is defined for every continuous map of X. For the relation between Lefschtez number and fixed point index we have the following theorem. T h e o r e m 10.5.2 (Lefschetz Axed point formula). Suppose X is a compact ENR and f : X -+ X is a continuous map. Then I ( f ) = L(f ). For the proof, see Dold [Dl, D2]. From Theorem 10.5.2 and Property 10.4.1 it follows that if f : X + X is a continuous map of a compact ENR then

R e m a r k 10.5.3. Let f: Sn -+ Sn be a continuous map of an n-dimensional sphere. Then we have

where deg( f ) is the mapping degree of f .

...

R e m a r k 10.5.4. Let T" be an n-torus. Then H1(Tn) E Z $ $ Z (the direct sum of n-times). Let {el, ,en) be a system of generators of HI(Tn). For a continuous map f: Tn -t T" we have f,(ei) = Cy=laijej (1 5 i 5 n). If A1,. ,An are eigenvalues of the matrix (aij), then the Lefschetz number L( f ) is expressed as

--

n

This is checked as follows. For 1 5 i 5 n let wi be a cycle as in Remark 10.2.18 and let ei = [wi]. Then, we can consider the set {ei, x ei, x x ei, :

CHAPTER 10

332

---

1 5 il < if < < ik 5 n ) as a system of generators of Hk(lIn) (see Remark 10.2.16), and the following relation is easily checked:

Therefore, we have

where I denotes the identity matrix.

$10.6 Orientability of manifolds

Let X be a connected locally path connected Hausdorff space and suppose the following properties hold: for all x E X

and there is a connected open neighborhood V, of s in X , called a canonical neighborhood, such that for some 0, E H,(X, X - V,), i,, (0,) is a generator of H,(X, X - z) for z E V, where i, : (X, X - V,) -+ (X, X - z) is the inclusion. Here 0, is a fundamental homology class of Hn(X, X - V,). Since X is Hausdorff, it follows that if 0: E Hn(X, X - V,) is another fundamental homology class, then we have either iZ,(Oh) = i,,(O,)

(Vz E V,)

or

i,,(O:)

= -i,,(O,)

(Vz E V,).

< Let w : [O, 11 + X be a path. Then there is a sequence so = 0 < sl < s, = 1 such that w([si, si+l]) c Vw(,i) for 0 5 i 5 n - 1 where V,(,,) is a

510.6 Orientability of manifolds

333

canonical neighborhood of w(si). Let Ow(,i) be a fundametal homology class of H(X, X - Vw(,i)) and define an isomorphism wit : Hn(X,X - ~ ( s i ) -)+ Hn(X,X - w(si+l)) by wit 0 iw(,i)*(O,(,i)) = iw(,i+l)*(Ow(,i)). Then the composite wu = ~

n - 1 0 "~' ~ 1 8 0

wop : H n ( X , X - ~ ( 0 )-t ) Hn(X,X - ~ ( 1 ) )

does not depend on the choice of {VW(,,))and {Ow(,i)). It is easy to see that (PI) if w is a constant path then wp is the identity map id, (P2) if two paths wl and wz satisfy wl(0) = wz(l), then (wl wz)p = Wzp 0 Wit. (P3) if wl is homotopic t o WZ, then wlp = wzp. We say that X is orientable if wl = id for all closed path w in X . In this case, there exists the family of generators

(1, E Hn(X,X - x) : x E X ) , called an orientation of X , such that ~ p ( l ~ (=~ ) ) for all paths w in X . Let X and Y be topological spaces as above, and let f: X --t Y be a covering map. Since f is a local homeomorphism, the following (P4) is easily checked from defintion : (P4) f, o wu = (f OW)^ o f* for any path w in X where f, denotes the induced isomorphism. From the following Lemma 10.6.1 we can say whether or not a connected toplogical manifold without boundary is orientable.

Lemma 10.6.1. Let M be a topological n-manifold and let U be a connected open subset of M such that there is a homeomorphism h: U -+ Wn. Let V = h-'((2 E Wn : IlzJ1 < 1)) where )I 11 denotes the euclidean norm. If x E V, then the inclusion i : (M, M - V) + (M, M - x) induces an isomorphism i, : Hj(M, M - V)

-t

Hj(M, M - x)

( j 2 0).

Proof. This is clear from Lemma 10.2.12 (2) together with the excision isomorphism theorem. Remark 10.6.2. Let M be a connected topological manifold without boundary. If M is non-orientable, then there is a double covering map p : M -+ M such that M is orientable. Indeed, let A = {[w] E nl(M) : w, = id). Then A is a subgroup of ?rl (M) with index 2. By Theorem 6.2.12 we can find a double covering map p : M -t M such that p,(al(&f)) = A. Then by (P4) for any path w with [w] E ?rl(M). Since [pow] E A, we have p, o wp = p,, and so w8 = id. Thus, M is orientable.

CHAPTER 10

334

Let M be an orientable closed topological n-manifold. Then, Hn(M) r Z and if OM is a generator of Hn(M) then i+(OM) is a generator of Hn(M, M-x) r Zfor all x E M where i+ : Hn(M) -+ H,(M, M x) denotes the homomorphism induced by the inclusion. Such an element OM, called a fundamental homology class of M , corresponds to an orientation of M. For the details, see Dold [D2] or Spanier [Sp]. For a continuous map f : M + M the mapping degree, deg(f ) E Z,is defined by f+: Hn(M) Hn(M). OM) = deg(f )OM

Remark 10.6.3.

+

If, in addition, f is a self-covering map, then the absolute valute I deg(f)l is consistent with the covering degree of f . Indeed, let b E M and put F = f -'(b). Then the inclusion i : M + M induces a homomorphism i, : Hn(M) + Hn(M, M - F). Let (1, E Hn(M, M - x) : x E X) be an orientation of M corresponding to OM. Since Hn(M, M - F ) r ecEFHn(M,M - c), we have i,(OM) = CBCE~lc.On the other hand, from (P4) it follows that there is a constant k = f1 such that f,(l,) = k l f ( for all x E M where f, : Hn(M, M - x) -+ Hn(M, M - f (x)) =! is the induced isomorphism. Hence f + ( e c E ~ l = c )kUFlb. Since the diagram

1

1

Hn(M)

Hn(M, M - b)

f*

commutes,

we have deg(f)lb = kdFlb. Therefore, )deg(f)l = UF.

Remark 10.6.4. Let M be a closed topological manifold. If a continuous map f : M + M is homotopic to a homeomorphism (or more generally, a self-covering map), then f is surjective. Indeed, suppose M is orientable and let OM be a fundamental homology class of M. If there is x E M such that x $ f (M), then we have a commutative diagram

and so f + o i*(OM) = deg(f)i+(OM). Since f (M) C M - x, it follows that f, o i*(OM) = 0, thus contradicting. In the case M is non-orientable, by Remark 10.6.2 we take a double covering (M))) = p,(nl (M)), map p : M --, M such that M is orientable. Then f+(p,(.~rl and so there is a lift f of f by p (see Remark 6.3.5) and f is homotopic to a

$10.6 Orientability of manifolds

homeomorphism (or a self-covering map). Therefore,

335

j is surjective, and so is

f In the remainder of this section we shall discuss the (homological) dimension of generalized foliations on connected topological manifolds without boundary. Proposition 10.6.5. Let M be a connected topological manifold without boundary and suppose 3 is a generalized foliation on M . Then each leaf of 3 is an ENR under the leaf topology.

Proof. Let L be a leaf of 3 and let x E L. By definition there is a local coordinate cp : D x K -, N at x. Since N is open in M , N is an ENR (by Theorem 10.3.7) and so is D. Since D is open in L and the leaf topology of L has a countable base, from Theorem 10.3.6 it follows that L is an ENR.

Lemma 10.6.6 (cf. Bredon [Br]). Let X and Y be non-trivial topological spaces. If the product topological space X x Y is a connected topological nmanifold without boundary, then (1) there are integers p, q > 0 with p q = n such that for x E X and

+

YGY

( 2 ) each point of X has a canonical neighborhood in X and the same fact holds for Y .

Proof. First we claim that if A and B are abelian groups and the tensor product A 8 B is isomorphic to Z, then both A and B are isomorphic to Z. Indeed, let cp : A @ B + Z be an isomorphism. Then we can take

such that p ( ~ ai ~8 bi) l = 1. Let $"A denote the direct sum of m-times and define a homomorphism c p :~$"A -+ Z by m

~ ~ ( x l + . - . + x , ) = c p ( ~ x i @ b i for ) X~+...+X,E$~A. i=l

Then VA($"A) = cp(A 8 B), and so e m A g Ker(cpA)$ Z. Similarly, e m B g Ker(cp~)$ Z. Thus it follows that

CHAPTER 10

336

On the other hand, since A @ B r Z, we have

from which Ker(cpA) and Ker(cpB) are finitely generated. Therefore, A and B are isomorphic to Z. To show (1) let (2, y) E X x Y. Since X x Y is a manifold, X and Y are Hausdorff, and hence {X x (Y - y), ( X - x) x Y) is an excisive couple in X x Y. By Kunneth formula we have the following split exact sequence: 04

c

Hi(X,X-x)@Hj(Y,Y

- y) + H k ( ( X , X - x )

x (Y,Y - y))

i+j=k

Since dim(X x Y) = n and X x Y has no boundary, we have

From this together with the fact that is a torsion group, it follows that

xi+

j=k-l

Hi(X, X - x) * Hj (Y, Y - y)

>

and so by the first claim we can find p 0 and q 2 0 with p + q = n such that (10.4) holds. Then, p # 0 and q # 0 since X and Y are non-trivial. Since X x Y is connected, it follows that p and q are independent of the choice of points in X x Y. It remains to show (2). Let p and q be as in (1) and let x E X and y E Y. As representatives of generators of Hp(X,X - x) and H,(Y, Y - y) take cycles c, E Ap(X) and d, E Aq(Y) respectively. If 8 is a boundary homomorphism, then ac, E Ap-l(X - x) and so there exists a neighborhood D, of x in X such that ac, E Ap-l(X - z ) for all z E D,. Similarly, there exists a neighborhood D, of y in Y such that ad, E Aqdl(Y - z) for all z E D,. Choose a neighborhood V of (x, y) in X x Y as in Lemma 10.6.1. Then Hn(X x Y, X x Y - V) 2 Z. As a representative of its generator we take a cycle a v E A n ( X x Y). Then by Lemma 10.6.1, [av] is a generator of Hn(X x Y, X x Y - z) for all z E V. Kunneth formula shows that [c,] x [d,] is a generator of Hn((X, X - X) x (Y, Y - y)) % Z. On the otherhand, by Corollary 10.2.15 we have that [c,] x [d,] is represented by a cycle c,Vd, where V is the Eilenberg-MacLane map. Thus, there exist

510.7 Orient ability of generalized foliations

337

+

a E A,+l(X x Y) and b E A,(X x Y - (2, y)) satisfying c,Vd, - EU = Ba b where E is either 1 or -1. Let V' be a neighborhood of (x, y) in X x Y with V' c V such that b E A,(X x Y - z) for all z E V', and choose neighborhoods DL and DL of x in X and y in Y respectively, such that Dk c D,, DL c D, and Dk x DL c V'. Then, Dk is a canonical neighborhood with [c,] a fundamental homology class. The similar fact holds for DL. Proposition 10.6.7. Let M be a connected topological manifold without bounda y and let 3 be a genemlized foliation on M. Then there exists an integer p with 0 < p < dim(M) such that for every leaf L E 3 and for evey x E L

Proof. Let cp : D x K -t N be a local coordinate. By Lemma 10.6.6 we can find 0 < p < dim(M) such that for y E D

Take z E N and let L E 3 be the leaf through z. Then for some y E D Indeed, if D' is the connected component of z in N fl L, then there is z' E K such that cp( ,z') : D + D' is a homeomorphism, from which where z = cp(y, z'). Since D' is open in L, by applying the excision isomorphism theorem the above relation are obtained. Since N and z E N are arbitrary and M is connected, we obtain the proposition. Let 3and p > 0 be as in Proposition 10.6.7. We say that the dimension of 3is p, and write dim(3) = p. If 3and 3' are transverse generalized foliations on a manifold M , then the following equality holds:

$10.7 Orientability of generalized foliations

The purpose of this section is to discuss orientabiity for generalized foliations. Let M be a connected topological manifold without boundary. Suppose 3 is a generalized foliation on M . We prepare seven claims to mention how to introduce orientability for 3. Let dim(3) = p and denote as L(x) the leaf through x of 3.

CHAPTER 10

338

Claim 1. Take and flx a E M. If cp : D x K 4 N is a local coordinate at a, then N is expressed as the disjoint union N = UxEKD, of subsets D, where D, = cp(D, x). By Lemma 10.6.6 (2) there is a canonical neighborhood of a in D , written as V. Then R = cp(V x K ) is connected and open in M. It is clear that R c N and R = UxEKV,. Here V, = cp(V,x) for x E K.

Figure Let X E K . Thencp(a,x)=xandcp( , x ) : ( D , D - V ) + ( D , , D X - V , ) i s a homeomorphism. Hence, V, is a canonical neighborhood of x in D,. Take a fundamental homology class 0 E Hp(D, D - V) and put

We write IK={O,:XEK) and fix K. For x1,xz E K define a homeomorphism q!JXl,,, such that the following diagram commutes :

Then we have q!Jxi,+a*(Oxi)= V( r ~ 2 ) 0* (P( , X I ) , ' ( O ~ ~ ) = cp( ,x2)*(0) = o x , where Ox,, Ox, E K.

810.7 Orientability of generalized foliations

339

Claim 2. For q E R take x E K with q E V,. Then a homomorphism

is induced by the inclusion

Since 0, E Hp(D,, D, - V,) n IK is a fundamental homology class, the image iq,(O,) is a generator of Hp(D,, D, - q). Note that D, is the connected component of q in N fl L(q). Since D, is open in L(q), the inclusion j, : (D,, D, - q ) -t ( L ( q ) ,L(q) - q) induces an isomorphism

Therefore, j,, o iq,(O,) is a generator of H,(L(q), L(q) - q).

Claim 3. Let ql, qz E R , then there are qz E V,,. We now define an isomorphism

by *:,~(jq,*

0

XI, 22

E K such that ql E Vxl and

iq,+(Oxl)) = jqa+0 iqa+(Oxa). Then the following (10.5) holds:

{

(10.5)

@::,,

= id,

@::,R

= @::,R O

(aqa qlVR1-l

=

*Z,R,

aql q2,R'

Claim 4. Let a E M be as above and take b E M. Let cp' : Dt x K t --+ N t be a local coordinate at b and choose a canonical neighborhood V' of b in Dl. As seen above, N' = cpt(D' x K t ) and Rt = cp'(Vt x K ) are expressed as the disjoint unions N' = D& R' = V& yEKf u€K where D& = cpt(D',y) and V; = cpt(Vt,y). Let 0' E HP(D1,Dt - V t ) be a fundamental homology class and put

U

U

As above we write

K' = ( 0 ; : y E K t } and fix F.For yl, yz E K t define a homeomorphism

by $&l,y,= cp'(

,~

2 0 )cpt(

,y ~ ) - ' . Then @&l,ya+(O&l) = O&, holds.

CHAPTER 10

340

Claim 5. As in Claim 3, for ql, 92

E

R' we define an isomorphism

@:,R' : H p ( L ( ~ l ) , L ( q l-) 91)

+

Hp(L(q2)~ L(q2) - 92).

Claim 6. Suppose R n R' # C#J and let ql, 92 E R f l R'. If there exists a path y : [O, 11 + R n R' such that ~ ( 0=) ql and y(1) = 92, then (10.6)

,

=,

on Hp(L(ql),L(91) - 91).

Figure 40 We need the following seven steps to check (10.6). We first show the existence of t 2 > 0 such that

S t e p 1. Let cp and cp' be as above and let p ~ : D x K --, K and D' x K t + K t be the natural projections. For t E [0,11 define

~ K :I

Then a ( t ) E K and a l ( t )E K t for t E [O, 11. Since V c N and V,(t) = p(V, a ( t ) ) , we have y ( t ) E V,(q. For, since cp : D x K + N is a local coordinate at a, clearly a ( t ) = cp(a,a(t)) E cp(V,a(t))= V,(;). Note that y ( t ) E R = cp(V x K ) for t E [ O , l ] . If pv : V x K --, V is the natural projection, then cp-' o y ( t ) = (pv o cp-I o y(t),pK o (P-l o ~ ( t and ) ) hence

because p v o cp-l o ~ ( tE )V . Therefore y ( t ) E Va(t). Similarly, y ( t ) E V;,(,) holds for t E [ O , l ] .

510.7 Orientability of generalized foliations

341

Step 2. For simplicity, we denote as t and t' the symbols a(t) and al(t) respectively, when they are in subscripts. Then, by the above step we have for t E [O, 11

Since Dt and D:, are open in L(?(t)), it is clear that Dt nD:, is an open neighborhood of ~ ( t in ) L(y(t)). In particular, Do n Dh, is an open neighborhood of $0) = ql in L(ql). Since the leaf L(q1) is locally compact, we can find a connected open neighborhood C of ql in Do n Dh, such that the closure of C, cl(C), is compact c ~ ( c )c DOn D;,. Then cl(C) C N f l N'. Notice that N n N' is open in M. Since +0,0 : DO+ Do is the identity map and +o,t = +o,t : Do -+ N is continuous, there is to > 0 such that for all t E [0,to]

Since +o,t(cl(C)) is connected in L(-y(t)), for some x' E K t we have +o,t (cl(C)) C Dk, because N' = UzlEKI DL,. Then ~ ( t E ) DL, = Di,, from which +o,t (cl(C)) c Di,. Hence Pt(cl(C)) c $ ~ I , o l ( D = ~ lDkl )

for all t E [0,to]

where Pt

Since

pt

: cl(C)

4

= $lI,Ol

0

$o,t.

Dh, varies continuously with respect to t E [O,to] and

Po :cl(C) + Do n Dh, is the inclusion, from the fact that Do n Dh, is an open neighborhood of cl(C) in Dh, it follows that for some 0 < tl 5 to

Step 3. Since Vo r l Vd, and C are open neighborhoods of ql in Do n Do,, there exists a canonical neighborhood W of ql in Do n Dh, satisfying

cl(W) is compact cl(W) C C cl(W) c Vo n vd,.

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342

By continuity we can find 0

< tz 5 tl such that for all t E [0,t z ]

By (10.7) and (10.9) we have a continuous map

pt

:( C , C

-W )

4

(Do n Dkl, Do n Dkt - 41)

for all t E [0, t z ] . Since Pt is homotopic to Po, clearly

Hence, for each t E [0, to] we have the following commutative diagram :

where i, is a homomorphism induced by the inclusion. Step 4. Since W relation :

c Vo by (10.8), by the inclusions we obtain the following

HP(Do,DO - VO)4 Hp(Do,Do - W ) r Hp(C,C - W ) . To avoid complication, we consider 00E Hp(Do,Do - Vo)nK as a fundamental homology class of Hp(C,C - W ) . From the relation

the element Ohl E Hp(Dhl,Db, - Val) n F is also considered as a fundamental homology class of Hp(Pt(C),Pt(C) - P t ( W ) ) . Similarly, for t E 10, t z ] we identify fundamental homology classes as follows :

$10.7 Orientability of generalized foliations

Step 5. mutes :

By the definition of

pt

it is clear that the following diagram comPi -----+

(C,C - W )

(tJt(c),tJt(c -W))

+o,t,c \

J

(*o,t

Hence, letting

+o,t

+L,t1pi('.)

(C),*o,t (C - W ) )

= +o,t lc and

Denote inclusions as follows:

iql : ( C , C - W )

=

lpi ( ) we have

--

( C ,C - q l )

w)

ibl : ( P t ( c ) , P t ( c >( p t ( c ) , p t ( c )- 91) jql : ( C ,C - 91) ( L ( q l ) ,L(ql) - q l ) jil : ( P t ( c ) , P t ( c )- 91) ( L ( q l ) , L ( q l )- q l ) -+

and

Then we have that for all t E [O,tz]

Step 6. We now are in a position to prove that for t E [O,tz]

To do this it is enough to show that if

then

jql* 0 iql*(OO)= jil* 0 ibl*(O;)), and otherwise, that is, if one has

343

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344

then

jql+ 0 iql+(00) = In the case of (10.10) we have

0

ihl,(Oi, ).

i,(t)*(Ot) = i,(t)*(o:,) since j,(+ is an isomorphism. Notice that Ot and O:, are fundamental homology classes of H,($o,t(C),$o,t(C - w)).Since ~ ( tE) $ o , ~ ( W )it, follows that for z E $ o , ~ ( W ) iz,(Ot) = iz,(O:,) where i,, denotes the homomorphism induced by the inclusion i z : ($o,t(C),+o,t(C - W

))

+

($o,t(C),$o,t(C) - t.).

Hence, letting zl = $;,,,,( q l ) we have = izl*(0:t). Take qi E W such that zl = $o,t(qi), and write iz1*(0t)

4 = $0,tIC If i,: : ( C ,C - W ) + relation is obtained :

c - 9:)

($o,t(C),+o,t(C)- n). ( C ,C - 9;) denotes the inclusion, then the following :( C ,

+

izl*0 $o,t*(Oo) = 4, 0 i,;+(Oo).

-

Similarly, let -1

$ = +i.,t~,,(,, : (Pt(C),Pt(C)- 91) (40,t(C),$o,t(C)- z,), then we have i,,, o + ~ l , t l ( O=~ l ) o ibl,(O&). Therefore o i,;+(Oo)= o ib,,(Obl).

4:

4,

4:

On the other hand, letting ;: (Pt(C),Pt(C)-91) denote the inclusion, we have

Pt,

- -

= i+o

o

-'

(Dor l D;,, D o n D;, -91)

4+o iq;+

from which

Pt.(Oo) = +; o 4:-I o 4, o iq;.(Oo) = 2, o ibl+(O;,). Since pt, = PO, on Hp(C,C - W ) , it follows that and therefore

jgl+0 iq1+(Oo) = jil, 0 ibl+(O&). In the same way as above, we obtain the similar result for the case of (10.11). Since t is arbitrary in [0,t 2 ] ,we proved that for all t E [0,t 2 ]

510.7 Orientability of generalized foliations

Step 7. Let 4 be the set of s E [O,1] such that

for t E [O,s]. It is easily checked that 4 is open and closed, which shows 4 = [O, 11. Therefore, (10.6) holds. Claim 7. Let w : [O, 11 -+ M be any path. Then we can find a sequence

and a sequence XI,.

..,x,

of points in M such that

where Rxi = cpxi(Vxi x K,,), cp,, : DXi x KXi -+ Nni is a local coordinate at xi, and Vxi is a canonical neighborhood of xi in Dxi. Define an isomorphism

By (10.5) and (10.6), w, is well defined. Furthermore, the following properties are easily checked : (Ll) if w is a constant path then w, = id, (L2) if two paths wl and w2 satisfy wl(0) = w2(1), then (wl w2), = w2* 0 W l * , (L3) if wl is homotopic to w2, then wl, = ~ 2 , .

.

By making use of the family {w,) of the isomorphisms w, defined as above, we can introduce orientability for generalized foliations as follows. A generalized foliation 3 on M is said to be orientable if w, is the identity map for all closed paths w. Let dim(3) = p. We call an orientation of 3 the family of generators

for any path w in M. If 3 is orientable, then there are ) if ~ , ( l , ( ~ ) = exactly two orientations for 3. L e m m a 10.7.1. Let Mi (i = 1,2) be a connected topological manifold without boundary and let 3i be a generalized foliation on Mi. If p : MI -+ M2 is a covering map and 3 1 is the lift of 3 2 by p, then for any path w of MI

Proof. This is clear.

346

CHAPTER 10

Proposition 10.7.2. If F is non-orientable, then there exists a double covering map p : M + M such that ( 1 ) if? is the lift of 3 by p then ? is orientable, ( 2 ) i f f: M + M is a self-covering map and 3 is the lifi of 3: by f then there is a l i f t f : M -t M o f f byp.

Proof. Let A = { [ w ]E nl ( M ) : w, = id). Obviously A is a subgroup of nl ( M ) with index 2 , and then there exist a topological manifold M and a double ) )A. Since covering map p : M -t M such that p , ( n l ( ~=

for [w]E T ~ ( M )we , have p, o w, = p, because [ p o w] E A, and so w, = id. Thus, ? is orientable. For [w] E A, we have

and so ( f o w ) , = id, from which [ f o w] E A. Hence, we have f,(A) C A. This implies the existence of a lift f of f by p.

510.8 Fixed point indices of expanding maps

The purpose of this section is to show the following fked point index theorem for TA-covering maps belonging to ?EM. Theorem 10.8.1 (Fixed point index theorem). Let M be a closed topological manifold and suppose f : M 4 M is expanding. If M is orientable, then there is C > 0 such that for every m 2 C all fked points of f m have the same index 1 or - 1.

Proof. As before we denote as M f the space of the inverse limit system. Let d be a metric for M and let e > 0 be an expansive constant for f . From Remark 5.3.3 ( 1 ) we can find 6 > 0 such that for x = ( x i ) E M f , U6(xo)C W,"l,(x) where U6(xo)= { y E M : d(xo,y ) < 6). Since M is a closed manifold, there exists an atlas {(U,h)) of M with the property that { U ) is finite, each diameter of U is less than 5, and h: U + Rn is a homeomorphism where n = dim(M). Let 6 > p > 0 be a Lebesgue number of { U ) . By Lemma 2.4.1 ( 2 ) we can take C > 0 such that for m 2 C and ( x i ) , (yi) E M f if d(xi,yi) 5 e for all i 5 0 then d(xiYm,~ i - 5~p )for all i 5 0. Fix m 2 C and let a E Fix(f m). Then the sequence

$10.8 Fixed point indices of expanding maps

347

is an m-periodic orbit of f . Choose (U, h) E {(U, h)) such that B,(a) c U where B,(a) = {y E M : d(a, y) 5 p ) . Since diam(U) < 6, by the choice of 6 we have U c Ua(a) c W,"l,(a). Let V denote the set of points y in M such that there is (yi) E Mf satisfying Y-, = Y, yo E U and d(yi,ai) 5 e/4 for all i 5 0. Then a = a-, E V and V C B,(a) since m 2 1. Write Fa = f p for simplicity. Since f is expanding, Fa : V + U is bijective, and hence it is a homeomorphism. Notice that V is open in M. Since a is exactly one fixed point of Fa, to obtain the conclusion it is enough to show that the fixed point index of Fa is 1 or -1 and it does not depend on the choice of a E Fix(f ,). Since M is orientable, we fix an orientation of M and denote it as (1, E Hn(M,M - x) : x E M). Since f is a self-covering map, from (P4) of $10.6 it follows that there is a constant E, = f1 such that

for all a E Fix(f), where iv : (V, V - a) -+ (M, M - a) and iU: (U, U - a ) + (M, M - a) are inclusions. Choose an orientation (1, E Hn(Rn,Rn - z) : z E Rn) of Rn such that h: U + Rn is orientation preserving, and define a map

Since each 1, is a fundamental homology class of z, by the definition of fixed point index GI* 0 i,'(lh(a)) = I(Fa)fo where i : (h(V), h(V) - h(a)) hand, letting

-+

(Rn,Rn - 0) is the inclusion. On the other

we have G2r(lh(a)) = I ( F ~ l ) l ~ and G1 = ( ' o ~ ~ o h o ~ , o h $ where (' : Rn + Rn is a map defined by ('(z) = -z. Hence GI,

0

i ~ ' ( l h ( ~= ) )('I 0 G2r 0 hr

0

Far 0

h&

i;'(lh(a))

CHAPTER 10

348

where (,(lo) = clo and c = f1. Thus, I(Fa) = e,I(F;'). Since : U -t V and V C B,(a) c U, we can define a homotopy Ht : U + U by Ht(z) = h-'(th o F r l ( z ) ) . Then HI = F;' and Ho is a constant map. Since V C B,(a), a subset

Frl

{z E U : Ht(z) = z for some t E [O,l]) C B,(z) is compact, and so I(F;l) I(Fa) = CE,.

= 1 by Properties 10.4.3 and 10.4.5. Therefore,

510.9 Fixed point indices of TA-covering maps In this section we show the following fixed point index theorem for TAcovering maps belonging to 'Td \ PEM.

Theorem 10.9.1 (Fixed point index theorem). Let M be a closed topological manifold. Suppose f : M -t M is a self-covering map belonging to I d \ P E M . If M is orientable and the family 3; of stable sets in strong sense is orientable, then there is 1 > 0 such that for every m 2 C all fixed points of f has the same index 1 or - 1. For the proof we need some preparations.

Lemma 10.9.2. Let U be a neighborhood of 0 in Rn and denote as A the diagonal of Rn x Rn. Then for N e d a E Rn the map ia : (U, U - 0) + (Rn x Rn,Rn x Rn \ A),

i(z) = (z

+ a,a)

is a homotopy equivalence. Proof. Consider maps

Za

(Rn x Rn,Rn x Rn \ A ) , d : (Rn x Rn,Rn x Rn \ A) + (Rn,Rn - O), : (Rn,Rn - 0)

-t

-a

i (z) = ( z + a,a), d(z,w) = z - w.

Clearly d o ia = id. A homotopy from id to P o d is given by Ht(z, w) = (z t(a - w), w t(a - w)) (0 5 t 5 1). Hence is a homotopy equivalence. Since the inclusion (U, U - 0) + (Rn, Rn - 0) is a homotopy equivalence, we obtain the conclusion.

+

+

aa

mf a.

Let .rr : -t M be the universal covering and define as in (6.3) of 56.5. By Theorem 6.7.4 for each u E the families 7' and 7: of stable sets and unstable sets are transverse generalized foliations on Fix an orientation of M as (1, E Hn(M, M - x) : x E M ) where n = dim(M). Then we can choose an orientation of so that the covering map n

af

510.9 Fixed point indices of TA-covering maps

--

349

z}.

is orientation preserving, and denote it as ( 1 , E H n ( M ,M - x ) : x E Let ( 1 , E Hn(Rn,Rn - z ) : z E Rn) be an orientaion of Rn. When V is an open set of Rn and z belongs to V , we denote as

the inclusion map. It is clear that

is an isomorphism and i;;l(l,) is a fundamental homology class of z. Let U be a coordinate domain at a point in with respect to and and define a continuous map TU : U x U 4 U as in (6.8) of $6.7. Choose a and continuous injection 4 : U + Rn which preserves the orientations of 1 " . We note that + ( U ) is open in Rn and 4 : U 4 4 ( U ) is a homeomorphism. For fixed x E U define a continuous map T;,+ : + ( U ) -r Rn by

z

x,

z

for z E 4 ( U ) . Then T;,+(z) = 0 if and only if z = + ( z ) . Thus, we have a map

From the following lemma we can find a constant c E Z, called the intersection number of Faand 7:, which depends only on the family Fa.

,

Figure 41

and let 4 : N --t Rn L e m m a 10.9.3. Let N be a connected open subset of be a continuous injection which preserves the orientations. Then there exists a constant c E Z such that for any coordinate domain U in N and for any XEU

CHAPTER 10

350

Moreover, the number c does not depend on the choice of N and 4, nor on the choice of u E

mf.

Proof. To show the first statement, let x E U and choose 6 > 0 such that 4(U) > B6 where Ba = {z E Rn : llQ(x) - z11 1 5 ) . Let Ua12 be the open ball of $(x) with radius 612. Then there is E > 0 such that if z E B6 - U6/2 then IIT;,4(z)II 2 8. Since ~ u q5, and 4-I are continuous, we can find 0 < /I< 612 such that for y E U if 4(y) E Bp then I(T;,4(z) - T;,+(z)ll I e l 2 for z E Ba. Let y E U with 4(y) E Bp and define Ht : B6 + Rn by

Then we have for z E B6 - U6/2

and hence T;,+ and Ti,+ are homotopic continuous maps from (B6,B6- U6p) to (Rn, Rn - 0), from which T;,41Bs = Ti,4lB6* on Hn(B6, B6 - U6/2). Therefore T;,4* = Ti,+*: Hn(4(U), 4(U) - U6/2) Hn(Rn, Rn - 0)+

+

Since 4(z), $(y) E UaIz, by the fact that UaI2 is a canonical neighborhood it follows that

Hence we can find c E Z such that (10.13) holds for all y E UaI2. Since U is connected, the integer c does not depend on the point x E U. Let U and U' be coordinate domains in N. If U n U' # 4, then a coordinate domain U" in U n U' exists and 7u11= Y U ~ U I I ~ U= I I ~ U ~ ~ ~ Iholds. I ~ U Hence I I for x E U" +(=)-I T;,+, 0 i4(u)* (14(,)) = T&,+* O 2.4(2)-1 4 ( ~ ' )(l4(2))9 * which shows that the number c does not depend on U, because N is connected. We proved the first part of the lemma. Next, let N be a connected open set of and let 4,4' : N + Rn be orientation preserving continuous injections. Let U be a coordinate domain in N. For fixed x E U denote as D" and D u the path connected components of x in U n Ws(x) and U n Wu(x; u) respectively. Then 7~ : Ds x Du + U is a homeomorphism. Hence the composite maps

810.9 Fixed point indices of TA-covering maps

351

are homeomorphisms. For a E Rn let Ta denote the translation on Rn defined by Ta(x) = x - a. Consider the following diagram

where j and j' denote inclusions, and i+(") and iQ(") are the maps defined as in Lemma 10.9.2. It is easy to see that the two squares of the above diagram commute. Since the restrictions of the map

are homotopy inverses of i&(") and i+'(") definition we have

Hence, from the fact that follows that

4' o 4-'

: 4(U)

(see the proof of Lemma 10.9.2), by

-+

#(U) is orientation preserving it

which shows that the number c does not depend on the choice of 4. Let N and N' be connected open sets of and let 4 : N + Rn and 4' : N' -+ Rn be orientation preserving continuous injections. Since is a connected manifold, there are a connected open set N" and an orientation preserving continuous injection 4" : N" -+ Rn such that N n N" # 0 and N' n N" # 0. Applying the above result to 4,4" : N n N'' -+ Rn and I$', 4" : N' n N" -+ Rn, we obtain that the number c does not depend on the choice of N and 4. To see that the number c does not depend on a point u in take and fix x E and choose a chart (V, h) at n(x) such that h: V + Rn is an orientation preserving homeomorphism. Then, for some T > 0, B,(n(x)) c V where B,(T(x)) = {w E M : d ( ~ ( x )w) , 5 r ) and d is a metric for M. For and u E Mr let E, : D8(y) x T ( y ; U) -+ N(y; U) be a local product yE structure as in Theorem 6.6.5. Then we may suppose that T ( y ; u ) c %(y)

m,

mf,

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352

mf.

for all y E li? and u E Here B,(y) = { w E : Z(y, w) 5 r ) and Z is a metric for li? as in Theorem 6.4.1. By Theorem 6.6.5 (4) there is a constant p > 0 such that &(y) c Z ( y ; u). Choose 6 > 0 such that h(Bp(x)) > B6 where B6 = {z E Rn : llh(x) - zll 5 6). Since V is evenly covered by n, we can take an open neighborhood U of x Put 4 = h o n. in li? such that the restriction n : U + V is a homeomorphism. Then 4 : U + Rn is orientation preserving, because so is n : M -t M. Let u E &if. Since n is locally isometric, r ( x ; u) c U. We note that F i x ; u) is a coordinate domain with respect to and 'Sz (see Remark 6.7.5). Hence, TX is defined as in (10.13), and N(z;u),9 4(Z(x; u)) = h 0 n(Z(x; u)) = h(Nru(z)) 3 h(Bp(n(x))) 3 B6

-

where ru : M -t M f denotes the map defined in (6.5) of 56.5 and Nru(,) is a neighborhood of n(x) as in Theorem 5.2.1. Let S,(,) : NTU(,) -t M f be a continuous section satisfying the condition (B) of 55.2 and let a,(,) : Da(n(x)) x DU(rU(x))+ NTu(,) be a local product structure on M (see Theorem 5.2.1). Then we have for z E h(N,,(x))

TZ N(,;~),,$ = 4 0 7X(=;u)(4-1(z)9

- 4 0 %(a;")(X,

4-'(4)

- h 0 %"(=)(n(x), h-'(z)) = h 0 [STu(,)(h-'(z)), n(x)l - h 0 [7u(.), h-l(z)I. = h 0 a,(z)(h-'(z),n(x))

Let Ualr be the open ball of 4(x) with radius 612. Since B6 C ~ ( Z ( Xu)), ; we can choose E > 0 such that ((TZ (z)ll E for all z E Ba - U6/a. N(z;u),O Let u' E and let k > 0 be an integer. Then we can find a covering transformation Pk : li? t li? for n such that pi o ru(x) = pi o T ~ I ( P ~ ( X for) )i with lil I k, where pi : M ft M is the projection to the i-th coordinate. Put yk = Pk(x). Since Pk is orientation preserving, clearly 40~;' : Pk(U) t Rn is an orientation preserving homeomorphism. By the fact that Pk is an isometry is under 2, we have T ( y k ;u') C Br(lk) C Pk(U). Hence, TXk N(ur;uf),OoP;' defined and for z E h(Nrul(pk(Z)))

>

mf

TE ~(~~;~~),+,-,p;l =4o

o~ v ~ ~ ~ ; ~ o 4-'(')9 l ) ( P kyk)

- 4 P;'

%(yk;u~)(~k,Pk O #-'(z))

= h ar,,~(~k)(h-~(z), ~ ( y k )) h arU,(uk)(n(yk), h-'(z)) = h [SGf(gk)(h-'(z)),n(~)] - h [~u1(~k),h-'(z)]. Since B,(T(x)) c N u .)nNru,(,,), by the choice of 6 we have B6 c h(Nry(,))n ! h(NrU,(,,)). From thls and the fact that pi o ru(x) = pi o ru1(yk) for z with (i(5 k, it follows that if z E B6 then for i with lil 5 k Pi

0

S,(z)(h-'(~))

= Pi 0 sT",(y,)(h-'(z)).

810.9 Fixed point indices of TA-covering maps

Hence, by Lemma 5.1.9 (4) if k is large enough then for z E B6

Define a homotopy Ht : B6 - U6/2-$ Rn, 0 5 t

5 1,by

Since for z E B6 - UaIz

we have

from which

Therefore, the conclusion was obtained. We take an atlas {(U,g ) ) of M such that {Ui) is finite and each g: U + Rn is a homeomorphism. Let So > 0 be a Lebesgue number of { U ) . By Theorem 5.2.1 we can construct a local product structure a, : D s ( x o )x D u ( x ) + N, for each x = ( 2 ; ) E Mf with the property that each diameter of N, is less than So. By (3) of Theorem 5.2.1 there is po > 0 such that Bp,(xo) c N , for all ( x i ) = x E Xf. Choose a finite open cover { V ) of M such that each diameter of V is less than po and for any V there are an open set V' and a homeomorphism h: V' + Rn so that V C V' and h ( V ) = { z E Rn : Ilzll < 1). We recall that such a V is canonical neighborhood (see Lemma 10.6.1). Let po > S1 > 0 be a Lebesgue number of { V ) . Then we can find 0 < pl < b1 such that for ( x i ) = x E Mf if y E B P l ( x 0 )then d(pu 0 a,(y),xo) < b1 where pu : D s ( x ) x D u ( x ) + D u ( x ) is the projection. This is ensured by compactness of M. Next, we choose an atlas { ( W ,h ) ) of M with the property that each diameter of W is less than p l , { W ) is finite, and each h: W + Rn is a homeomorphism. Let pl > y > 0 be a Lebesgue number of { W ) . Then for x E M there is W E { W ) such that

Choose e > 0 as in Theorem 5.2.6 and fix m 2 P . Let x E F i x ( f m ) . Then there exists x = ( x i ) E Mf such that xo = x and a m ( x ) = x where

CHAPTER 10

354

u : M f + M f denote the shift map. As in Theorem 5.2.6 we define D s ( x ) , D u ( x ) , K S ( x )and K u ( x ) , and write

if there is no confusion. From Theorem 5.2.6 it follows that

Fa,, = fF. :Ds

_t

K" and F,,, =

fG,: K U

+ DU

are both homeomorphisms. Since

Ds

c W:",( x )

and K u

c DU c W,Uo( x )

where eo is an expansive constant for f , by Lemma 2.4.1 the point x is only one fixed point of F,,, and F,,,. By Theorem 5.2.1 it is clear that Dd and DU are ENRs. See also Lemma 5.2.5 and Proposition 10.6.5. By Theorem 5.2.6 (s2) and (u2), K" is open in Do where u = s,U.

Lemma 10.9.4. I ( f m , x ) = I(F,,,). I(F,,,). Proof. From Theorem 5.2.6 (su) together with Property 10.4.6 we obtain the conclusion.

Lemma 10.9.5. For every x E F i x ( f m ) , ( 1 ) I(Fs,,) = I(Fu,i) = 1, ( 2 ) I(Fu,,) is either 1 or -1 and so is I(F$). Proof. (1): Let {(W,h ) ) be the atlas choosen above. Since -y is a Lebesgue number of { W ) , there is (W,h ) E { ( W ,h ) ) such that B,(x) c W c B,,(x) c N,. By Theorem 5.2.6 (sl) we have K s c B,(x). Define a homotopy Ht : D" + D", 0 t L: 1, by

<

where p" : DS x DU + D s denotes the projection. Then Ho is a constant map and H I = F,,,. Since { y E D" : H t ( y ) = y for some t E [O, 1 ) ) is contained in a compact set pa o a;'(Bp1(x)), by Property 10.4.5 we have I(Fs,,,x) = 1. Also I ( F c 5 , x ) = 1 because Fa,, and F;: are mutually in dual relation. Since x is arbitrary in Fix(fm), ( 1 ) was obtained. (2): Let {(U,g ) ) and 60 > 0 be as above. Since 60 is a Lebesgue number of { U ) and diam(Nx) < 60, there is (U,g) E { ( U , g ) ) such that Nx c U . Write $ = o a,,

K = $ ( K s x K U ) and a = $ ( x , x ) .

Since $ : Ds x Du + Rn is injective, it follows that K is open in Rn. By the definition of fixed point index we have

510.9 Fixed point indices of TA-covering maps

355

where G = id - 11, o (F$ We consider a map

x FUt2)o

: (K, K

- a) + (Rn, Rn - 0).

GI = id - 11, 0 (Fa,, x F;) o 11,-' : (g(Nx), g(Nx) - a) + (Rn, Rn - 0). Since 11, o is a homeomorphism, we have x FU,=)o where

C

G = ( 0 G1 0 11, 0 (F:; X F,,,) 0 11,-I : Rn + Rn is a map defined by ( ( z ) = -r. Hence

I ( F 2 x FU,2)= fI(F8,2 x F):; since I(F8,. x F,;:)lo = Glc o i,D(Nx)c-'(l,) by definition. On the other hand, I(F8,, x F):; = I(F,,,)I(F;:) = 1 by (I), and hence f1 = I(F;: x F,,,) = I(F,T,)'I(F,). Therefore, I(FU,,) is either 1 or -1 and so is I(F,S,'). Since f : M + M is a self-covering map, we can find a constant such that for all x E M

E,

= f1

fF oi,'(lz)

=~mlfm(z) where f m : (U, U - x) + (M, M - fm(x)), i : (U, U - x) + (M, M - x) is the inclusion and U is a small neighborhood of x. Proof of Theorem 10.9.1. By Lemmas 10.9.4 and 10.9.5 it is enough to show that I(Fu,,) is a constant for every x E Fix(fm). Let p = dim(3:). Since 3; is orientable, there is an orientation of 3;

{I: € H ~ ( W ~ ( Xw)8, ( x ) - x) : x E M}, and by Lemma 10.7.1 we can find a constant c8 = f1 such that for x E M Let x E Fix( f m,

f;lw.(,)+(l:) = cSljm(,). and denote the inclusion maps as in : (D8,D 8 - x) + (w8(x), w8(x) - x),

jU: (KU,K U- x) + (DU,D U- x). Then in,j8 and ju induce isomorphisms id,,jf and jp respectively. For simplicity write NO = Nx, N' = ax(D8 x K"), N~ = a x ( K 8 x D"), N~ = ax(K8 x KU). Since N Q is open in M , the inclusion kNq : (NQ,N Q - x) + (M, M - x) induces an isomorphism kNq+: Hn(Nq,N Q- X) + Hn(M, M - X) where q = 0,1,2,3. Since a, : D8(x) x Du(x) + No is a homeomorphism, by Kiinneth formula there exists a generator 1; E Hn-,(DU, D u - x) such that

CHAPTER 10

356

Claim 1. There ezists a constant cU = f1 such that for each x E Fix( f m ,

Proof, Let x E Fix(f m ) . Then by (10.14) we have

If we operate f r m the both sides of the above equation, then by Theorem 5.2.6 (su) the left hand side equals

and the right hand side equals

fFm 0 k ~ f , ( l ==) ~ m k , ; , ( l z ) . h o m the fact that

(10.15)

a,*(id,-'(l:) x j:-'(l;))

= j,t,(l,),

it follows that for x E Fix(f " )

from which the conclusion is obtained. Let 4 : N o -+ Rn b e an orientation preserving continuous injection and let pu : D nx DU + DU denote the projection. Then

are an euclidean neighborhood retraction and Fix(+, o F,,, Hence, letting

by the definition of fixed point index we have

(10.16)

G I *0

Define a map G 2 : + ( N o )

-+

.&(=)-I

=I(Fu,2)l~.

Rn by

G2 = G 1 o

o ( i d x F;:) o 4;'

00= {+(%)I holds.

$10.9 Fixed point indices of TA-covering maps

357

where = $ 0 ax : D dX DU + $ ( N O ) . Then for z E $ ( N o ) we have that G 2 ( z )# 0 if and only if z G 2 induces a homomorphism $1

# $(x).

Hence

Ga+: H n ( $ ( ~ O $) ,( N o )- $ ( x ) )+ Hn(Rn,Rn - 0). Since

GI. o i:i;?;):(l+(z)) = G I *0 $1.0 (id x F::.) =1 =G

0

442)-1

i*No).(i~z))

(by (10.14)) o ( i 1 ( 1 )x c u j u l ( l ) ) (by Claim 1) 0

$ 1 ( ~ ( 1X ()

4 0 6' 1

)

by (10.16) we have For simplicity write then we have

(10.18)

G 2 ( z )= $1 o (id x F;:) 0 $,'(z) - A'(%). Let { ( W ,h ) ) be the atlas as above. Since 7 is a Lebesgue number of { W ) , for some ( W ,h ) E { ( W ,h ) ) Define A: : $ ( N o ) + Rn,O

< t 5 1, by

and next a homotopy Ht : $ ( N o )-t Rn by where pd : Dd x DU + Dd denotes the projection. Let { V ) be a finite cover of M choosen above. Since 61 is a Lebesgue number of { V ) , there is V E { V ) such that B ~ , ( x C) V C B,,(x). Let TEo,+: $ ( N o ) -,Rn be defined by Then we have a map

TEo,+ : ( 4 ( N 0 ) , 4 ( N 0) $(x)) Compare with (10.12).

+

(Rn,Rn - 0).

CHAPTER 10

358

Claim 2. (1) Let c E Z be the intersection number of the families where u E Then the following equality holds:

mf.

and 7;

( 2 ) Let Ht : 4(N0) -t Rn be as above. Then

and Hl = Gz. and Ho = Tgo,+ Proof. (1) is easily checked from the definition of intersection number. To see (2), we show first that Ht maps 4(N0) - 4(V) into Rn - 0. Let t E [O,l]. For 2 E +(No) - 4(V) put z1 = h-' o Xi (z). Then zl E BPI(x). Since pl < 61,

Figure 42 clearly zl E B ~ , ( x ) .We note that Ht(z) is expressed as

Let p* 04C1(z) # x. Since XO(z)E 4(DU), we have Ht(z) # 0. If pa o q5r1(z) = x, then by the choice of pl, pUo a i l ( z l ) E B6,(z), and hence

Since z

4 4(V), it follows that

Therefore, Ht(4(N0) - 4(V)) C Rn - 0.

510.9 Fixed point indices of TA-covering maps

359

Let z E + ( N o ) . Then we have

and

( z ) , p U0 ax1 0 F$o pU 0 &;'(E)) Hl ( z ) = $1 (pdo = 41(pd0 + ; l ( z ) , F;: 0 P" O 4,'(2)) - X 0 ( 4 = o (id x F;:) o +,'(z) - XO(z) =G~(z).

- XO(z)

Therefore, ( 2 ) holds. Since x E V C B,,(x) C N o and V is a canonical neighborhood of x, from Claim 2 ( 2 ) it follows that

Gz* = T ~ o , +: Hn(4(N0), , + ( N o )- + ( x ) ) + Hn(Rn,Rn- O), and hence by (10.17)

From this fact together with Claim 2 (1) we obtain that C " I ( F , , ~ is ) the intersection number of 7"and Therefore, I(FU,=)does not depend on the chioce of x E Fix( f m).

z.

CHAPTER 11 Foundations of Ergodic Theory

In this chapter we shall discuss measure-theoretical aspects of continuous surjection of a compact metric space, and develop a part of ergodic theory to topological dynamics. 511.1 Measure theory

In this section we recall some fundamental notions from measure theory. Let X be a set. A o-algebm of subsets of X is a collection B of subsets of X satisfying

Then (X, B) is called a measurable space. A measure space is a triple (X, B, m) where X is a set, B is a a-algebra of subsets of X and m is a function m : B -r R satisfying (1) m(B) 2 0 for B E B, (2) m(UF=l B,) = EF!lm(Bn) if {B,) is a pairwise disjoint sequence of sets of B. We say that (X, B, m) is a probability space if m(X) = 1. A collection A of subsets of X is an algebm if A satisfies

Theorem 11.1.1 (Kolmogorov extension theorem). Let X be a set and A an algebm of subsets of X . Suppose that m : A -,R is a function satisfying (1) m(A) 2 0 for A E A and m(X) = 1, (2) m(U, A,) = C,m(A,) if {A,) is a painuise disjoint sequence such that each A, E A and U, A, E A.

$11.1Measure theory

361

Then there is a unique probability measure 13 defined on the a-algebra generated by A such that *(A) = m(A) for A E A.

Let (Xi, Bi, mi), i E Z be probability spaces. Then their direct product

is defined as follows. (a) X = U i ~ z X i , (b) Let nl < n t < < n, be integers and let A,, E Bni for 1 5 i 5 r . Then we define a measumble rectangle to be a set of the form {(xj) E X : z,, E Ani, 1 5 i 5 r } . Let A be a collection of all finite unions of measurable rectangles. Then A is an algebra. In fact, (1) and (3) are clear. (2) follows from the fact that

.- -

and that A is closed under finite intersections. Let B be the a-algebra generated by A. (c) Each element of A can be written as a disjoint finite union of measurable rectangles. Now we define

and then extend m to A. Since m satisfies the conditions of Kolmogorov extension theorem, we can extend m to B. Therefore a probability space (X, B, m) is obtained. Let (X, 23, p ) be a measure space. A function ( : X -t R1 is a measurable if for all c E R1, ( c , oo) E B. If X is a topological space and B is the u-algebra generated by the open subsets of X , then each continuous function ( : X 4 R1 is measurable. We say that B is a Borel class and each member of B is a Borel set. A measure on the Borel class is said to be a Borel measure. A simple function is a function of the form Cy=lailA,, where ai E R1, the sequence {Ai) are disjoint elements of B and lAidenotes the characteristic function of Ai. The simple functions are measurable. We define the integral for simple function by

Suppose

< : X -+ R1 is measurable and non-negative, and take


{ (i

-1

2

if (i - 1)/2n 5 ((x) if <(x) 2 n.

< i / 2 n , l 5 i 5 n2n

CHAPTER 11

362

Then we define J (dp = limn,, J 4,dp. This definition is independent of the choice of ((,I. For each measurable function ( we have that ( = (+ - (where

We say that ( is integrable if J [ + d p < oo and J ( - d p < oo. We define J ( d p = J (+dp - (-dp. Observe that ( is integrable if and only if 141 is integrable. Theorem 11.1.2 (Lebesgue dominated convergence theorem). Let {(,) be a sequence of measumble functions (X, B,p) 4 R1. Suppose 4, -,( p-a.e. and there is an integrable function q such that Itn(x)l 5 q ( x ) p-a.e. for n 1. Then ( and each of En are integrable and J (,dp -+ J (dp.

>

Let G be a topological group. If there is a measure m defined on the Borel sets of G such that m ( x E ) = m ( E ) for all x E G and all Borel sets E , then m is called a Haar measure of G. Theorem 11.1.3 (Pontrjagin[Po]). Let G be a compact topological group. Then there exists a finite Haar measure of G. Let S1 = { z E C : lzl = I), then S1 is a compact topological group. Let m denote a normalized circular Lebesgue measure of S1. Then m is a Haar measure. In fact, m ( a E ) = m ( E ) for all a E S1 and all Borel sets E. Theorem 11.1.4. If m and p are finite Haar measures of compact topological group G, then m = cp for some c > 0. If m ( G ) = p(G) = 1, then m = p. Remark 11.1.5. If U is a nonempty open set of G, then U has positive Haar measure. This follows from the fact that G = glU U g2U U .gnu for some 91,. ,g, E G by compactness.

..

.

Remark 11.1.6. Let X be a locally compact group with a left invariant Haar measure p (it need not be right invariant). In ergodic theory, the question of whether the Haar measure of a non-compact locally compact group can be ergodic under a group automorphism was raised in Halmos [Ha3]. For the subclasses of abelian groups and connected groups, the question was answered as follows. If an automorphism is ergodic with respect to p and X is abelian, then X is compact ([Ra], [Y2]). Moreover it was shown in [Ka-R], [Wu], [Y2] that X is compact for the case when X is connected. Let Xo be the connected component of the identity in X. Then the factor group X/Xo is totally disconnected. It may be shown that X is compact if the automorphism is ergodic and X / X o is compact. Thus the question is deduced to a problem of whether the Haar measure of a non-compact locally compact totally disconnected group can be ergodic under an automorphism.

311.2 Measure preserving transformations

363

This problem was solved in Aoki [A071 with the help of tools in topological dynamics.

311.2 Measure preserving transformations Suppose ( X i ,Bi, mi),i = 1,2 are probability spaces. A transformation f : X 1 + X 2 is measurable if f-'(B2) C B1(B2 E B2 + f - l ( B 2 ) E Bl). f is measure preserving if f is measurable and m l ( f - ' ( B 2 ) ) = m 2 ( B 2 )for all B2 E B2. We say that f : X1 -t X 2 is an invertible measure preserving transformation if f is measure preserving, bijective and f -' is also measure preserving. Because we wish to study iterations f ", we are mainly interested in the case ( X l , B l , m l ) = (X2,132,m2). Remark 11.2.1. As examples of measure preserving transformations we can describe the following : ( 1 ) The identity map of ( X ,23, m ) is measure preserving. ( 2 ) Let m be the Haar measure of S 1 and define f : S 1 -t S 1 by f ( t )= a t where a is a fixed element of S 1 . Then f is measure preserving. (3) The map f ( x ) = ax defined on a compact group G ( a is fixed) preserves the Haar probability measure. (4) Let Yk = (0, I , . . . ,k - 1 ) and give a number pi to i E Yk such that pi = 1. Then the shift map a ::Y +:Y preserves the measure of each measurable rectangle. Thus it preserves the measure of sets which are finite disjoint unions of measurable rectangles. From this we can easily check that a is measure preserving. (5) Let X 1 be a set and let ( X 2 ,D2,m 2 )be a probability space. For any surjection f: X 1 -t X 2 we can choose a a-algebra B1 and a measure ml of X 1 to make f measure preserving. In fact, Bl = f-'(B2) and define ml by ml ( f -' B ) = m2 ( B ) for B E B2. Conversely, let X 2 be a set and ( X I ,Bl, m l ) a probability space. For f : X 1 -+ X 2 any surjection we can choose a 0-algebra B2 = { B C X 2 : f ( B ) E B1) and define m 2( B ) = ml ( f - ' B ) for B E B2. (6) Define a map f: [0,1) + ( 0 , l ) by

where { y ) denotes the fractional part of y. Then f preserves the Gauss measure m of [O,1]which is given by

for Bore1 sets A. We shall introduce some general concepts in ergodic theory.

364

CHAPTER 11

Theorem 11.2.2 (PoincarB recurrence theorem). Let f be a measure preserving transformation of a probability space ( X , B , m ) . Let E E B such that m ( E )> 0. Then all points of E return infinitely often to E under positive iteration by f.

-'

Proof. For N > 0 let EN = Ur=Nf -"(E). Clearly, f ( E N )= EN+',EN C EN-1 C C Eo and r n ( E ~ + l= ) m ( f - ' ( E ~ )=) EN). Therefore, EN) = m(E0) for each N , and m(n;=, E N )= m(E0). The set EN = f -"(E) is the set of all points entering E infinitely often under E N )consists of all points of positive iteration by f . Moreover, F = E n E which enter E infinitely often under positive iterates of f . Since EN C Eo and both sets have the same measure,

nGZo

n;=,u:,,

(ngZo

n;=o

To see that a point of F returns to F infinitely often, let x E F. Then there is a sequence 0 < nl < nz < such that f " i ( 3 ) E E for i. We first show that f " 1 ( x ) E F . Since f ( f "1 ( 2 ) )= f "i ( x ) E E for i > 0, f " 1 ( x ) enters E infinitely often under iterates of f "=-"I,f """1,- . .. This implies that f n l ( x )E F. Similarly we can show f "'(3)E F for i.

---

A transformation f : (X, B, m ) + (X, B, m ) is ergodic if for B E B, f -'(B) = B then m ( B )= 0 or m ( B )= 1. Theorem 11.2.3. Let f : X -+ X be a measure preservang tmnsfonnation. Then the following are equivalent. (1) f is ergodic, ( 2 ) m ( f - ' ( B ) A B ) = 0 for B E B + m ( B )= 0 or 1, (3) For A, B E B with m(A),m ( B ) > 0 there is n > 0 such that m(f -"(B) A B ) > 0. Here AAB = (A\ B ) u (B\A).

Proof. ( 1 ) + ( 2 ) : Suppose m ( f - ' ( B ) A B ) = 0 and let B, = f -i(B) E B. Since

n:=oUEn

we have fW1(B,) = B, and m(B,) = m(B). Thus m(B,) = 0 or 1 and therefore m ( B )= 0 or 1. ( 2 ) + ( 3 ) : Let m(A) > O,m(B) > 0 and suppose ( 3 ) is false. Since m(f -"(A) n B ) = 0 for all n > 0, we have m((U,",, f -"(A)) n B ) = 0. Let A' = Ur=l fVn(A). Then f-'(A') C A' and m(At) = m(f-'(A')). Thus m(f-'(A')AA1)= 0. By ( 2 ) we have m(At) = 0 or 1. But f-'(A) c A' and f is measure preserving. Thus m(At)= 1. This contradicts the fact that m(At n B ) = 0.

$11.2 Measure preserving transformations

365

( 3 ) + ( 1 ) : Suppose ( 1 ) is false, i.e. f - ' ( B ) = B and 0 < m ( B ) < 1 for some B E t?. Then m ( f-"(B) f l ( X \ B ) ) = 0 for n > 0, which contradicts (3).

A characterization of ergodicity is given as follows. Theorem 11.2.4. Let f: ( X ,13, m) -t ( X ,B, m ) be measure preserving. Then the following statements are equivalent. ( 1 ) f is ergodic. ( 2 ) Let ( be a measumble function of X . If ( o f ( x ) = ( ( x ) for x E X , then ( is a constant a.e.. ( 3 ) Whenever ( is measumble and ( o f ( x ) = f ( x ) a. e. then ( is a constant a. e.. ( 4 ) Whenever ( E L 2 ( m ) and ( o f ( x ) = ( ( 3 )for x E X then ( is a constant a. e.. ( 5 ) Whenever f E L 2 ( m ) and ( o f ( x ) = ( ( 3 ) a.e. then ( is a constant a.e.. Here L 2 ( m ) is a collection of square integmble functions. Proof. ( 1 ) ) ( 3 ) : Suppose ( is measurable and ( o f = ( a.e.. Define

Then f - ' ( x ( k , n ) ) A X ( k ,n) C { x : €

0

f

(2)

# ((2))

and thus m ( f - ' ( X ( k , n ) ) A X ( k , n ) ) = 0. By Theorem 11.2.3(2) we have m ( X ( k , n ) ) = 0 or 1. Fix n. Then U k E Z X ( k , n )= X which is a disjoint union. For each n there is a unique k, such that m ( X ( k , , n ) ) = 1. Let Y = w X(k,, n). Then m ( Y )= 1 and ( is a constant on Y. Thus f is a constant a.e.. It is clear that ( 3 ) + ( 2 ) + ( 4 ) , ( 5 ) + ( 4 ) and ( 3 ) + (5). Thus it only remains to show that ( 4 ) + (1). Suppose f - ' ( E ) = E for E E 13. Then 1 ~ E ~ ~ (and m 1)~ o f ( x ) = l E ( x ) for x E X . By ( 4 ) , l E is a constant a.e.. Thus m ( E ) = jl E d p = 0 or 1.

n,=,

Remark 11.2.5. Let X be a compact metric space and let m be a Bore1 probability measure of X which give positive measure to every nonempty open set. If f: X + X is continuous, measure preserving and ergodic, then m ( { x : Of ( x ) is dense in X ) ) = 1 where of( x ) = { f n ( x ): n > 0). This is checked as follows. Let {U, : 1 5 n < oo) be a base for the topology of X . of ( x ) is dense in X if and only if x E UEof-k(Un). Since f -'(Ur=o f --"(u,)) c UFOf --"(u~) and f is measure preserving and ergodic, we have m(UEof -k(Un)) = 0 or 1. Since UFOf f - k ( ~ , , ) is nonempty open set, we have m(Ur=of -"Un)) = 1.

n:=l

366

CHAPTER 11

Remark 11.2.6. Let G be a compact metric abelian group and let A : G + G be an endomorphism. Then A is ergodic with respect to the Haar probability measure if and only if yoAn = 7 for some character 7 then 7 = 1. Remark 11.2.7. Let T n be the n-torus. Then an endomorphism A : Tn + Tn is ergodic with respect to the Haar probability measure if and only if a lift A : Rn -t Rn of A has no roots of unity as eigenvalues. The proof was given in Remark 1.1.4 of Chapter 1. Remark 11.2.8. Let m be the measure of: Y constructed as (4)of Remark 11.2.1. Then the shift map a :: Y +:Y is ergodic with respect to m. Let A be the a-algebra generated by finite unions of measurable rectangles. Suppose a-'(E) = E for E E B. Let E > 0 and choose A E A such that m ( E A A ) < E . Then

Choose n so large that B = a-"(A) depends on different coordinates from A. Then m ( A n B ) = m ( A ) m ( B )= m ( ~ )Since ~ . m ( E A B )= m(u-"(E)Au-"( A ) ) = m ( E A A ) < E and since E A ( A n B ) c EAAUEAB, we have m ( E A ( A n B ) ) < 2 ~ Thus . Im(E) - m ( A n B)I < 2~ and

since m ( A ) 5 1 and m ( B ) 5 1. Since E is arbitrary, we have m ( E ) = m ( E ) 2 and so m ( E ) = 0 or 1.

$11.3 Ergodic theorems

The purpose of this section is to introduce the major result proved in 1931 by G.D.Birkhoff [Birl]. Theorem 11.3.1 (Birkhoff ergodic theorem). Let f : ( X ,B, m ) + ( X ,B, m ) be measure preserwing (we allow ( X ,B, m ) to be a-finite) and E L1(m) where L1(m) is the collection of all integrable functions. Then l l n ~ r i ; < ( f i x )converges a.e. to <* E L1(m),and [*o f = (* a.e. and / <*dm= J
<

$11.3 Ergodic theorems

367

Remark 11.3.2. I f f is ergodic, then (* is a constant a.e. and if m(X) < oo

&

then (* = J (dm a.e.. We define the time mean of ( to be

and the space mean of ( to be

If f is ergodic, then the ergodic theorem implies these means are equal. As one of applications of Birkhoff ergodic theorem we have the following result. Let f: [O, 1) -t (0,l) be defined by f (x) = 2%mod 1. Then f preserves Lebesgue measure and is ergodic (by Remark 11.2.1(2) and 11.2.7).

Theorem 11.3.3 (Borel's theorem). Let m be Lebesgue measure of [0, 1).

'.{ n

the number of 1's in the first n digits of the binary expansion of x E [O,l)

}

1

a.e..

+ + ...

8 has a unique binary expansion. Then Proof. Suppose x = . . Let ((x) = 111/2,1)(x). Then f(x) =

+++

+.

Thus the number of 1's in the first n digits of the dyadic expansion of x is ((fix). Dividing both sides of this equality by n and applying Birkhoff ergodic theorem, we have

Let f : (X, U, m) + (X, f3, m) be measure preserving and define an operator Ur on Lp(m) by (Uft)(.) = ((fx). Then Uf(LP(m)) = LP(m) and, since f is measure preserving, IIUf(llp = ll(llp where lltllp = { J x ItIpdm}llP. To show Birkhoff ergodic theorem we need the following lemma.

CHAPTER 11

368

Lemma 11.3.4 (Maximal ergodic theorem). Let U : L1(m) + L1(m) be a positive linear operator (i.e. 4 2 0 + U( 2 0) which has a norm 5 1. For N > 0 an integer define

and put

FN= max

OSnsN

4".

Proof. We use the technique due to Garsia. Obviously FN E L1(m). Since FN2 (,, for 0 5 n 5 N, we have UFN > Utn by positivity. Thus UFN 4 tn+l and so

+ >

~ F N ( x+ ) ((2) 2

( when FN(x) > 0)

= max (,(x) OsnsN

= FN(x).

Thus 4 2 FN- UFN on A = {x : FN(x) > 0) and

2

FNdm -

/x

UFNdm (since UFN 2 0 when FN2 0)

2 0 ( since llUll = sup{IIU[II

t

: E ~ ' ( m ) II4IIi , = 1) 5 1).

Lemma 11.3.5. Let f : (X, 23, m) -+ (X, 23, m) be measure preserving. If L1(m) and 1 B, = {x E X : sup C(fmx) > a), "21 m=O

then

4E

knA

Cdm 2 am(& n A)

if f-'(A) = A and m(A)

< oo.

Proof. We first prove this result under the assumptions m(X) < oo and A = X. If 17 = 4-a, then we have B, = Uy=o{x : FN(x) > 0). Thus JB, qdm > 0

§ 11.3 Ergodic theorems

SBp

369

by Lemma 11.3.4. Therefore ( d m 2 am(B,). In the general case, use in the place of f . Then we have that qdm 2 a m ( A n B,).

fiA

SAnB

Proof of Theorem 11.3.1. For ( E L 1 ( m )we set

-1 ( * ( x )= lim n n

n-1

C (( f'x),

(.(x) =

i=O

1 n-l

-

C((fix). i=0

Then we have (* o f = (* and (, o f = t,. This follows from the fact that

For

p < a let

Then f

E,,p = { x E X : &(z) < /3

< a < (*(z)).

(E,,p) = E,,p and

To use Lemma 11.3.5 we show that m(E,,p) < oo. Suppose a > 0 and let C C E,,p with m ( C ) < oo. Then q = ( - a l c is integrable and by Lemma 11.3.4

where HN is defined analogously to the function FN in Lemma 11.3.4. Since C U;=,{X : H N ( x ) > 0 ) , we have ltldm 2 a m ( C ) . Therefore m ( C ) 5 Q Jx [ ( [ d m for every subset C of E,,p with finite measure and thus m(E,,p) <

c

Sx

'

00.

If a < 0, then /3 < 0 and so we can apply the above with -( and replacing 5 and a to obtain rn(E,,@) < oo. Let us put

By Lemma 11.3.5

LSs

(dm = L,pnBp

t d m L am(E,,p n B,) = am(E,,p).

-p

CHAPTER 11

370

Replacing f , a and

-

/3 by -f ,

* =-

-/3 and -a respectively, we have that

J

(-0. = -c*,

( d m 2 pm(E.,p).

E- ,@

However, since a > p, we have m(E,,p) = 0, and since

6

U

X : I*(%)< C*(x))C

Eag.

a,p:rational we have m ( { x : ( , ( x ) < ( * ( x ) ) )= 0, i.e. ( * ( x ) = ( , ( x ) a.e.. ( ( f i x )converges a.e.. To show (* E L 1 ( m )let

Therefore

then

mCn

so that !igJ<,dm < oo. By Fatou's lemma = [(,I is integrable. (Here Fatou's lemma says that for integrable functions G , if (, 2 0 a.e. then Jx h ( , d m 5 @ Jx (,dm). Finally we show that J ( d m = J ( * d m if m ( X ) < oo. To do so we set k k+l D; = { x E X : - (*(z)< -1, k € Z, n 2 1. n

For

E

<

> 0 small we have D; n B(i-,)

l;

(dm 2 (

Since E is arbitrary, we have

J

;'J

n

= D;. By Lemma 11.3.5

k

-E ) ~ ( D ; ) .

( d m 2 : m ( ~ : ) . Then

kfl ('dm _< -m(D;) n Summing over k, we have

Thus Jx (*dm 5 Jx ( d m since m ( X ) Jx(-t)*dm 5 Jx(-()dm, i.e.

-

ez, - - -

1

5 -m(D;) n

< oo.

(.dm 5 -

L

+

Replacing -( by (, we have

(dm.

Since (, = (* a.e., we have that 1, (,dm 2 JX ( d m and therefore J ( * d m = J (dm. As a corollary of Birkhoff ergodic theorem we have

511.3 Ergodic theorems

371

Corollary 11.3.6. Let f : (X, D,m) -t (X,D,m) be a measure preserving tmnsfonnation of a probability space. Then f is ergodic if and only if for A,BEB

.

lim

n+ao

An

x

n-1

m(f -'(A)

n B) = m(A)m(B).

i=O

Proof, =+ ) : Suppose f is ergodic. Putting have

E

= 1~ in Theorem 11.3.1, we

and multiplying by 1~

If we integrate the above, by the dominated convergence theorem we have

( 1 ) : Let f - ' ( E ) = E , E e B a n d l e t A = B = E . Theni~:itrn(E)= m(E)2 and so m(E) = m ( ~ ) Therefore, ~ . m ( E ) = 0 or 1.

T h e o r e m 11.3.7 (LP-ergodic t h e o r e m of Von Neumann). Let 1 5 p < oo. Let f: (X, 8,m) + (X, 8,m) be measure preserving on a probability space. If [ E LP(m), then there ezists (* E LP(m) such that o f = a.e. and

c*

<

c*

Proof, If is bounded and measurable, then ( E LP(m) for any p 2 1. By Birkhoff ergodic theorem, we have $ <(fix) + <*(x) a.e. and [* E LoO(m).Thus <* E LP(m). Clearly, I$ Cyit <(f - <* (z)lp t 0 a.e. and by the bounded convergence theorem

~yzt

CHAPTER 11

372

Let q E LP(m) and M n ( q ) ( ~ )= q(f%) for n 2 1. Then {Mn(q)) is a Cauchy sequence in LP(m). In fact, we can choose E Lm(m) such that JJv- [/Im < ~ 1 4 Then .

<

llMn(7.7) - Mn+k('V)IIp L IIMn(77) - Mn(<)llp+ IIMn(0 - Mn+k(q)llp

+ IIMn+k(€) - Mn+k(q)llp

< E l 4 + &/2+ E l 4 = E when n is sufficiently large and k > 0. Since

-

we have q* o f = q* a.e.. Let f : (X,8,m) (X, 8,m) be measure preserving on a probability space. Then f is said to be weakly mixing if for A, B E B 1 N-l N i = O lrn(f-'(~) n B ) - m(A)m(B)l

-

-

0,

and to be strongly mixing if for A, B E B

m(f - N ( ~ n) B )

+ m(A)m(B).

By definition we see that if f is strongly mixing then it is weakly mixing, and that weakly mixing is stronger than ergodicity. Remark 11.3.8. There are examples of weakly mixing which are not strongly mixing (Kakutani, Maruyama, Chacon, Katok-Stepin). We say that a subset J of Z+ satisfying fl(Jn{O,l,.-- , n - 1)) -0 asn+oo n is a set of density zero. Here Z+ denotes the set of all natural numbers. Theorem 11.3.9. Let f: ( X ,B, m) 4 (X, B,m) be measure preserving on a probability space. Then the following are equivalent. (1) f is weakly mixing. (2) For A, B E B there is a density sen, set J(A, B ) C Z+ such that lim

m ( f d n ( A ) n B ) =m(A)m(B).

n+m

n@J(A,B)

( 3 ) For A, B E 8,

lim

n-oo

n

lm(f -'(A) n B) - m ( ~ ) m ( ~ )=l '0. i=O

This follows from the following lemma.

8 11.3 Ergodic theorems L e m m a 11.3.10. If {a,) following are equivalent.

373

is a bounded sequence of real numbers, then the

:

( 1 ) zygt jail -t 0, ( 2 ) There is J c Z+ of density Zen, such that limn a n = 0 if n @ J .

---

Proof. For M C Z+ let a ~ ( ndenote ) the cardinality of ( 0 , 1, ,n - 1 ) n M . ( 1 ) +- ( 2 ) : Let Jk = { n E Z+ : lanJ 2 %)for k > 0. Then J1 c Jz c and each Jk has density zero since lai[ : ) a J,(n).Therefore there are integers 0 = 10 < el < C2 < such that for n ek

.

:

>

>

Define J = U z o { J k + l n [ek,ek+l)). Then J has density zero. In fact, if e k _< n < Ik+l, then J

n [O,n)= { J n [o,ek))u { J n [ e k , n ) > c {Jk n [O,ek))u {Jk+l n [O, 4

)

and therefore

Thus ; a J ( n ) -t 0 as n -t oo, i.e. J has density zero. If n > ek and n 6 J , then n @ Jk+l and lanl < Therefore, lim,,,,,g~ lan[ = 0. ( 2 ) +- ( 1 ) : Suppose lan/ _< K for n. Let E > 0. Then there is Nc > 0 such that n 2 N, and n $! J imply lanl < E , and Me > 0 such that n Me implies < E . For n 2 max{Nc, M,), n

&.

>

( 1 ) e ( 3 ) : This follows from the fact that lim lan[ = O u nlihlan12 = O . n4J n4J

n-oo

Let f : ( X ,B , m ) -t ( X ,B,m) be measure preserving on a probability space. Define a transformation f x f : X x X -,X X X by ( f x f ) ( x , y ) = ( f ( x ) ,f (Y)). Then f x f is a measure preserving transformation on ( X x X , l? x B, m x m).

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374

Theorem 11.3.11. If f is measure preserving on a probability space ( X ,B, m ) , then the following are equivalent. ( 1 ) f is weakly mixing. ( 2 ) f x f is ergodic. ( 3 ) f x f is weakly mixing. Proof. ( 1 ) + ( 3 ) : Let A, B E B and C , D E B. Then there are J1 and Jz of density zero such that lim m ( f - " ( A ) n B ) = m ( A ) m ( B ) , n4J1 lirn m ( f a n ( C )n D ) = m ( C ) m ( D ) . n-+m n4Ja

n+m

Then we have lim m x m ( ( f x f ) - " ( A x C ) n ( B x D ) ) n+m n4JiuJa = lim m ( f - " ( A ) n B ) m ( f - " ( C ) n D ) n;;

= m ( A ) m ( B ) m ( C ) m ( D= ) m

x m ( Ax C ) mx m(C x D)

The above relation holds for finite disjoint unions of rectangles, which form an algebra 3. Notice that 3 generates the a-algebra B x B. By Lemma 11.3.9 we have

. n-1

for A , B E 3,and the result holds for A , B E ( 3 ) =+ ( 2 ) : This is clear. ( 2 ) + ( 1 ) : Let A, B E B. Then we have

1 n-l

B x B.

1 n-l n a=O .

-E r n ( f - ' ( ~ )B n ) = - E m x m ( ( f x f)-'(A i=o

x X ) n ( B x X))

and also

1 n-l 1 n-l E(m(f-'(~) n B))' = -Em x m((f x n n i=o

i=O

f ) - ' ( ~ x A) n ( B x B))

511.4 Probability measures of compact metric spaces

Thus 1 n-l

- x{m(f-'(A)

n B ) - m(A)m(B)}'

i=o

Therefore f is weakly mixing by Theorem 11.3.9. 511.4 Probability measures of compact metric spaces Let X be a compact metric space. A Borel probability measure ,u is said to be regular if for E > 0 and a Borel set B there exist a closed set C, and an open set U, such that C, c B c U, and p(U,\C,) < E . Theorem 11.4.1. Every Borel probability measure is regular. Proof. Let p be a Borel probability measure of X and define

> 0 there are a closed set C, and an open set U, such that C, c A C U, and p(U,\C,) < E

for E

If all closed subsets of X are members of M and if M is a a-algebra, then we have M = B,and therefore ,u is regular. We first show that M is a a-algebra. Since X is open and closed, we have that X E M . If A E M, for E > 0 we have C, C A c U, and p(U,\C,) < E as in definition. Thus, X\C, > X \ A > X\U, and (X\C,)\(X\U,) = U,\C,, and so ,u((X\C,)\(X\U,)) < E . This shows that X \ A E M . It remains to show that if A1, A',. E M then Un A, E M. Let E > 0. Then C,,, c An c UC,, and p(U,,,\C,,,) < €12, for certain closed set C,,, and open set U,,,. Let U, = U, U,,,, clearly U, is open. But C: = U, C,,, is not closed. Choose k > 0 such that ,u(CL\ C,,,) I €12 (by the following Lemma 11.4.2) and k define C, = U,=, C,,,. Then C, is closed and

.

u:=,

00

c, c

U A, k=l

and moreover

c U,,

CHAPTER 11

376

We have shown that M is a a-algebra. Finally we show that M contains all closed subsets of X. Let C be a closed subset of X and define a sequence of open sets Un by

Un = C. For E > 0 fix k > 0 such that Clearly, Ul > Uz > > p(Uk\C) 5 E and put U, = Uk and C, = C. Then C, c C c U, and so C E M. Therefore M contains all closed subsets.

L e m m a 11.4.2. E B. Then

Let p be a Bore1 probability measure of X and let Al, Az,

.

- -

(1) A1 C A2

* p( U An) = lim p(An),

c.

n-m

n=l m

(2) A1

> Az

3

- * p( n An) = lim p(An). n+m

n=l

Proof. (1) : Define a sequence of measurable sets Bn by B1 = A1,Bn = An\An-l,n = 2,3, -.. . Then An = UL1Bi, U,"==,An = U:=l Bn and {Bn) are pairwise disjoint. Thus

= lim p ( U Bi) = lim p(An). n+oo

n-00

i=l

.

( 2 ) : Put Bn = A1\An,n = 1,2,... Then B1,Bz,... E Band AnnBn = 8 for n 2 1. Since A1 = An U B,, we have 1 2 p(An) p(Bn). Since A1 = An) U ( U Z 1 Bn),

+

(n:=P=,

n

n=l

u m

m

p(Al) = I"(

An)

+ p(

n=l

00

= p(

An)

n= 1

0 An) + lim { p ( A l )- p(An)) n=l

n+w

m

and thus p(n:=l

n 00

Bn) = p(

An) = limn+- p(An).

+ lim p(Bn) n+m

511.4 Probability measures of compact metric spaces

377

Remark 11.4.3. Let p be a Borel probability measure of X . For any B E we have

B

p(B) = sup{p(C) : C C B and C is closed ), p(B) = inf{p(U) : B C U and U is open ). This follows from Theorem 11.4.1. Theorem 11.4.4. Let p and u be Borel probability measures of X . If

for all ( E C(X), then p(B) = u(B) for all B E B. Proof. If p(C) = u(C) for any closed subset C, by Remark 11.4.3 we have p ( B ) = u(B) for every B E B. Thus it sufficies t o show that p(C) = u(C) for closed sets C. Since p is regular, for E > 0 there is an open set U such that C C U and p(U\C) < e. Define a continuous function ( : X -, R by

d , ) d(x,X\U + d ( + , C )

ifxgU if u.

Clearly, ((x) = O for x E X\U, t(x) = 1 for x E C and 0 I ((x) 5 1 for x E X . Since p(U) 5 p(C) E, we have

+

Since E is arbitrary, u(C) I p(C). By replacing p by v we have p(C) I v(C), and therefore p(C) = u(C).

.

Theorem 11.4.5. Let X be a compact metric space and let plyp2, ,p be a sequence of Borel probability measures. Then the following conditions are equivalent :

(2) lim sup pn(C) 5 p(C)

( closed set C c X),

(3) lim inf pn(U) 2 p(U) n--+oo

( open set U

n+w

(4) nlim pn(A) = p(A) +w

Proof. (2)

(3) is clear.

c X),

(p(BA) = 0 where a A is the boundary of A).

CHAPTER 11

378

(1)

+ (2) : Let C be a closed set. For k 2 1 define

nk

Then Uk > Uk+l for k 2 1 and Uk = C. Thus p(Uk) -+ p(C) by Lemma 11.4.2(2). Since X is compact, we can find continuous functions (k satisfying

{0

i f x ~ C i f x @ Uk.

1

tk(x) = Thus lim sup pn(C) n-00

I lim sup n+cn

Since k 2 1 is arbitrary, we have

(2), (3) + (4) : Let A E B and denote as int(A) the interior of A. If p(aA) = 0, then we have p(int(A)) = p(A) = p(cl(A)) and thus lim sup pn(A) I lim sup pn(cl(A)) I p(cl(A)) n-m

n+m

P(A),

liminf pn(A) 2 liminf pn(int(A)) 2 p(int(A)) = p(A). n+m

n-oo

Therefore, limn,, pn(A) = p(A). (4) =+ (1) : For 5 E C ( X ) define m = min{t(x) - 1 : x E X ) ,

M = max{((x)

+ 1:x E X}

and u(E) = p((-'(E)) for Borel sets E of the interval [m, MI. Notice that v is a Borel probability measure of [m, M] and the set A = {t E [m, M] : u({t)) = 0) is dense in [m, MI. For E > 0 choose a finite sequence to, tl, ,tk such that each t j belong to A and

.. .

For 1 <_ j 5 k the set Aj = {x E X : tj-1 I ((2) < tj} is an element of B and ,Ak) is a partition of X . Since dAj c cl(Aj)\int(Aj) and {Al,

we have p(aAj) = 0 for 1 I j I k. Thus

311.4 Probability measures of compact metric spaces

379

k zj=o tjlAj(x)

for sufficiently large n. Since a sequence of simple functions converges uniformly to for sufficiently large k and n we have

t,

which shows (1). Let C(X) be the collection of all real valued continuous functions. We define a norm for C(X) by II
>

for all

E C(X).

Proof. See Halmos [Ha2]. Let X be a compact metric space and M(X) a collection of all Borel probability measure on X. Since C(X) is a Banach space with the norm, there exists a countable set it1,t2, . ) which is dense in C(X). For p, v E M(X) define

.-

The11 D is a metric on the space of Borel probability measures, which gives a topology, called the weak topology. Theorem 11.4.7. M(X) is compact.

Proof. It sufficies to show that if {p,) is an infinite sequence of Borel probability measures in M(X), then there are a sequence {pni} and p E M(X) such that D(pn,,p) + 0 as i + oo ; equivalently

CHAPTER 11

for all ( E C(X). To do so we write p(() = SX (dp when [ E C(X) and p is a Borel probability measure. Choose a sequence C1, t2, that is dense in C(X). We consider the sequence of real numbers {pn(Cl)). This sequence is bounded by II(1II and so has a convergent subsequence, say { p k l ) ( ~ ) ) . Consider the sequence {p?)(&)). This is also bounded and so has a convergent subsequence (2) {pn (€2)). Notice that {p?)((l)) also converges. Continue this manner, and for each i 2 1, construct a subsequence {p?)) of {pn} such that

..

.

Then, for € = tl,t2, ,Ci the sequence {p?)(()) converges. Taking the diagonal {pt)}, the sequence {p?)(&)) converges for all i. Thus {p?)(()) converges for all ( E C(X). Define J(() = lirn,,,p?)(() for E C(X). Then J : C(X) -t R is 0 when ( 2 0, by Riesz linear and bounded. Since J ( l ) = 1 and J ( ( ) Representation Theorem there exists a Borel probability measure p on X such that J(<) = (dp for all ( E C(X). Therefore,

>

Sx

for all ( E C(X). Remark 11.4.8. Let f: X + X be a homeomorphism of a compact metric space. Then f and the inverse f-' are B-measurable, i.e. f (E), f-'(E) E B for E E B. Indeed, the families M = {E E B : f-'(E) E B) and M' = {E E B : f ( E ) E B} are a-algebras containing open sets of X. Thus we have that are Bmeasurable. B = M = MI, i.e. f and f

-'

Theorem 11.4.9 (Krylov-Bogolioubov's theorem). Let f: X -+ X be a homeomorphism of a compact metric space. Then there exists a Borel pmbability measure p such that

for all Borel sets B. Proof. Fix x E X. For

< E C(X) and n 2 0 define

$11.4 Probability measures of compact metric spaces

381

Then Jn : C ( X )+ R satisfies the conditions of Riesz Representation Theorem. Thus there exists a Borel probability measure p, on X such that

for all ( E C ( X ) . Since M ( X ) is compact, there exists a subsequence { p n j ) and a Borel probability measure p on X such that

for all ( E C ( X ) . Since

Sx

S,

Sx

-' Sx

we have ( o f dp = ( d p and thus (dpo f = (dp for all ( E C ( X ) . By Theorem 11.4.4 we have that p ( f - l ( B ) ) = p ( B ) for all Borel sets B . Similarly we can prove that p(f ( B ) )= p ( B ) for all B E B.

A Borel probability measure p is said to be f -invariant if p(f - ' ( B ) ) = p ( B ) for all B E B when f is a continuous map, and if p( f - ' ( B ) ) = p(f ( B ) )= p ( B ) for all B E B when f is a homeomorphism. Let M f (X) be the collection of all f -invariant Borel probability measures.

Theorem 11.4.10. Let f : X + X be a homeomorphism of a compact metric space. Then ( 1 ) M ( X ) is nonempty, ( 2 ) M ( X ) is a closed subset of M ( X ) , ( 3 ) M ( X ) is a convex set. Proof. ( 1 ) is clear by Theorem 11.4.9. (2) : Suppose {p,) C M ( X ) converges to p. Then

for all ( E C ( X ) . Thus p is f-invariant. ( 2 ) is obvious. Let p be an element of M ( X ) . The Borel probability measure p is said to be an edremal point of M f ( X )if p # apl (1 - a ) p 2for all p1, p2 E M ( X ) and all a with 0 < a < 1.

+

CHAPTER 11

382

Theorem 11.4.11. Let m E M (X). Then m is an extremal point of M (X) if and only if m is ergodic with respect to f . Proof. Suppose m is not ergodic. Then there exists a Borel set E such that f-'(E) = E m-a.e. and 0 < m(E) < 1. Define measures ml and m2 by

for B E

B. Then ml # m2 and ml ,ma are in M (X), and moreover

Thus m is not an extremal point of M (X). Conversely, suppose m E Mf (X) is ergodic and m = am1 (1 - a)m2 where ml,m2 E Mf (X) and 0 Ia I1. Since ml << m(ml is absolutely continuous with respect to m), Radon-Nikodym's theorem tells us that there is a function %(x) satisfying ml(E) = JE %(x)dm. Thus

+

Therefore %(f-ly) = %(y) m-a.e.. Since m is ergodic, we have a constant = k m-a.e.. Therefore 1 = ml(X) =

J.

2=

kdm = km(X) = k.

Since k = 1,ml = m and therefore m2 = m = ml.

Theorem 11.4.12. Let f: X -t X be a homeomorphism of a compact metric space and let R(f) be the nonwandering set of f . Then p(R(f)) = 1 for all CL E M f (XI. Proof. It sufficies to show the theorem for the case X # R(f). Take x E X\R(f), then there is an open neighborhood U of x such that U n f "(U) # 0 for all n # 0. Since { f -n(U) : n 2 0) is a pairwise disjoint sequence, we have

Since p is f-invariant, we have p(U) = 0. If p(E) > 0 for a Borel set E with E C X\R(f), then there exists a closed set C such that C c E and p(C) > 0 (since p is regular). Since C n R( f ) = 0, for x E C we have fn(Ux)nUz = 0 (all n > 0) for some open set U, containing z. Since C is compact, there exists a finite sequence U,, , ,U,, such that C C U,, U U,, Then p(Uzi) > 0 for some i, since p(C) > 0. This cannot happen. Therefore, p(X\R(f)) = 0. U

s

.

.

.

.-

$11.4 Probability measures of compact metric spaces

383

Theorem 11.4.13. Let f : X -t X be a homeomorphism of a compact metric space and let R ( f ) be the recurrent set of f , i.e. R( f ) = { x E X : x E a ( x )U w ( x ) } . Then R( f ) is a Borel set and p(R( f )) = 1 for all p E M ( X ) . Proof. Let A be a Borel set of X and define

for integers n, k. Then p(Ao,,) = 0 for all p E M f ( X ) . Indeed, since f (Ao,,) = f ( A o rl A,) = A , r l An+l for n 2 1, we have

and by induction f ( A k , m ) = all k 2 0, we have

for k 2 1. Since Ale,- rl A ~ + I=, 8~for

Therefore, p(Ao,,) = 0. Similarly we have p(Ao,-,) = 0. Let {u(")} be a countable basis of the topology of X and define as above for ~ ( " 1 . Obviously p ( H - ) = 0, where H - = UnUi$, for all

u:$

p E M f ( X ) . To see x E a ( x ) when x E X\H- fix u:") with x E Uin). If x $! u?) for all m 2 1,x E UAn)\ UZl = u:% c H - , which contradicts x E X\H-. Thus x E u$) = fm(Uin)) for some m 2 1, and so f-"(2) E Uin),i.e. x E a ( x ) . Similarly we have p(H+) = 0, where H+ = u$_, for all p E M f ( X ) and therefore x E w ( x ) whenever x E X\H+. Therefore, X\(H- U H + ) c R ( f ). If we establish X\(H- U H+) = R ( f ) ,then R( f ) is a Borel set and p ( R ( f ) )= 1 for all p E M f ( X ) . Let x E R ( f ) , then there is ~ ( j with ) x E ~ ( 3 ) . Since f n i ( x ) E ~ ( j for an increasing sequence ni, we have x E f-"i(U(j)) for all i 2 0 and so x E U , f-"(U(j)). Thus x $4 U(j)\ U n f - " ( ~ ( j ) ) = u:!!,. This is true for all U ( j )with x E U ( j )and therefore x $! U , u,$:?, = H-. In the similar way we have x $! H+. Therefore, x E X\(H- U H + ) .

UP)

UE,

Let f : X -+ X be a homeomorphism of a compact metric space. A point x E X is said to be quasi generic for the measure p E M ( X ) if there exists an increasing sequence of integers Nk > 0 such that

)

CHAPTER 11

384

for all

< E C ( X ) . A point x E X

is said to be generic for p E M ( X ) if

<

for all E C ( X ) . It is clear that a measure p belonging to M ( X ) has a quasigeneric point then p is f-invariant. Every point of X is quasigeneric for some Borel probability measure, but not necessarily generic. A point x E X is said to quasi regular with respect to f if it is generic with respect to some Borel probability measure. We denote as Q ( f ) the set of all quasiregular points.

Remark 11.4.14. A point x E X belongs to Q ( f ) if and only if <*(x)= < ( f j ( x ) )exists for all ( E C ( X ) . In this case, the linear limn+m functional H <*(x)on C ( X ) corresponds to a measure that is denoted by pz. Thus the point x is generic for p,. In general the map x -, p, from Q ( f ) into M ( X ) is neither injective nor surjective.

zj": <

Theorem 11.4.15. The set Q ( f ) is a Borel set and p ( Q ( f ) ) = 1 for all P E M f( X I . Proof. For ( E C ( X ) and N > 0 define I N = & { x E X : limN+m t N ( x )exists ). Then we have

z,"=i1 < and put A ( ( ) = ofj

A ( ( ) = { x E X : { t N ( x ) )is a Cauchy sequence)

i}

is open, A(<)is a Borel set. By Birkhoff's Since { x E X : I t N ( x )-cM(x)I < ergodic theorem we have p(A(<))= 1 for all p E M ( X ) . Let {<,) be a dense A(<,). Thus Q ( f ) is a sequence in C ( X ) . Then Q ( f ) coincides with Borel set and p ( Q ( f ) )= 1 for all p E M ( X ) .

nr=l

Let f: X -r X be a homeomorphism of a compact metric space. We denote as G, the set of all points which are generic for p E M ( X ) .

Theorem 11.4.16. The set G , is a Borel set. Proof. The proof is similar to that of theorem 11.4.15. Theorem 11.4.17. If p E M I ( X ) is ergodic, then p-almost all points of X are generic for p, i.e. p(G,) = 1. Proof. Let {<,} be a dense sequence of C ( X ) . We denote as G(<,) the set of points x satisfying

$11.4 Probability measures of compact metric spaces

385

If p is ergodic, then we have p(G((,)) = 1 by the ergodic theorem. Obviously G, = G((,,) and therefore p(G,) = 1.

n,

Theorem 11.4.18. If p

E M f ( X ) is non-ergodic, then p(G,) = 0.

Proof. This follows from the next theorem.

We define the set E ( f )=

U{G,:p is ergodic in M f ( X ) } .

Clearly x E E( f ) if and only if x E Q ( f ) and p, is ergodic.

Theorem 11.4.19. The set E ( f ) is a Borel set and p ( E ( f ) ) = 1 for all 11 E M f ( X ) .

Proof. For ( E C ( X ) we define tN= &

xZi1(

E ( ( ) = { x E Q ( f ): lim t N ( y )= N+oo

I

o

f for N

> 0 and put

€dpZfor pz-a.e. Y ) .

For x E Q( f ) we have

and therefore

J I l i m t N ( y )- l i m € N ( x ) 1 2 d p x (=~0). )

E(O = { x E ~ ( f : )

That E ( ( ) is a Borel set follows from the following calculation ;

1

= limlim lirn N

M K-00

K

I t N ( f j y ) - (M(x)12 p.-a.e. j=o

Let p E M ( X ) . Then we claim that p ( E ( ( ) )= 1, i.e.

( lim t N ( y )- lim (M(x)12dpz(y)= 0 N+oo

M+w

y.

CHAPTER 11

386

holds for p-a.e.x in Q(f). Since p(Q(f)) = 1, (11.1)

J{JI

lim (N(y) - lim ~M(x)12dpz(y))dp(x) = 0.

N+w

M+w

Let 11 11 and < , > be the norm and the scalar product in L2(p) respectively. Since p is f -invariant, we have

By von Neuman's ergodic theorem

Since (tN)* = t*,

Now (11.1) is equivalent to

x IItN

K-1

limlimlim M K K j=o

o

f i - tMI) =0

and thus to l i m l~i m /ItN ~ - tMII= 0 by von Neuman's ergodic theorem. Let {tn) be a dense sequence in C(X). Clearly E ( f ) = E((,), and therefore E( f ) is a Bore1 set and p(E(f)) = 1.

nr=l

Let f : X -+ X be a homeomorphism of a compact metric space. Theorem 11.4.19 tells us that if v E M f ( X ) then v-almost every point y E X is quasiregular and generic for an ergodic measure p,, i.e.

511.5 Applications to topological dynamics

for all ( E C(X). This means, by ergodic theorem, that

for all ( E C(X). For all y E X such that p, is ergodic, the set

is an invariant Borel set (f(r,) = r,) and p,(I',) = 1. The set r, is said to be the ergodic fiber of y. Denote as 23, the restriction of t3 to r,. Then we obtain a probability space (r,, B,,p,) such that fir#: I?, + I?, is ergodic and measure preserving. Obviously we have the following remark.

Remark 11.4.20. Let f : X + X be a homeomorphism of a compact metric space. If v E M (X), for all ( E C(X) (1) ( is p,-integrable for v-a.e.y, (2) the function y + J (dp, is Bore1 measurable, (3) J tdv = J ( J t d ~ u ) d v ( ~ ) , (4) Limn,, ((fjy) = J (dp, v-a.e..

~jn=i

$11.5 Applications to topological dynamics Let f : X + X be a homeomorphism of a compact metric space. First we investigate what the homeomorphism means in topological situation when f is ergodic with respect to a Borel probability measure. Theorem 11.5.1. Let f : X -t X be a homeomorphism of a compact metric space and let p be a f -invariant Borel probability measure on X . If p has positive measures for all nonempty open sets and f is ergodic with respect to p, then p({x E X : cl(Of(x)) = X)) = 1.

..

Proof. Let {Ul, U2,. ) be a basis of the topology of X and define En = " fm(X\Un) for n 2 1. Then f (En) = En and each En is a Borel set. By Remark 2.2.1 we have

nm=-,

Since each Un is open, p(Un) > 0 by the assumption and so p(E:) ergodicity ensures that p({x E X : cl(Of (x)) = X ) ) = 1.

> 0. The

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388

Remark 11.5.2. Let R ( f ) be the nonwandering set of a homeomorphism f of a compact metric space X. If f has POTP, then there exists a probability measure p E M ( X ) such that p has positive measures for all nonempty open sets. Let {Ul ,U 2 , . ) be a basis of the relative topology of R( f ). For U j let V be a closed subset with the interior points such that U, C V . For E such that V > U c ( x ) where U c ( x ) = { y E R ( f ) : d ( x ,y) < E ) , take a sufficiently small 6(0 < 6 < E ) satisfying the property in the definition of POTP. Since R ( f ) is a nonwandering set, there exist z E R ( f ) and no > 0 such that z , f n o ( z ) E U6/2(~). Now, define an no-periodic 6-pseudo orbit { z , ) by Znok+i = f i ( z ) for k E Z and 0 5 i 5 no - 1. Since f l n ( f ): R ( f ) + R ( f ) has POTP, there is y E R ( f ) satisfying d ( f j ( y ) ,z j ) < E for all j E Z. For all m 2 1 define

..

where 6, denotes a Dirac measure, i.e. for a Borel set B

Then each pm is a Bore1 probability measure on R ( f ) such that p m ( V ) 2 p m ( U c ( ~ )2) 1/2no. Since M ( X ) is compact by Theorem 11.4.7, a suitable subsequence { p m i )converges to a Borel probability measure p ( j ) . By Theorem 11.4.5 1 /.&J)(v)lim sup pni ( V ) 2 i-a, 2n0

>

and so p ( j ) ( u j )2 p ( j ) ( ~>) 0 . Next we show that p(j) is f-invariant. For any Borel set B we have

and similarly

511.5 Applications to topological dynamics

389

Let ( E C(R(f)). If ( 2 0 then ( is uniformly approximated by an increasing sequence of simple functions

where

Thus we have .-

((f (x)) = n-+m em

C ai~f-l(,,~)(x)

(uniformly),

i=O n2"

((f -'(x)) = 1

n-rm

1 f B , i ( x ) (uniformly). i=O

Since 0 5 ( ( 2 ) - (,(x) 5 1/2n (x E (11.2) and (11.3)

Fi n and take the limit

n(f)) for sufficiently large n, by using

of the above inequality. Then

S

S

S

Since n is arbitrary, we have ( o f dp(j) = (dp(j). Similarly, ( o f -Id@) = tdp(j). These two equality hold for general continuous function ( since it is expressed as ((x) = (+(x) - 5-(x) where (+(x) = max{((z), 0)

S

and (-(x) = ma{-((x),O). Therefore we have ~ ( j ) (-'(B)) f = p ( j ) ( ~= ) p(j)(f (B)) for all Borel sets B. The measure ~ ( jis) depend on the open set Uj, and so we put m

for Borel sets B. Then p is f-invariant and p(Uj) > O for all Uj.

390

CHAPTER 11

Theorem 11.5.3. Let f : X -r X be a homeomorphism of a compact metric space. Suppose X is connected and not one point. Iff is minimal, then f does not have POTP. Proof. Let L be the diameter of X , i.e. L = max{d(x, y) : x, y E X ) and put E = 113. I f f : X + X has POTP, by the following lemma, for x E X there exist y E X and k > 0 such that cl(Ofh (y)) C U.(x). Thus, X = cl(Of. (fj(y))). From connectedness and minimality we have cl(Ofr (y)) = X. For instance, if k = 3, then Aj = cl(of3 (fj(y))), j = 0,1,2, are not pairwise disjoint because X is connected. Thus A. n A1 # 0, from which we take a point z. Then z, f-'(z) E A. and so we have Bo U f2(Bo) = X where BO= cl(Ofs(z)). Since Bo n f2(Bo) # 0, we take w E Bo n f2(Bo) and then w, f-2(w) E Bo. Since Or.(w) c [Bo n f2(Bo)I U {f-'(~),f-~(w),f-~(w),-.-) C [Bon f2(Bo)l U f 2(BO)= f 2(BO),we have cl(Ofs(w)) = X. Since w E Bo = cl(Of3(z)) C A. = cl(Of 3 (y)), consequently cl(Ofs (Y)) = X. But l 2.5 since Ofa(y) c Uc(x), thus contradicting.

:u::

<

Lemma 11.5.4. If a homeomorphism f has POTP, given E > 0 and x E R(f) there ezist y E X and k = k ( x , ~ )> 0 such that cl(Ofh(y)) C Uc(x). Proof. For ~ / > 2 0 let 6 > 0 be a number satisfying the definition of POTP. Take sufficiently small 6 such that 0 < 6 < E. Since x E R(f), there exist z and k > 0 such that z , fk(z) 6 U6/2(x). Construct a 12-periodic 6-pseudo orbit { z i : i E Z) by z,k+i = fi(z), n E Z and 0 5 i 5 k. Then d(f i(y), zi) < ~ / 3i, E Z for some point y E X . Since d(f nk(y),znk) < &/3, we have d(f nk(y),x) < 5 ~ / 6for n E Z, i.e. cl(Ofh (y)) C Uc(x). Let f : X + X be a homeomorphism of a compact metric space. The homeomorphism f is said to be distal if for x, y E X

then x = y. Obviously an isometry is distal. Let A be a subset of Z. Then A is syndetic if there is a finite set K of Z with Z = K A. Let x E X , then x is an almost periodic point if { n E Z : fn(x) E U) is a syndetic set for all neighborhood U of x.

+

Theorem 11.5.5 (Aoki[Ao8]). Let f : X + X be a homeomorphism of a compact metric space. Suppose X is not one point. I f f is distal, then f does not have POTP. Proof. Suppose f has POTP and let diam(X) = L. Let E = L/9. By Lemma 11.5.4, for yo E R ( f ) there exist y E X and k > 0 with cl(Ofh(y)) c Uc(yo). We write g = f k for simplicity. Note that g: X + X has POTP and is distal. Let 0 < 6 < E be a number in the definition of P O T P for g. Since X is

511.5 Applications to topological dynamics

391

compact and connected, there is a finite sequence pl = y,p2,. . , p such ~ that d(pi,pi+l) < 6/2(1 5 i 5 N - 1) and X = Ua(pi). By the next Theorem 11.5.7 each point x E X is almost periodic. Thus there exist c(i), 1 5 i N , with d(pi,gc(i)(pj)) < 612, and so we define a sequence {xi : i E Z) by

,u:

xi = gi(y)

if i

<

< 0,

ifOSi
if 0 2 i 5 ~ ( 2 ) 1,

Obviously {xi) is a 6-pseudo orbit and is &dense in X. Since g has POTP, there is z E X with d(gi(z), xi) < E for all i E Z, and so for w E X there is w' Og(z) with d(w, w') < 2 ~ For . simplicity write

Then we have

Thus

Letting 0; = {gi(z) : i 5 0) and Of = {gi(t) : i 1 01, we have g(0;) c U,(C~(O,(~))) and gc(O$) C U,(cl(O,(y))). Since cl(igi(z) : i E Z)) = 0; U Of, by Baire theorem one of O;, Of has interior points in 0, = ~ l ( { ~ ~:( z ) i E Z)). Since g: 0, + 0, is a homeomorphism, there is a g-invariant Bore1 probability measure p on 0,.Since 0, is a minimal set for g by the next

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Theorem 11.5.8, we have that p(U) > 0 for any nonempty open set U. This follows from the facts that Urmgi(U) = 0, and p(0,) 5 Crrnp(U). If 0; contains interior points of 0,, then 0, = g(0,). Indeed, suppose g-l(O;) # 0;. For the case when V = r)k,,lg - k ( ~ ; ) has interior points of 0, we have g-j(z) E V for some j 2 0. Thus 0; C V since g ( V ) C V . Therefore, g-l(O;) = O;, a contradiction. For the case when V has no interior points we have that p(O;\V) > 0 since O;\V is nonempty and open. Since 0; = Uk,,O g - k ( ~ )U V , we have p ( W ) > 0, Put W = O;\g-'(0;). thus contradicting ~ ( 0 ;5 )1. If O$ contains interior points of 0,, we have also g ( 0 f ) = O f . In any case, 0, = O f or 0, = 0;. Thus 0, c Uc(cl(Og(y))). Since cl(Og(y))C Uc(yo),

Since { x i ) is &dense in X and d(&a), xi)

< e(i E Z ) , we have

and thus X = UaC(O,)= U4,(yo). Therefore, diam(X) = I dicting.

< 8 ~thus , contra-

A homeomorphism f : X -t X of a compact metric space is said to be uniformly almost periodic if for E > 0 there is a syndetic set A such that f n ( x ) E U,(x) for all x E X and n E A. Theorem 11.5.6 (Ellis[El]). I f f : X f is distal.

-t

X is uniformly almost periodic, then

Proof. We first prove that if f is uniformly almost periodic then { f n : n E Z) is equi-continuous. Let A be a syndetic set such that f n ( x ) E UclS(x)for all x E X and n E A. Then Z = A K for some finite set K = { n l , . . . ,ne). Take 0 < 6 < €13 such that

+

d ( x ,y) < 6

* max{d(f"' (x),f "' ( y ) ): 1 5 i 5 1 ) < -. 3

Then, for n E A and m E K we have that if d(x,y)

€

< 6 then

and so { f n : n E Z ) is equi-continuous. Suppose i n f { d ( f n ( x )f,n ( y ) ) : n E Z ) = 0. For j > 0 there is n j E Z with d ( f n j ( x ) f, n j ( y ) ) < llj. Thus d ( f n j ( x ) f, n j ( y ) ) < 6 for all j large than a certain positive integer J. Thus we have

$11.5 Applications to topological dynamics

from which d ( x , y) x=y.O

393

< E . Since E is arbitrary, we have d(x, y)

= 0 and then

Let g: X -t X be a continuous map of a compact metric space. The collection i g ( x ): x E X ) will be considered to be a point of the product topological X , where X , = X for x E X . Obviously xXis a comspace xX = pact Hausdorff space. Let f : X -+ X be a homeomorphism. The closure of { f n : n E Z) in xXis said to be an Ellis group and denotes it as E ( X ) . A nonempty subset I of E ( X ) is called a right ideal if g o h E I for all g E E ( X ) and all h E I .

nzEx

Theorem 11.5.7 (Ellis[El]). If a homeomorphism f : X each x E X is an almost periodic point.

-t

X is distal, then

Proof. Before starting with the proof we prepare claims. Claim 1. Let I C E ( X ) and I # 0. If I is a right ideal of E ( X ) , then there is u E I such that u2 = u. Indeed, a map Rg : E ( X ) -+ E ( X ) defined by R g ( h )= h o g is continuous. This follows from the fact that if a discrete sequence { h a ) of E ( X ) converges to h E E ( X ) , then h,(gx) = h(gx) for all x E X and thus R g ( h a )-t Rg(h). Let M denote the family of closed nonempty subsets S c I with S o S c S . Then M is the partially ordered set under the inclusion and is inductive. Thus a minimal element S exists in M by Zorn's lemma. If g E S , then Sog = R g ( S )# 0 and (Sog)o(Sog)c SoSoSog C Sog. Since S is minimal, we have Sog = S and then there is h E S with hog = g. Obviously R;l ( S ) n S # 0. Since flog = frog = gfor f l , f 2 E R y l ( S ) n S ,wehave ( f l o f 2 ) o g= flog = g and so f 1 0 f 2 E R i l ( g ) n S . Therefore, ( R ; l ( g ) n S ) ~ ( R ; l ( ~ ) ncS )R T ' ( ~ ) ~ S . Minimality ensures that R Y ~ (n~S )= S. This shows that g2 = g. Claim 2. Let M be a closed subset of E ( X ) . If f n o g E M for g E M and n E Z, then M = E ( X ) . Indeed, E ( X ) o M C M , by Claim 1 there is g E M with g 2 = g. Put y = g(x). Then g(x) = g(y). Since g E E ( X ) , we have that f n -t g for some sequence { f " ) and

Since g(x) = g(y), obviously d ( f n ( x ) ,f n ( y ) ) -+ 0 as n -+ m. Thus inf{d(fm( x ) , fm(y)) : m E Z) = 0. Since f : X -+ X is distal, we have x = y and then g is the identity since x is arbitrary. Therefore, f n 0 id = f E M for all n E Z. This shows that M = E ( X ) . Now we are ready to prove the theorem. Note that the identity id is contained in E ( X ) . Let U be an open neighborhood of id in E ( X ) . We first show that there is a syndetic set A C Z such that f n E U for all n E A. Fix h E E ( X ) and define a map Lh : E ( X ) -+ E ( X ) by Lh(g) = h o g . It is easily checked that if h is continuous then so is Lh. Thus Lfn is continuous for n E Z, and Lfn o L f -n = L f -n o Lfn = id, from which Lfn is injective

CHAPTER 11

394

and bicontinuous. Thus Lfn(U) = f o U is open in E ( X ) and so W = E(X)\(U:f o U) is closed in E(X). IfWf0,thenwehave f n o g € W f o r g ~ W a n d n ~ Z T. h u s W = E ( X ) by Claim 2, which contradicts W # E(X). Therefore, W = 0, from which E(X) c U a: f n o U. Since E ( X ) is compact, we have E ( X ) C fni o U for some finite set K = {nl , ,nt). Let A = {n E Z: f E. U). Then there is ni E K with f n E f n' o U for all n E Z. Thus, f "-"'E U, i.e. n - n; E A and so n E A K. Since n is arbitrary, we have Z = K A and then A is a syndetic set. To show that each x E X is an almost periodic point define a continuous map 0, : E ( X ) + X by 02(g) = g(x). Let V be a neighborhood of x in X . Then there is a neighborhood U of id in E ( X ) such that O,(U) c V. Replacing this U by the above U, for n E A

u:=~

-

+

+

fn(,) = f n

0

Ox(id) = 0 2 ( f n o id) = 0 2 ( f n ) E Ox(U) c

and therefore x is an almost periodic point of f .

Theorem 11.5.8. Iff: X is a minimal set.

-+

X is distal, for each x E X , the closure of Of (x)

Proof. By Theorem 11.5.7 each x E X is an almost periodic point. For U an open neighborhood of x define A = {n E Z : fn(x) E U). Thus Z is expressed as Z = A K for some finite set K = I n l , ,nk). Thus we have

+

..

from which cl(Of (y)) n cl(U) # 0 for all y E cl(Of ( x ) ) . Here the integer no implies no = 0. This follows from the facts that f -"'(y) E cl({ f n(x) : n E A)) c cl(U) since y E cl{fn+"i(x) : n E A) for some integer ni. Since U is arbitrary, we have x E c l ( 0 (y)) and so cl(Of (x)) = cl(Of (y)), i.e. cl(Of (x)) is a minimal set. Let f : X -+ X be a homeomorphism of a compact metric space. The homeomorphism f is said to be semisimple if there exists a sequence {En)of closed subsets such that f (E,) = E a , X = U Ea and each Ea is a minimal set. Obviously, E, n Ep = 0 if a # P.

Theorem 11.5.9. If f : X + X is distal, then f is semisimple. Proof. {cl(O (x)) : x E X )is a cover of X. By Theorem 11.5.8, each cl(Of (x)) is minimal. Thus we have the conclusion. Let f : X -t X be a homeomorphism of a compact metric space. The homeomorphism f is said to be uniquely ergodic if M (X) is one point set. Obviously uniquely ergodicity implies ergodicity.

$11.5 Applications to topological dynamics

395

Theorem 11.5.10. Let f : X -, X be a uniquely ergodic homeomorphism. Then f is minimal if and only i f the f -invariant Borel probability measure p has positive measure for all nonempty open sets. Proof. + ) : Suppose f is minimal and p(U) = 0 for some nonempty open set U. Since X = Ura fn(U), we have p(X) = 0, thus contradicting. -+ ) : Suppose f is not minimal, i.e. f ( F ) = F # X for some closed set F # 0. By Krylov-Bogolioubov theorem, flF-invariant Borel probability measure p~ exists. Define pt(B) = pF(F n B) for Borel sets B of X. Then pt is f-invariant Borel probability measure and pl(X\F) = 0 holds. Thus p # pt. Since f is uniquely ergodic, we have a contradiction. 0

Theorem 11.5.11 (Oxtoby[Ox]). Let f : X -,X be a homeomorphism of a compact metric space. Then the following conditions are equivalent : (1) f is uniquely ergodic. (2) For each ( E C ( X ) the sequence ( o f j converges uniformly on X to a constant. ( 3 ) For any x E X there exists p E M f ( X ) such that x is generic with respect to p.

k x;i1

Proof. (1) =$ (2) : Suppose that (2) is false, i.e. there exist ( E C ( X ) and > 0 such that for any N > 0 there are n N and xn E X so that

>

E

By (I), p is only one f-invariant. Fix the point xn and define a linear map Jn: C(X) + R by n-1

By Riesz Representation Theorem

for some pn E M ( X ) . Since M ( X ) is compact, there is a subsequence {pni) and v E M ( X ) such that

For any q E C ( X )

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396

Thus we have

I)

I)

-'

from which v o f = v by Theorem 11.4.4. Similarly, v o f = v and therefore v E M ( X ) . However,

which shows that p # v . This is a contradiction. ( 2 ) + ( 3 ) : Since the constant referred to the statement of ( 2 ) is J t d p for some p E M ( X ) , ( 3 ) is concluded.

Figure 43

( 3 ) + ( 1 ) : By the assumption, for a: E X there is p E

M f( X ) such that

1 n-l lim - E [ ( f i x ) = n

n-oo

for all

i=O

< E C ( X ) . Thus, for any v E M f ( X )

By Lebesgue Convergence Theorem we have

for all [ E C ( X ) , and therefore v = p. This shows that set.

M

( X ) is one point

$11.5 Applications to topological dynamics

397

Let f : X -t X be a homeomorphiim of a compact metric space. The homeomorphism f is said to have specification if for any E > 0 there exists M = M ( E )> 0 such that for any finite sequence of points

and for j with 2

< j < It, choosing any sequence of integers < <

such that aj - bj-1 2 M ( 2 j k ) and an integer p with p 2 M there exists a point x E X with fP(x) = x SO that

<

+ (bk - a l ) ,

k. for aj 5 i 5 bj and 1 5 j If a point x E X is a periodic point with period p, then p, = 6f.(z) is an f -invariant Bore1 probability measure and f is ergodic with respect to p,. We denote a s E(f) the set of all measures p, constructed as above. Obviously every element of E ( f ) is an extremal point in M j ( X ) by Theorem 11.4.11.

c::

Theorem 11.5.12 (Sigmund[Sig2]). Let f : X -t X be a homeomorphism of a compact metric space. I f f is a topologically mixing TA-homeomorphism, then E( f ) is dense in M j ( X ) . Proof. Let p E M ( X ) be given. Any neighborhood of p contains a set of the form

~ ( p=){ V E M ~ ( x:)1

J ~ -c J ( d u l <

for all

c E F)

where F is some finite subset of C ( X ) . We may assume II(II 5 1 for all ( E F. Thus we must show that W ( p ) contains an element of E ( f ) . If N is large enough, for ( E F and x E Q ( f ) (the set of quasiregular points) we have

where ['(x) is the time average of <. By Birkhoff Ergodic Theorem we have / ( d p = /(*d,u. Since p ( Q ( f ) ) = 1, this implies

Since t* is bounded and F is finite,

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398

+

is a positive number. For I; E F and j = 1 , 2 , . - - ,[8A/e] 1 ([ ] denotes the Gauss symbol), define = {Qi(I;), ' ' ' ,Q[BA/~]+I ( I ; ) ) a partition of Q(f) into Borel sets such that

Since F is finite, {nA€ : At E ac for I; E F ) = {QI, .. . ,Qs) is a Borel finite partition of Q( f ). For x j E Q j we have

and so

From this inequality and (11.4)

for sufficiently large N. For any 6 > 0 choose a natural number m with l l m ,nk such that nl,

.

and by taking n j or n j

+ 1 as m j

Thus, from (11.5)

for sufficiently small 6 > 0.

< 6.

Then there exist

$11.5 Applications to topological dynamics

Now we have

and so define a sequence {un : 1 5 n sequence runs

5 mN} of points in X as follows. The

... ,f N-l

ml times through {XI,f (XI),

(31I),

next mz times through

...

(22,

f (xz), ... ,f N-1(~2)},

finally mk times through {xk, f (xk),

.. ,f N-l(~k)},

Then we have

for all ( E F. Since F is finite and each [ E F is continuous, there exists 5

> 0 such that

If there exists a periodic point z with period m N such that

for 1 5 n 5 m N , then we have

Unfortunately the existence of such periodic points is not ensured. But by specification property there exists y E X which almost behaves that way. Since f is a topologically mixing TA-homeomorphism, by the following Theorem 11.5.13, f has specification. Thus we can use this property to obtain the conclusion.

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400

Indeed, let M = M ( 6 ) be a number in the definition of specification. Here 6 is as above. Define

Define yt = x j if mi-1 5 t

< m j ( l 5 j 5 12).

By specification property there exists y E X with period p = m ( N such that

. ,fp-'(y))

The sequence { y , f ( y ) , CP:~ 6 p c , l . Thus

;

+ M - 1)

defines an f -invariant Bore1 measure p, =

Let E = Z n ~ z ~ ' [bi]. a Then ~ , E has m N elements and just as in (11.6) we have

x;='=( (,f i y )is just the sum CiEE t(f i Y ) ,plus m M - 1 extra terms. Since ))t)) 5 1, the two sums therefore differ by at most m M - 1. This implies

But

If N is sufficiently large, then we have 2MIN

for all ( E F, i.e. p, E W ( p ) .

< €14. Thus

511.5 Applications to topological dynamics

401

Theorem 11.5.13. Let f : X + X be a homeomorphism of a compact metric space. If f is a topologically mixing TA-homeomorphism, then f has specification. Proof. Since f is a TA-homeomorphism, f is expansive and has POTP. Let c > 0 be an expansive constant. For E with 0 < E < c/2 let 6 > 0 be a number in the definition of POTP. Let u = {Ui) be a finite open cover of X such that each U; has the diameter < 612. Since f is topologically mixing, for Ui, U j E u there is Mi,, > 0 such that for all n Mi,,

>

Let M = m a x { M i f ) and take any finite points

and any integers

a1 5 bl
> M for 2 5 j

Define

ak+l = bk+l = p

< ... < a & 5 bk

5 k, and an integer

+ a l , xh+l

= fa'-aL+'

( 3 1 ).

For any z E X we denote as U ( z ) an open ball U in u satisfying z E U . Since aj+l - bj M , by topologically mixing we have

>

from which there is yj E u ( f b j x j )such that

Define a sequence { z ; ) in X by

< bj), zi = fi-bj(yj) (bj 5 i < aj+l), zi = f i ( x j )

(aj 5 i

zi+p = zi

(i E Z).

Then { z ; ) is a 6-pseudo orbit. Since f has POTP, there is x E X such that d ( f i ( x ) , x i ) < E for all i E Z. Since z ; + ~= zi for i E Z, we have d ( f i + ~ ( xzi) ) , < E for i E Zand so d ( f i of p ( x ) , f a ( x ) )< 2~ < c. By expansivity we have fP(x) = x. Therefore f has specification.

REFERENCES

[A] R.Anderson, On rasing flows and mappings, Bull. Amer. Math. Soc. 69 (1963), 259-264. [Anl] D.V.Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967), 1-235. [An21 D.V.Anosov, I.U.Bronshtein, S.Kh.Aranson and V.Z.Grines, Smooth Dynamical Systems, Dynamical Systems I, Encycl. Math. Sciences 1, Springer-Verlag, 1988, 149-233. [Aol] N.Aoki, Topological stability of solenoidal automorphisms, Nagoya Math. J. 90 (1983), 119-135 (correction to: Topological stability of solenoidal automorphisms, ibid 95 (1984), 103). [A021 N.Aoki, Expanding maps of solenoids, Mh. Math. 105 (1988), 1-34. [A031 N.Aoki, Topological Dynamics, Topics in General Topology (K. Morita and J. Nagata, eds.), North-Holland, Amsterdam, 1989, 625-740. [A041 N.Aoki, On homeomorphisms with pseudo orbit tracing property, Tokyo J. Math. 6 (1983), 329-334. [A051 N.Aoki, The set of Axiom A diffeomorphisms with no cycles, Bol. Soc. Bras. Mat. 23 (1992), 21-65. [A061 N.Aoki, A simple proof of the Bernoullicity of ergodic automorphisms on compact abelian groups, Israel J. Math. 38 (1981), 189-198. [A071 N.Aoki, Dense orbits of automorphisms and compactness of groups, Topology and its Appl. 20 (1985), 1-15. [A081 N.Aoki, Homeomorphisms without the pseudo-orbit tracing property, Nagoya Math. J. 88 (1982), 155-160. [A-D] N.Aoki and M.Dateyama, The relationship between algebraic numbers and expansiveness of automorphisms on compact abelian groups, Fund. Math. 117 (1983), 21-35. [A-D-K] N.Aoki, M.Dateyama and M.Komuro, Solenoidal automorphisms with specifications, Mh. Math. 93 (1982), 79-110. [A-H] N.Aoki and K.Hiraide, The linearization of positively expansive maps of tori, The Theory of Dynamical Systems and its Applications to Nonlinear Problems, World Sci. Publ., Singapore, 1984, 27-31. [A-A] V.I.Arnold and A.Avez, Ergodic Problems in Classical Mechanics, Benjamin, 1968. [B-S] J.Banks, J.Brooks, G.Cains, G.Davis and P.Stacey, On Devaney's definition of chaos, Amer. Math. Month. 99, April (1992), 332-334. [Bill P.Billingsley, Ergodic Theory and Information, John Wiley and Sons, 1960. [Bin] R.Bing, The cartesian product of a certain nonmanifold and a line is E4, Ann. of Math. 70 (1959), 399-412.

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INDEX

abelian group 15 absolutely continuous 13 act 183 Alexander-Whitney map 317 algebra 360 almost periodic 390 a-limit set 53 a-related 96 annihilator 222 Anosov diffeomorphism 6, 19 Anosov differentiable map 7, 21 arc 33 arcwise connected 33 atlas 50 automorphism 16 Axiom A 8

base space 193 basic set 8, 9, 102, 107, 120 Bernoulli measure 12 Bernoulli shift 12, 13 Birkhoff center 107, 122 Birkhoff ergodic theorem 366 Borel class 361 Borel measure 361 Borel set 361 Borel's theorem 367 boundary 50, 230 boundary homomorphism 310 boundary operator 304 branch set 230 Brouwer theorem 51 bundle projection 193

canonical coordinate 214

canonical neighborhood 149, 332 Cantor set 65 Cantor subinterval with r a n k r 66 centralizer 293 c-expansive 7, 22, 57 chain complex 304 chain homotopic 305 chain homotopy 305 chain homotopy equivalence 305 chain homotopy inverse 305 chain map 304 chain recurrent 9, 96 character 222 character group 222 chart 50 classical solenoid 223 closed 50 closed path 170 closed smooth manifold 19 closed topological manifold 28 compact covering domain 196 connected metric 190 connecting homomorphism 306 continuous family of semi-conjugacy maps 292 contractible 313 converging semi-orbit 53 coordinate domain 214 coordinate function 193 coordinate neighborhood 50 cover 16 covering degree 177 covering map 10, 31, 149 covering space 149 covering theorem 179 covering transformation 183 covering transformation group 183

INDEX

cross product 318 curve 1 cycle 105, 304

6-pseudo orbit 8, 22, 78, 81 diagonal set 41 diffeomorphism 1 differentiable 1 differentiable dynamical system 3 differentiable manifold 1 differentiable map 1 dimension of 3 337 distal 390

Eilenberg-Maclane map 317 Eilenberg's constant 31 elementary set 102, 120 EIIis group 393 endomorphism 16 €-chain 148 €-chain of s-direction 147 &-traced 8, 22, 78 €-tracing point 81 equivalent 13, 177 equivalence set 96 ergodic 12, 17, 364 ergodic fiber 387 ergodic measure 12 euclidean neighborhood retract (ENR) 218, 321 Euler characteristic 325 evenly cover 149 exact sequence 305 excision isomorphism theorem 316 excision theorem 315 excisive couple 314

existence theorem 180 expanding 7, 10, 16, 20, 50 expanding factor 163 expanding tord endomorphism 16 expansion 231 expansive 7, 22, 36 expansive constant 22, 36, 40, 57 exponential topology 97 extremal point 13, 381

face 324 factor 13, 25, 90 factor group 15 fiber 193 fiber bundle 193 filtration lemma 112 finite type 86 finitely generated 227, 324 finitely generated under 2 227 five lemma 307 fixed point 2 fixed point index 326, 329 fixed point index theorem 346, 348 flow 1 foliation 21 1 fundamental domain 108 fundamental group 171, 172 fundamental homology class 326, 332,334

y-dense 133 generalized foliation 21 1 generator 37 generic 384 group 15

INDEX

Harr measure 362 Hausdorff metric 98 homeomorphism 1 homology class 304 homology exact sequence 312 homology group 304, 311 homomorphism 173 homotopic 168, 173, 293, 312 homotopy 168, 173, 293, 312 homotopy class 288 homotopy equivalence 174, 313 homotopy group 216 homotopy inverse 174 homotopy lifting property 174 hyperbolic 5, 16 hyperbolic metric 47 hyperbolic set 6 hyperbolic toral automorphism 16 hyperbolic toral endomorphism 16

identity 222 identifying space 170 immersion 230 infra-nil-endomorphism 29 infra-nil-manifold 29 integrable 362 intersection number 349 intrinsic topology 153, 198 invariant 11, 381 inverse limit space 56 inverse path 169, 293 invertible measure-preserving transformation 363 isolated 101 isomorphic 13 isomorphism 304 isotopy 288 isotopy class 288

Kolmogrov extension theorem 360 Krylov-Bogolioubov's theorem 380 Kiinneth formula 310, 319 Kupka-Smale 28

leaf 211 leaf topology 213 Lebesgue dominated convergence theorem 362 Lebesgue number 32 Lefschetz fixed point formula 331 Lefschetz number 331 lift 16, 149, 187, 215 lifting of local product structure 205 linearly independent 223, 324 local coordinate 50, 211 local coordinate system 50 local homeomorphism 31 local product structure 127, 155 local stable set 46 local unstable set 46 locally arcwise connected 34 locally closed set 320 locally compact 320 locally connected 34 locally contractible 180 locally finite 194 locally isometric covering map 61 locally path connected 34 locally uncountable 88 logistic map 4 lower semi-continuous 107, 289 LP-ergodic theorem 371 Lyapunov function 24

INDEX

mapping degree 329, 334 Markov partition 132 Markov subshift 3, 87 maximal ergodic theorem 368 Mayer-Vietoris exact sequence 314 measurable 361, 363 measurable rectangle 361 measurable space 360 measure preserving 363 measure preserving transformation 11, 17 measure space 11, 360 measure theoretically conjugate 13 minimal 13, 25, 54, 78 minimal set 55, 78

33 connected 33 lifting property 149 of homeomorphisms 288, 291 PEM 166 periodic 9 periodic point 2, 88 Poincar6 recurrence theorem 364 polyhedron 324 positively expansive 7, 22, 40 probability space 11, 360 proper fundamental domain 108 proper rectangle 132 properly discontinuous 184 pseudo orbit tracing property (POTP) 8, 22, 78

n-dimensional topological manifold 50 n-torus 15 no cycles 9, 108 neighborhood retract 320 nonwandering point 8, 92, 117 nonwandering set 8, 92, 117 normalizer 185

quasi generic 383 quasi regular 384 quasi-Anosov diffeomorphism 28 quotient complex 308 quotient topology 15

w-limit set orbit 2 orbit space order 86 orientable orientation

53 183 333, 345 333, 345

paracompact 194 parameters 4

path path path path

'

rank 223 rectangle 129 recurrent 13, 107 refinement 38 regular 375 regular map 6 regular point 6 related 96 retract 320 retraction 320 Riesz representation theorem 379 right ideal 393

INDEX

s-branch 146 s-injective 154 self-covering map 31 semi-flow 2 semilocally 1-connected 29, 180 semisimple 394 sensetive dependence on initial conditions 36 set of density zero 372 ~harkovskiiordering 4 shift map 3, 39, 56 short exact sequence 294, 305 u-algebra 11, 360 simple function 361 simplex 324 simplical complex 324 simplical decomposition 324 simply connected 171, 172 singular homology group 311 singular n-simplex 310 singular point 6 sink 105 smooth 1 solenoid 231 solenoidal group 57, 223 source 105 space mean 11, 367 special Anosov differentiable map 22 special topological Anosov map (special TA-map) 10, 28, 90 specification 397 split 294 stable manifold 8 S'TA 166 SS'TA 166 stable set 11, 46 stable set in strong sence 147 standerd n-simplex 310 strong transversality 9 strongly mixing 13, 372

strongly special topological Anosov map 11, 166 structurally stable 27 structure group for inverse limit system 195 subcomplex 307, 324 subgroup 15 subshift 3, 86 support 12 symbolic dynamics 3 symmetric 41 symmetric neighborhood 41 syndetic 390 system of coordinate neighborhoods 50

7 166 I d 166 TAX 166 tangent bundle 1 tangent space 1 tangent vector 1 tensor product 308 tent map 5 time mean 11, 367 topological Anosov homeomorphism (TA-homeomorphism) 90 topological Anosov map (TA-map) 10, 28, 89 topological decomposition theorem 101, 120 topological dimension 63 topological dynamical system 3 topological group 15 topological manifold 1 topological manifold without boundary 50 topological pair 311 topological subgroup 15 topologically conjugate 5, 25, 90

INDEX

topologically mixing 3, 101, 120 topologically semi-conjugate

weak topology 12, 379 weakly mixing 13, 372

5, 25, 90 topologically stable 8 topologically stable in the class of homeomorphisms 90 topologically stable in the class of self-covering maps 90 topologically transitive

3, 25, 36, 119 toral automorphism 5 toral endomorphism 5 torsion group 223 torsion product 309 total space 193 trace 331 transformation 195 transition matrix 87 transverse 214 types (I), (11) and (111) 11, 16 type K(7r, 1) 217

u-branch 146 uniformly almost periodic 392 uniformly local splitting theorem 161 uniquely ergodic 394 uniqueness of lifting 156 uniqueness theorem 179 universal covering 177 universal covering space 177 unstable manifold 8 unstable set 10, 46 upper semi-continuous 107, 289 u-subrectangle 140

weak generator 37

zero-charactor 222