Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors in Dynamics 0521475724, 9780521475723

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 35 EDITORIAL BOARD D. J. H. GARLING, T. TOM DIECK, P. WALTERS

HYPERBOLICITY

AND SENSITIVE

CHAOTIC DYNAMICS

AT HOMOCLINIC BIFURCATIONS

Already published 1 2 3 4 5

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26 27 28 29 30 31 32 33 34 35 37 38 39 40

W.M.L. Holcombe Algebraic automata theory K. Petersen Ergodic theory P.T. Johnstone Stone spaces W.H. Schikhof Ultrametric calculus J.-P. Kahane Some random series of functions, 2nd edition H. Cohn Introduction to the construction of class fields J. Lambek & P.J. Scott Introduction to higher-order categorical logic H. Matsumura Commutative ring theory C.B. Thomas Characteristic classes and the cohomology of finite groups M. Aschbacher Finite group theory J.L. Alperin Local representation theory P. Koosis The logarithmic integral I A. Pietsch Eigenvalues and s-numbers S.J. Patterson An introduction to the theory of the Riemann zeta-function H.J. Baues Algebraic homotopy V.S. Varadarajan Introduction to harmonic analysis on semisimple Lie groups W. Dicks & M. Dunwoody Groups acting on graphs L.J. Corwin & F.P. Greenleaf Representations of nilpotent Lie groups and their applications R. Fritsch & R. Piccinini Cellular structures in topology H Klingen Introductory lectures on Siegel modular forms P. Koosis The logarithmic integral II M.J. Collins Representations and characters of finite groups H. Kunita Stochastic flows and stochastic differential equations P. Wojta.szczyk Banach spaces for analysts J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in harmonic analysis A. Frohlich & M.J. Taylor Algebraic number theory K. Goebel & W.A. Kirk Topics in metric fixed point theory J.E. Humphreys Reflection groups and Coxeter groups D.J. Benson Representations and cohomology I D.J. Benson Representations and cohomology II C. Allday & V. Puppe Cohomological methods in transformation groups C. Soule et al Lectures on Arakelov geometry A. Ambrosetti & G. Prodi A primer of nonlinear analysis J. Palis & F. Takens Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations Y. Meyer Wavelets and operators C. Weibel An introduction to homological algebra W. Bruns & J. Herzog Cohen- Macaulay rings V. Snaith Explicit Brauer induction

HYPERBOLICITY

AND SENSITIVE CHAOTIC DYNAMICS

AT HOMOCLINIC BIFURCATIONS

Fractal Dimensions and Infinitely Many Attractors

Jacob Palis Professor of Mathematics, IMPA Rio de Janeiro Floris Takens Professor of Mathematics University of Groningen

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521390644 © Cambridge University Press 1993

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993 First paperback edition 1995 Appendix 5 is reproduced with permission of the Annals of Mathematics A catalogue record/or this publication is available from the British Library

ISBN-13 978-0-521-39064-4 hardback ISBN-IO 0-521-39064-8 hardback ISBN-13 978-0-521-47572-3 paperback ISBN-JO 0-521-47572-4 paperback Transferred to digital printing 2005

CONTENTS

Preface 0 - Hyperbolicity, stability and sensitive chaotic dynamical systems §1 Hyperbolicity and stability §2 Sensitive chaotic dynamics

Vll

1 - Examples of homoclinic orbits in dynamical systems §1 Homoclinic orbits in a deformed linear map §2 The pendulum §3 The horseshoe §4 A homoclinic bifurcation §5 Concluding remarks

11

2 - Dynamical consequences of a transverse homoclinic intersection §1 Description of the situation - linearizing coordinates and a special domain R §2 The maximal invariant subset of R - topological analysis §3 The maximal invariant subset of R - hyperbolicity and invariant foliations §4 The maximal invariant subset of R - structure §5 Conclusions for the dynamics near a transverse homoclinic orbit §6 Homoclinic points of periodic orbits §7 Transverse homoclinic intersections in arbitrary dimensions §8 Historical note

18

1 1 8

12 12 14 15 15

18

22 23 27 30 31 31

32

3 - Homoclinic tangencies: cascades of bifurcations, scaling and quadratic maps §1 Cascades of homoclinic tangencies §2 Saddle-node and period doubling bifurcations §3 Cascades of period doubling bifurcations and sinks §4 Homoclinic tangencies, scaling and quadratic maps

37 40 45

4 - Cantor sets in dynamics and fractal dimensions §1 Dynamically defined Cantor sets §2 Numerical invariants of Cantor sets §3 Local invariants and continuity

53 53 60 82

vi 5 - Homoclinic bifurcations: fractal dimensions and measure of bifurcation sets §1 Construction of bifurcating families of diffeomorphisms §2 Homoclinic tangencies with bifurcation set of small relative measure - statement of the results §3 Homoclinic tangencies with bifurcation set of small relative measure - idea of proof §4 Heteroclinic cycles and further results on measure of bifurcation sets

92 93 99 102 107

6 - Infinitely many sinks and homoclinic tangencies §1 Persistent tangencies §2 The tent map and the logistic map §3 Herron-like diffeomorphisms §4 Separatrices of saddle points for diffeomorphisms near a homoclinic tangency §5 Proof of the main result §6 Sensitive chaotic orbits near a homoclinic tangency

125 129 131

7 - Overview, conjectures and problems - a theory of homoclinic bifurcations - strange attractors §1 Homoclinic bifurcations and nonhyperbolic dynamics §2 Strange attractors §3 Summary, further results and problems

132 133 138 148

Appendix 1 - Hyperbolicity: stable manifolds and foliations

154

Appendix 2 - Markov partitions

169

Appendix 3 - On the shape of some strange attractors

178

Appendix 4 - Infinitely many sinks in one-parameter families of diffeomorphisms

180

Appendix 5 - Hyperbolicity and the creation of homoclinic orbits, reprinted from Annals of Mathematics 125 (1987) References Index

185 223 233

112 113 116 119

PREFACE

Homoclinic bifurcations, which form the main topic of this monograph , belong to the area of dynamical systems, the theory which describes mathematical models of time evolution, like differential equations and maps. Homoclinic evolutions, or orbits, are evolutions for which the state has the same limit both in the "infinite future" and in the "infinite past". Such homoclinic evolutions, and the associated complexity, were discovered by Poincare and described in his famous essay on the stability of the solar system around 1890. This associated complexity was intimately related with the breakdown of power series methods, which came to many, and in particular to Poincare, as a surprise. The investigations were continued by Birkhoff who showed in 1935 that in general there is near a homoclinic orbit an extremely intricate complex of periodic solutions, mostly with a very high period. The theory up to this point was quite abstract: though the inspiration came from celestial mechanics, it was not proved that in the solar system homoclinic orbits actually can occur. Another development took place which was much more directed to the investigation of specific equations: in order to model vacuum tube radio receivers Van der Pol introduced in 1920 a class of equations, now named after him, describing nonlinear oscillators, with or without forcing. His interest was mainly in the periodic solutions and their dependence on the forcing. In later investigations of this same type of equations, around 1950, Cartwright, Littlewood and Levinson discovered solutions which were much more complicated than any solution of a differential equation known up to that time. Now we can easily interpret this as complexity caused by (the suspension of) a horseshoe, which in its turn is a consequence of the existence of one (transverse) homoclinic orbit, but that is inverting the history ... In fact, Smale, who originally had focussed his efforts on gradient and gradient-like dynamical systems, realized, when confronted with these complexities , that he should extend the scope of his investigations. Seventy years after Poincare, Smale was again shocked by the complexity of homoclinic behaviour! By the mid 1960 he had a very simple geometric example (i.e. no formulas but just a picture and a geometric description) , which could be completely analysed and which showed all the complexity found before: the horseshoe. This new prototype dynamical model, the horseshoe, together with the investigations in the behaviour of geodesic flows of manifolds with

viii

Preface

negative curvature (Hadamard, Anosov), grew, due to the efforts of a number of mathematicians, to an extension of the gradient-like theory which we now know as hyperbolic dynamics, and which in particular provides models for very complex (chaotic) dynamic behaviour. Around 1980 this hyperbolic theory was used by Levi to reanalyse the qualitative behaviour of the solutions of Van der Pol's equation, largely extending the earlier results. He proved that, besides all the complexity we know from the hyperbolic theory, even the new and extreme complexity associated with homoclinic bifurcations, which we shall consider below, actually exists in the solutions of this equation. Homoclinic bifurcations, or nontransverse homoclinic orbits, become important when going beyond the hyperbolic theory. In the late 1960s, Newhouse combined homoclinic bifurcations with the complexity already avail. able in the hyperbolic theory to obtain dynamical systems far more complicated than the hyperbolic ones. Ultimately this led to his famous result on the coexistence of infinitely many periodic attractors and was also influential on our own work on hyperbolicity or lack of it near homoclinic bifurcations. These developments form the main topic of the present monograph, of which we shall now outline the content. We start with Chapter O that presents general background information about the hyperbolic theory and its relation to (structural) stability of systems, and discuss as well some initial aspects of chaotic dynamics; many results on stable manifolds and foliations are stated, and their proofs sketched, in Appendix 1. The later chapters, except the last one, do not depend on the results described in this chapter and are basically self-contained. In Chapter 1, we give a number of simple examples of homoclinic orbits and bifurcations. Chapter 2 discusses the horseshoe example and shows how it is related to homoclinic orbits. Then, in Chapter 3, we consider some preliminary and more elementary consequences of the occurrence of a homoclinic bifurcation, especially in terms of cascades of bifurcations which have to accompany them. In Chapters 4, 5, and 6, we come to our main topic: the investigation of situations where there is an interplay between homoclinic bifurcations and nontrivial basic sets, the sets being the building blocks of hyperbolic systems with complex behaviour. Since such basic sets often have a fractal structure, we start in Chapter 4 with a discussion of Cantor sets and fractal dimensions like Hausdorff dimension. In Chapter 5 the emphasis is on hyperbolicity near a homoclinic bifurcation associated with a basic set of small Hausdorff dimension. Then, in Chapter 6, we discuss types of homoclinic bifurcations which yields, in a persistent way, complexity beyond hyperbolicity. In this chapter we provide a new, and more geometric, proof of New house's result on the coexistence of infinitely many periodic attractors. Finally, in Chapter 7

Preface

ix

we present an overview of recent results, including specially Henon-like and Lorenz-like strange attractors. We also pose new conjectures and problems which may lead to a better understanding of nonhyperbolic dynamics (the "dark realm" of dynamics) and the role of homoclinic bifurcations. Summarizing, we deal with the following rather striking collection of dynamical phenomena that take place at the unfolding of a homoclinic tangency - transversal homoclinic orbits, which in turn are always associated to horseshoes (invariant hyperbolic Cantor sets): Chapters 1 and 2, - cascades of homoclinic tangencies, i.e. sequences in the parameter line whose corresponding diffeomorphisms exhibit a homoclinic tangency: Chapter 3, and for families of locally dissipative diffeomorphisms, - cascades of period doubling bifurcations of periodic attractors (sinks): Chapter 3, - cascades of critical saddle-node cycles: Chapter 7, - residual subsets of intervals in the parameter line whose corresponding diffeomorphisms exhibit infinitely many coexisting sinks: Chapter 6 and Appendix 4, - positive Lebesgue measure sets in the parameter line whose corresponding diffeomorphisms exhibit a Henon-like strange attractor: Chapter 7 and Appendix 3, - prevalence of hyperbolicity when the fractal (Hausdorff) dimension of the associated basic hyperbolic set is smaller than 1: Chapters 4 and 5 and Appendix 5, - nonprevalence of hyperbolicity when the above fractal dimension is bigger than 1: Chapter 7.

In our presentation we mainly restrict ourselves to diffeomorphisms in dimension 2 (which is the proper context to investigate classical equations e.g. the forced Van der Pol equation), although extensions to higher dimensions are mentioned; also we concentrate mainly on the general theory as opposed to the analysis of specific equations. Consequently a number of topics like Silnikov's bifurcations and the Melnikov method are not discussed. We hope that by putting this material together, rearranging it to some extent and pointing to recent and possible future directions, these results and their proofs will become more accessible, and will find their central place in dynamics which we think they merit. We wish to thank a number of colleagues from several different institutions as well as Ph.D. students from the lnstituto de Matematica Pura e Aplicada

X

Preface

(IMPA) who much helped us in writing this book. Among them we mention M. Benedicks, L. Carleson, M. Carvalho, L. Diaz, P. Duarte, R. Maiie, L. Mora, S. Newhouse, M. J. Pacifico, J. Rocha, D. Ruelle, R. Ures, J. C. Yoccoz and most especially M. Viana. Thanks are also due to Luiz Alberto Santos for his fine typing of this text.

CHAPTER0 HYPERBOLICITY, STABILITY AND SENSITIVE CHAOTIC DYNAMICAL SYSTEMS

In this chapter we give background information and references to the literature concerning basic notions in dynamical systems that play an important role in our study of homoclinic bifurcations. Essentially, the chapter consists of a summary of the hyperbolic theory of dynamical systems and comments on sensitive chaotic dynamics. This is intended both as an introduction to the following chapters and to provide a more global context for the results to be discussed later and in much more detail than the ones presented in this Chapter 0. In the first section we concentrate on hyperbolicity and emphasize its intimate relation with various forms of (structural) stability. In the second section we discuss several aspects of sensitivity ("chaos") and indicate how it occurs in hyperbolic systems.

§1

Hyperbolicity

and stability

These two concepts, hyperbolicity and (structural) stability, have played an important role in the development of the theory of dynamical systems in the last decades: the hyperbolic theory was mostly developed in the 1960's, having as a main initial motivation the construction of structurally stable systems; in its turn, the notion of structural stability had been introduced much earlier by Andronov and Pontryagin [AP,1937]. As conjectured in the late 1960's and only recently proved, it turns out that the two notions are essentially equivalent to each other, at least for C 1 diffeomorphisms of a compact manifold. OJ course, for stability one also has to impose the transversality of all stable and unstable manifolds or, for limit-set stability, the no-cycle condition; see concepts below. It is, however, the hyperbolicity of the limit set which is the main ingredient in this comparison. The solution of this well known conjecture and other related results that we state here go beyond what is needed to understand the next chapters of this book. It is, however, enlightening, in the study of bifurcations (meaning loss of stability), to be acquainted with the fact that the notions of stability and hyperbolicity are that much interconnected. The concept of (structural) stability deals with the topological persistence of the orbit structure of a dynamical system (endomorphism, diffeomorphism or flow) under small perturbations. (Notice the difference with the concept

2

0 Hyperbolicity, Stability f3 Sensitive Chaotic Dynamical Systems

of Lyapunov stability which concerns attracting sets of a given system). The persistence of the orbit structure is expressed in terms of a homeomorphism of the ambient manifold sending orbits of the initial system onto orbits of the perturbed one. If this can be done for any Ck-small (k ~ 1) perturbation, then we call the system Ck (structurally) stable or globally stable. Here we are mostly concerned with diffeomorphisms, in which case we require this orbit preserving homeomorphism to be a conjugacy. That is, if cp is the initial map, 0 there are points x 0 = p, x 1, x2, · · · , Xk = p such that d(f (xi-1), xi) < e for 1 ::::;i ::::;k, d being a distance function. If £+(cp) (or L-(cp)) is hyperbolic (see Chapter 2 and Appendix 1), then one can show that Per (cp) = L + (cp) (or L - (cp)), where Per (cp) indicates the set of periodic points of cp;one can then write as in [N,1972]:

where each A; is invariant, compact, transitive (it has a dense orbit) and has a dense subset of periodic orbits. This is called the spectral decomposition of L+(cp). Moreover, by [HPPS, 1970] (see also [N,1980], [B,1977] for a different and relevant proof using the idea of "shadowing" of orbits), each Ai is the maximal invariant set in a neighbourhood of it. This last fact is actually equivalent to what we call local product structure in Ai: there exist e > 0 and 8 > 0 such that if the distance between x, y E Ai is smaller than 8 then their local stable and unstable manifolds of size e (see Appendix 1) intersect each other in a unique point and this point is in Ai. Also, one can prove that if w(x) c Ai then x E W 8 (z) for some z E A;. In general, a set with the properties above is called a basic set for the diffeomorphism. If we assume that the nonwandering set n( cp)is hyperbolic and Per (cp)= f2(cp),then we say that cp satisfies Axiom A. In this case we have f2(cp)= L + (cp) and so we can write the nonwandering set as a finite union of basic sets. This is the content of Smale's spectral decomposition theorem [S,1970]; the corresponding version for the limit set as presented above appeared later in [N,1972]. Notice that if A1 U · · · U Ak is the spectral decomposition of L+(cp) (or n(cp)) then M = W 8 (Ai), where W 5 (Ai) = {y Iw(y) C A;} is

u i

called the stable set of A;; as discussed above W 8 (Ai)

=

LJW (x). Similar 8

xEA;

statements are valid for the unstable sets of the A;'s, corresponding to a spectral decomposition of L-(cp) or O(cp). Some W 5 (Ai) must be open; in this case Ai is called an attractor. (A more general definition of attractor is in the next section). Dually if wu(Ai) is open, then we say that Ai is a repeller. Finally, A; is of saddle type if it is neither an attractor nor a repeller. Another property of Axiom A diffeomorphisms: the stable sets of attractors cover an open and dense subset of M and the same is true for unstable sets of repellers. It is an interesting-fact that if cp is C 2 , then the union of the stable sets of attractors has total Lebesgue measure; see Ruelle [R,1976] and Bowen-Ruelle (BR,1975]. There are, however, examples of C 1 saddle-type

4

0 Hyperbolicity, Stability & Sensitive Chaotic Dynamical Systems

basic sets with stable sets of positive Lebesgue measure [B,1975b], which are detailed in Chapter 4. Another interesting fact about basic sets is that they are expansive: for each basic set A there is a constant a > 0 such that for each pair of different points in A, their (full) orbits get apart by at least a. Prom this it follows that hyperbolic attractors which are not just fixed or periodic sinks have sensitive dependence on initial conditions: for most pairs of different points in the stable set of such a., attractor A, the positive orbits get apart by at least a constant (which depends on the attractor). Most here means probability 1 in W 8 (A) x W 8 (A). The following relevant result concerning basic sets states that they are persistent under Ck-small perturbations (see Appendix l); in particular, hyperbolic attractors are persistent. THEOREM 1. If A is a basic set for a Ck diffeomorphism 0 and suppose unfolds into transversal homoclinic points for µ > 0. Given µ there are small pieces of "parabolas" (see the above discussion of homoclinic tangencies) rt cw; near q and rt cw; near that, for µ > 0, their position relative to ws and wu near indicated in Figure 3.3. µ µ

the tangen cy

> O near zero, on unfoldin gs r. We assum e q and r is as

wsµ Pµ

Figure 3.3 Now take µ = µ arbitrarily small. Clearly, if n > 0 is large then

0 much smaller than µ,we have, for th: sa ~e

3.2 Saddle-node and period doubling bifurcations

37

integer n, that 0, b > 0 and IP2I < 1 we have the following unfolding of the saddle-node: a sink and a saddle collapse and then disappear, as show n in Figure 3.5. (a)

(b)

(c)

--~µ-~-~µ-xi

x2

w~

µ =0

µ>0

Figure 3.5 The double arrows in the figure mean that the contraction in the norm al

3.2 Saddle -node and period doubling bifurcations

39

direction is stronger than along wi. If we consider, for µ ~ µo, the curves µ -+ xµ,µ -+ xµ of fixed points, we get Figure 3.6.

µ

Figure 3.6 Notice that the two curves are differentiable for Iµ - µol small, µ < µo. If we follow the curveµ-+ Xµ forµ/ µo, we can then return alongµ-+ Xµ with decreasing values of µ . So the two branches can naturally be oriented as above (or vice versa). In words: if we follow the curve of saddles for increasing values ofµ, up toµ= µ 0 , we then return along the curve of sinks for decreasing values of µ. This fact will play a role in the proof of the next theorem. Now we consider the expression (2) above corresponding to the eigenvalue p 1 = -1. Similarly to what we have done before in (1), we take a=/=-0 (which is a generic condition) and call the orbit a period doubling bifurcation (or flip); we say that it unfolds generically if b'{0) =/-0 (another generic condition!). When a > 0 and b'(0) < 0, we can easily show that there exists a unique fixed point which is a sink for µ < µo and a saddle for µ > µo (both with negative eigenvalues); for µ > µo there is also a period 2-sink (with positive eigenvalues). Thus the name period doubling bifurcation. The results are of course similar in the other cases, where a and b'(0) may have signs different from the ones above. Notice that period doubling which unfolds generically is isolated; the same is true for saddle-nodes. The assumptions and results are also similar for period doubling bifurcations of periodic orbits by just considering the power of the map equal to the period . For instance, a sink of period k may bifurcate into a saddle of period k {both with corresponding negative eigenvalues) and a sink with twice the period (and positive eigenvalues). For the period doubling bifurcation considered above, if in the set of periodic points we identify points in the same orbit we obtain a topological 1-complex-the curve of sinks forµ < µo, branching off into two topological 1-manifolds: one is formed by the curve of saddles and the other by the curve of sinks with twice the period (see Figure 3.7). Notice that the sink to the left and the saddle to the right both have the same period (and a

40

3 H omoclinic Tangencies

corresponding negative eigenvalue for df!, k being the period) . This remar k will be relevant in the next section.

sink

saddle

µ

p

a> 0, db > 0 dµ Figure 3.7

§3

Cascades

of period doubling

bifurcations

and sink s

We now discuss the definitions and assumptions of the next theorem show in g the existence of many sinks (or sources) and period doubling bifurcati ons while creating a horseshoe . The sinks that we exhibit in this chapter ari se from period doubling bifurcations; they occur for different values of the parameter. Let R be a rectangle in 1R2 and {IPµ} a family of diffeomorphisms o f R into 1R2 such that (a) 1P- 1(R) n (R) = ¢, (b) IPµJR is dissipative (area contracting) jdet(d1Pµ)j < 1 on R,

for -1

< µ < 1, th at is

(c) 1P1has periodic points and they are all saddles, (d) IPµ(R) n S1 = ,IPµ(R) n S2 = ,-1 ~ µ ~ 1, where S 1, S 2 ar e two opposite sides in the boundary of R, say the vertical sides,

IPµ(B) nR = ,-1 ~ µ ~ 1, where Tis the top side of (e) 1P 1,(T) nR = , R and B is the bottom side .

In this section we also assume th e following generic (residual or Bair e second category) condition on the family {IPµ}(f) IPµhas at most one nonhyperbolic periodic orbit for each -1 ~ µ s; 1 and this orbit must correspond either to a saddle -node or to a p eriod

3.3 Cascades of period doubling bifurcations and sinks doubling bifurcation which unfolds generically. contracting there is no Hopf bifurcation).

41

(Because IPµ is area

Although we did not formally require 1Pi to be a horseshoe mapping like in Chapter 2, that is precisely the situation we have in mind. In this case, we say that we have an area decreasing family creating a horseshoe, as in Figure 3.8. T

R

B R

((lo

((lo

(B)

((lo

(R)

(D

Figure 3.8 Before continuing the discussion, we want to point out that the above conditions are satisfied for a generic unfolding of a homoclinic tangency re placing IPµby IPi, taking < µ < instead of -1 < µ < +1 and choosing R appropriately. Of course, to get the area decreasing property, we assume the determinant of the Jacobian of the map at the fixed (or pe riodic) saddle with a homoclinic tangency to be less than 1 in absolute value . To see the creation of a horseshoe, let IPµbe such that (i) 1Pohas a fixed saddle p and jdet(d1Po)pj < 1,

-o

+o

(ii) there is a generically unfolding homoclinic tangency q associate top . We then claim that for each neighbourhood V of q there exists a rectangle R C V, a number o > 0 and an integer N > 0 such that IP:IR creates a < µ < o:take R to be a thin rectangle near q and parallel horseshoe for as in Figure 3.9 . to the local component of Let us see why we can choose R, o and N as wished. First, for µ small , we choose C 1 coordinates linearizing each IPµin a fixed neighbourhood of p containing an arc c W0 from p to q; these coordinates may be chosen

-o

w;

e•

42

3 Homoclinic

wu0

Tangencies

wu -0

___ R

vzzbz, _...,__,__ 'P°t (q)

wu0 R

R

vzz6?21

ws0

p

wu0 Figure 3.9 to depend continuously on µ (see Appendix 1). We then choose R to be thin and sufficiently close to W 0 so that its projection on W 0 parall el to W0 contains in its interior cpf (q) for some large N. Then cpf (R) will be a "curved box" close to an arc in W0 near q. For µ near zero, we then have the situation indicated in the figure . One can then apply arguments simil ar to those in Chapter 2 to show that 'PilR is area decreasing for -8::;; µ ::;;8 and that cpfJRhas its maximal invariant set hyperbolic with dense sub set of periodic orbits; see also Section 4. In fact, we observe that although t he (R) resembles the situation in Chapter II, the rectangl es configuration R, considered are quite different : there we had a long rectangle containing p and q; here the rectangle is contained in a small neighbourhood of q. We now return to the general discussion about creating horseshoes. For 2 2 I we replace cp by cp-1 and if A· a = I our construction does not work). Let q and r be poinfs on the orbit of tangency in the domain of the linearizing coordinates as indicated in Figure 3.14. So, for some N, cp{;(r) = q. For each sufficiently big n, we take a box Bn near q such that -1< 1, such that >.K is a neighbourhood of O in K,

- a map w : K-----.K having a C 1+c expansive extension to a neighbourhood of K, - a Mark0v partition {K 1 , ...

,

Kk}.

Observe that although in this chapter we will not make use of the scalin g property in the definition, this condition will play a key role in Chapter 5. EXAMPLES: In each of the examples below we define the Cantor set by a Markov partition and expanding map. Observe that we can always associate to a Markov partition {K 1 , ... , Kk} and an expanding map W the Cantor

4.1 Dynamically defined Cantor sets

nw-i (K1 U · · · U Kk). Further,

57

00

set K

=

we consider only examples where

i=O

WIK; is affine, i.e. has constant derivative. Our first example is the mid-a-Cantor

set. In this case

K1=

[o,~(1-a)],

K2=

[~(l+a),1],

and WIK; maps K; affinely to [O,1]; the scaling constant can be taken as >.= ½(1 - a). For a = 1/3 this is the most well known Cantor set; in any case, for this construction one needs O < a < 1. See Figure 4.2. 0

I I

Yf----1

I

___J

L-_

Yf----1

Yf----1 Figure 4.2

The second example, or rather a class of examples, covers the affine Cantor sets. They are defined by a sequence of intervals K 1 , ... , Kk with endpoints Kl, K[ so that O = Kf < K 1 < K4 < K 2 < K§ < · · · < Kk; also WIK; maps K; affinely onto [O,Kn; as scaling constant one can take >.= K[ I Kk. See Figure 4.3. Finally we define generalized affine Cantor sets. They are obtained as the affine Cantor sets, only now the image w(K;J may be smaller. If we denote the endpoints of K; as above by Kl and K[ with ... 0 such that for all q, q and n 2: 1 with (a) 1wn(q) - wn(ii)I ~ 8,

4.1 Dynamically defined Cantor sets

59

(b) the interval [wi(q), wi(q)] contained in the domain of \JI for all O ~ i ~ n-l,

we have llog l(\Jln)'(q)I - log l(wn)'(q)II ~ c(8). Moreover c(8) converges to zero when 8 --+ 0. PROOF: From the fact that \JI is expanding it follows that, for some (J > 1, wi(q) - wi(q) ~ 8. (Ji-n for i ~ n. Since \JI is Cl+E and '11'is bounded away from zero, log 1'11'1 is CE. Then

I

I

-It

llogl(W")'(q)l- logl(w")'(Q)II

logIW'(W'(q))I- logIW'(W'(O))i\

n-1

n-1

i=O

i=O

~LClwi(q) - wi(iiW ~ L C8E. for some constant C > 0.

(JE(i-n)

This proves the theorem by taking c( 8)

= D

Let us finish this section with some comments about the bounded distortion property. First, we present a geometric consequence of it. Let V be some small open interval intersecting K. Since \JI is topologically mixing (see definition of Markov partition), there is n 2: 1 such that wn(V n K) = K. Take q0 , q1, q2 E V n K close enough to each other, so that the intervals (wi (q0 ), wi(qj)) are contained in the domain of \JI for O ~ i ~ n - 1 and j = 1, 2. By the mean value theorem, there are q E (qo,qi), q E (qo, Q2) such that 1wn(qo) - wn(q1)I = lqo - qil. l(wn)'(q)I and 1wn(qo)- wn(q2)I = lqo - q2I· l(wn)'(q)I. Then, from the theorem above, we get

e-clqo - qil < lwn(qo) - wn(q1)I < eclqo - q1I lqo - q2I - 1wn(qo)- wn(q2)I lqo - q2I for some c > 0 independent of n, V and the points involved. So \JI" essentially preserves ratios of distances between close points: they change but not by more than a uniform, multiplicative constant. This means that, up to a bounded distortion, small parts of K are just reproductions of big parts of K in a smaller scale. Our second remark concerns the differentiability assumptions in the stateme nt and proof of the theorem above. Clearly we used the assumption that \JI is c 1+ 1, then one of the following three alternatives occurs: Ki is conta ined in a gap of K2; K2 is contained in a gap of Ki; Kin K2 =/=¢. PROOF: We assume that neither of the two Cantor sets is contained in a gap of the other and we assume that Ki nK2 = ¢,and derive a contradiction from this. If Ui, U2 are bounded gaps of Ki, K2, we call (Ui, U2) a gap pair if U2 contains exactly one boundary point of Ui (and vice versa); Ui and U2 are said to be linked in this case. Since neither of the Cantor sets is contained in a gap of the other and since they are disjoint, there is a gap pair. Given such a gap pair (Ui, U2) we construct: a point in Ki n K2; or a different gap pair (U{, U2) with f(U{) < f(Ui); or a different gap pair (Ui, U2) with f(U 2) < f(U2). This leads to a contradiction: even if we don't find a point in Kin K2 after apply ing this construction a finite number of times, we get a sequence of gap pairs (U?l, uJi)) such that £(U?l) or f(UJi)) decreases and hence, since the sum of all the lengths of bounded gaps is finite, it goes to zero. Assuming f( ufil) goes to zero, take Qi E ufil: any accumulation point of {qi} belongs to Kin K2. Now we come to the announced construction. Let the relative position of U1 and U2 be as indicated in Figure 4.6.

Figure 4.6 Let Cf and CJ be the bridges of Kj at the boundary points of Uj, j = 1, 2. Since T1 · r2 > 1, ~~~~? . ~i~~\ > 1. So £(Cr) > £(U2) or i(C~) > £(U1), or both. (See Figure 4.7). Therefore the right endpoint of U2 is in Cf or the left endpoint of Ui is in or both. Suppose the first. Let u be the right

ct

64

4 Cantor Sets in Dynamics and Fractal Dimensions

endpoint of U2. If u E K1 then we are done, since u E K2 anyway. If u €/.K 1. then u is contained in a gap U{ of K 1 with e(U{) < e(U1) and (U{, U2) is the c required gap pair. This completes the proof.

Figure 4.7 REMARK 1: Let now 11 and 12 be minimal closed intervals such that K 1 C 11 and K2 C h We say that K1 and K2 are linked if Ii and h are linked. If T(K 1 ) • T(K 2 ) > 1 and if K 1 and K2 are linked, then K 1 n K2 -/- ¢ (since neither can K 1 be contained in a gap of K2 nor K2 in a gap of K 1). Since being linked is an open condition, it follows that whenever T(K 1 ) ·T(K2) > 1. then K 1 - K 2 has interior points. THEOREM l. Let K 1 , K 2 be Cantor sets in JR with Hausdorff dimension h 1, h2. If h 1 + h2 > 1 then (K1 - >.K2) has positive Lebesgue measure for almost every>. E JR (in the Lebesgue measure sense). Before going into the proof of the theorem, we first observe that from the assumption on h 1, h2 it follows that HD(K 1 xK2) 2::HD(K1)+HD(K2) > 1 (see Falconer [F,1985]). Also, let us see how we can state this result in a similar but slightly different way. For >. E JR take 0 E ( -1r /2, +1r/2) such that >.= -tan 0. Let 1redenote the orthogonal projection of JR2 onto the straight line Le which contains ve = (cos 0, sin 0). If we identify JR with Le through JR 3 x 1--+ x · ve then 1re(k) = k · ve = cos 0 · k1 + sin 0 · k2, for k = (k 1, k2) E lR.2 . By our choice of 0 we get 1re(K1 x K2) = cos0(K 1 ->.K2)Since cos 0 -/- 0 this shows that the theorem above can be rephrased in the following (slightly stronger) form. THEOREM 2. Let K C lR.2 be such that H D(K) > 1 and 1re : JR2 -> JR be as above. Then 1re(K) has positive Lebesgue measure for almost every 0 E (-1r /2, +1r/2) (in the Lebesgue measure sense). This result was first proved by Marstrand [M,1954]. The argument that we present here, which uses ideas from potential theory, is due to Kaufman and can be found in Falconer [F,1985].

65

4-2 Numerical invariants of Cantor sets PROOF: Let d

= H D(K)

> l. We first assume that O < md(K) < oo and

that for some C > 0

(1)

2 and Q < r ~ l. Letµ be the finite measure on JR defined by for all x E JR. 2 µ(A) = md(A n K), for A a measurable subset of JR . For -1r /2 < 0 < 1r/2, let us denote by µe the (unique) measure on JR such ~hat J fd~e = JUO 1re)dµfor every continuous function f. The theorem will follow, 1f we show that the support of µ 0 has positive Lebesgue measure for almost all 0 E (-1r/2, 1r/2), since this support is clearly contained in 1re(K). To do this 2

we use the following fact. LEMMA1. Let 17be a finite measure with compact support on JRand fJ(p) = _1_ J+ooe~ixpd1J(x),for p E JR (fJ is the Fourier transform of 17). If O < -./2i -oo 2 dp < \f/(p)\ 00 then the support of 17has positive Lebesgue measure.

f~:

PROOF OF THE LEMMA: The assumption that fJ is square-integrable implies (Pla ncherel's theorem) that d(K). Take co > 0 so that for O < c :s;co N 0 (K) :s;c- /3, i.e. there is a covering of K by not more than c- /3 intervals of length c. For every R E nn, the inverse images by (\J!nlR) of these intervals form a covering of R by intervals of length at most c>.;¾. This means that N 0 »-1 (R) :s;c /3for O < c :s;co, or, in other words , n,R

1

N0 (R) :s;>.~ ~ · c /3for O < c :s;>-;:;,¾·co- Then N 0 (K) :s;E-/3(

L

>.~~) for

RERn 1 all O < E :s;>.;;co, where An = sup An,R· Repeating the argument we get

RERn

for all k ?: 1

N 0 (K)

:s;c- /3(

L

>.~~t

if

O< c

:s;>.;:;-kc 0.

RERn

This implies

and so, making

/3-+ d(K),

Since An > 1 this proves that

L

>.~.~K) ?: 1, that is, d(K) :s;f3n-

RERn

Now we derive a contradiction from the assumption that H D(K) < 0 11 • Take H D(K) < o < o,,. Then there are finite coverings U of K with arb itra ril y small diameter for which Ha(U) is also arbitrarily small . We assum e that every element of U intersects at most one R E R,11 • This will be the case if we require that Ha(U) :s;co for some co = co(n , o) > 0. We denote UR = {U E U I Un R i=-qi}. Let , as above, k ?: 0 be such that

70

4 Cantor Sets in Dynamics and Fractal Dimensions

wk+l(Ki n K) = K for all Ki E R 1 . Then, if H 0 (U) and hence diam(U) is sufficiently small, (wn+k I R)(UR) is a well defined covering of K for all RE Rn. Note that

(since a< O'.n< f3n < ~)- We claim that

for some

.RoE nn. Otherwise

we would have

~ (c- 1 which is a contradiction,

I:

A~,R) · Ho(U)

RE'R,n

since, by assumption, a < an and so

In this way we construct, from the initial finite covering U, a new covering U' = (wn+klRo)(UR0 ), with fewer elements than U and such that H 0 (U1) :S c::0 . Repeating this argument we eventually obtain a covering of K with no elements at all. This is the required contradiction. Finally, to prove that (!3n- an)n -, 0 we first note that, by the bound ed distortion property there is a > 0, such that An,R :S a · An,R, for all n ?: 1 O'.nlog a + log C . and RE Rn. Take 8n =------,where)..= mf 1'11'1 > l. Then - log a + n log )..

'"'

L,_, RERn

)..-(on+On)< a(on+on)'"' A-On . A-On n,R L,_, n,R n,R RE'R,n :S a(on+on).)..-n·On. A~.~f RE'R,n = a(on+6nl. >-.-non. = l,

I:

c

by definition of Dn- It follows that f3n :S O'.n+ Dn, i.e.

an log a+ log C H D(K) · log a+ log C < -- -------. {3n - O'.n< ------ n log )..- log a n log )..- log a

4.2 Numerical invariants of Cantor sets

71

This implies the convergence we have claimed and completes the proof of □ the theorem. The above theorem is a consequence of the regularity of dynamically defined Cantor sets. It makes the propositions on the measure of the difference of two Cantor sets, in terms of limit capacity and Hausdorff dimension, cover, for dynamically defined Cantor sets, almost all cases the exceptions being d(K 1 ) + d(K2) = 1 and K 1 - >-.K2for exceptional values of >-..Before proceeding with our discussion on the relations between the invariants (dimensions) of a Cantor set, let us explore some consequences of the ideas involved in the proof of this theorem. First we recall that in the heuristic proof we have the following formula for the Hausdorff dimension and the limit capacity. If K is an affine Cantor set (see the examples in the previous section) with Markov parthen tition {K 1 , ... , Kk} and )..i denotes the (constant) value of l'11'1K.I, H D(K) = d(K) = d, d being the unique number such that I: >-.;d= l. We use this formula to compute the precise value of H D(K) = d(K) in a particular case. Take K to be an affine Cantor set with Markov partition {K 1 , ... , Kk} such that all the Ki have equal length, say {3 • diam K for O < /3 < l/k. Since we are assuming that K is affine (and not just generalized affine), \JImaps each Kin K onto K, so we must have )..i = {3-1 for all i. Therefore H D(K) = d(K) = log k/ log(/3- 1). For k = 2, since {3 = (l - a)/2, we get the formula stated at the beginning of this section. Incidentally, this shows that the dimension of a dynamically defined Cantor set can take any value between O and l. Also, for any p E (0, 1), there are diffeomorphisms exhibiting a saddle point p and a basic set A with p E A such that HD(A n W 5 (p)) = p. Our second remark concerns the role played by the bounded distortion property. Although we made use of it in the last part of the proof this is not strictly necessary for the theorem above. In fact this result is still true for Cantor sets defined by expanding maps which are only C 1 (and so may not have this property); see [T,1988]. Even more so, if O,

b b f3n- - :S an :S d(K) = H D(K) :S f3n :S O'.n+ -, for all n > l. n n We want to explore some important consequences of this estimate.

(2) First

72

4

Cantor Sets in Dynamics and Fractal Dimensions

observe that, denoting A

= sup IIJ!'Iand d = H D(K) = d(K),

and so ~ L., REnn

>.n,R -d < - Ab < oo , for all n >_l.

(3)

In a similar way,

L A;;,t 2: cA-b

> 0, for all n 2: 1.

(4)

REnn

Using these facts we prove the following proposition. PROPOSITION 3. Let K C IR be a dynamically defined Cantor set and let d = HD(K). Then, 0 < md(K) < oo . Moreover, there is c > 0 such that, for all x E K and O < r '.S1, -1

C

'.S

md(Br(x) d r

n K)

'.SC.

(5)

We point out that the bounded distortion property is fundamental her e: contrary to the theorem above, this last proposition wouldn't hold in general if \JI were only C 1. PROOF: We keep the notations from the proof of the above theorem. Observe that by the mean value theorem and condition (3),

Hd(Rn)

=

L

[f(R)]d '.S

REnn

L (>.;;-}·f(K))

d '.SAb(f(K))d,

RERn

where f(K) denotes the diameter of K. Since diam(Rn) this proves that md(K) '.SAb(f(K)l < oo.

----> 0

as n

----> oc.

Proving that md(K) is positive requires a little more effort. First , we claim that for some a 1 > 0 we have

(f(U)l

2: a1 · ~ L., A-d n,R

(6)

for every interval U intersecting K and n 2: 1 sufficiently large dependin g on U. To show this we fix o > 0 such that the a-neighbourhood of K i::, contained in the domain of \JI. Take k = k(U) 2: 0 minimal such that

4.2 Numerical invariants of Cantor sets

73

Let n > k. Then S E nn-k intersects wk(U) if and only if S = wk(R) for some R. E nn intersecting U. Moreover, in such case we have

An,R= sup l(wn)'IRI 2: inf l(wk)'IRI. sup l(wn-k)'lsl

2: inf l(wk)'luuRI. An-k,S· On the other hand, by the mean value theorem we have

Observe that, by construction, IJ!J(U UR) is contained in the domain of \JI for all O '.Sj '.Sk - l. The bounded distortion property implies that

where a is some positive number independent of U, R and k. From all this and the fact that d 2: On-k, we obtain

I:

(e(U))d 2: c- 1

A;;-~k .s · e(u)d

SEnn-k Snwk(U)#ef>

L

2: c- 1

A~~k,\Jik(R) · (f(wk(U))/ sup l(wk)'luuRl)d

HER" RnU#

L

2: c- 1od

A~~k,\Jik(R).a-d. (inf l(wk)'luuRl)-d

RERn RnU-f-¢,

> _ C

-1

O

~

d -d

a

L., REnn

A-d n.R·

RnU-f-¢,

· w1. ·th a1 = c- 1o da -d . This· proves t he c1aim Let now U be any finite covering of K. Take n 2: 1 such that (6) holds for all U EU. Then, by (4),

Hd(U) = L(f(U))d lfEU

2:

L a1 ( L A;;-,t) UEU

RERn RnUN

2: a 1

L A~.t2: a1CA-b. 1/ERn

Since U is arbitrary, this proves

4

74

Cantor Sets in Dynamics and Fractal Dimensions

Now we deal with the second part of the proposition. To make the argument more transparent we first derive an estimate for the d-measure of the intervals RE Rn. For some a 2 > 1, depending only on Kand Ill, we have (7)

for all R E Rn and n 2 1. To show this we observe that wn-imaps R diffeomorphically onto some K; E R 1. From the definition of Hausdorff measure, we have

On the other hand, by the mean value theorem, we have An-1,R

·

£(R) ::; £(K;) ::; An-1,R

·

£(R).

Finally, by the bounded distortion property, it follows that

with a > 0 as above depending only on K and Ill. From all this we get

Clearly, £(K;) can be uniformly bounded from zero and infinity, so to prove (7) we only need to show that the same holds for md(K; n K). The upper bound is trivial since md(K; n K) ::; md(K) < oo. The lower bound follows easily from the fact that, for some k 2 0, wk+ 1(K; n K) =Kand so, again by the definition of Hausdorff measure,

Now we prove (5). For x E K and O < r ::; 1, we let q = q(x, r) 2 0 be minimal such that where, as ·before, a > 0 is such that the domain of Ill contains the oncighbourhood of K. Then, arguing as above with Br(x) and '11qin the place of Rand wn-1 , respectively, we obtain

a

-d m O for all RE RP. It follows that

□

and this completes the proof of the proposition.

Finally, we prove a two-dimensional version of this proposition, which had been remarked following the proof of Marstrand's theorem relating Hausdorff dimension and measure of the difference set. It applies to hyperbolic basic sets of diffeomorphisms on surfaces. 4. Let K 1, K2 be dynamically defined Cantor sets and Jet di= HD(K1), d2 = HD(K2), d = d1 + d2 and K = K 1 x K 2 in JR.2. Then, for some c > 0, (a) 0 < md(K) < oo, PROPOSITION

(b) c- 1 :=;rnd

(K~Br(x))

:=;c for all x

E K and O < r

:=;1.

Takeµ to be the product measureµ= md, x md2 on K. Clearly, (a) and (b) hold ifwe replace there md byµ. Therefore it is now sufficient to show that µ is equivalent to md in the sense that for all Borel subsets A c K µ(A)/md(A) is bounded away from zero and infinity. We consider Markov' partitions R1,R2 for K1,K2 respectively and denote by Rf, i = 1,2, the PROOF:

76

4

Cantor Sets in Dynamics and Fractal Dimensions

family of connected components of ourselves to Borel sets of the form

w;(n-1\Lj),

LJ E R;. We may restric t

since these sets generate the Borel a-algebra of K. Let U = {Ui,j x U2.j l ::; j :S m} be any finite covering of A = R 1 x R 2 by cubes. Fix x;, 1 E U;.J n R ; . i = 1, 2, 1 :S j :Sm (obviously, we may assume U;,J n R; =I=0). Then J

µ(U1,j x U2,j) = md, (U1,j) x md 2 (U2,j) :S md, (B1,J) x md2 (B2,j) 1

:,S:C1C2· (f(U1,j)/

where B;,j denotes the ball in K; centred in Therefore m

·

(f(U2,j))d2

x;,j

and with radius f(U i,J)-

m

I)diam(U1

,j x U2,J))d;:::L(f(Ui

j=l

,J))d' x (f(U 2,j))d2

j=l

;:::(c1c2)- 1

L µ(U1,j x U2,j) j

;:::(c1c2)- 1µ(A). Since U is arbitrary

this proves

To obtain an inequality in the opposite direction we construct coverings Um, of A = R1 x R2, m >> n, as follows. Fix U1 E R 1 , U1 contained in R 1. For each x2 E R2..:_take m(U1, x2) maximal such that if U2(U1, x 2) denot es 1 2 (U,,x ) containing x 2, then f(U 2(U 1, x 2)) ;:=:f(Ui). Clearl y, the element of {U2(U1,x2): x2 E R2} contains a finite covering of R 2 by disjoint int erval s. Since these U2( U1, X2) are elements of Markov partitions R~, j ;:=:1, two o f them either are disjoint or have one contained in the other. Thus, we can extract a finite subcovering by disjoint elements. We now define Urnto be th e family of sets U1 x U2(U1, x2) obtained in this way for all U1 E R 1 contain ed in R1. This is a covering of R1 x R2 by disjoint cubes. Moreover, it is not difficult to deduce from the bounded distortion property that there is O < b < 1 (dependin~ only on K2 and llf2) such that, denoting by Vi(U 1, x 2) the ele2 . x2, we h ave €(U'( U , x ) ) ;:=:b€(U (U , x )) . men t o fR 2rn(U,,x )+l th a t contams 2 1 2 2 1 2 By definition of UHU1,x2), we also have f(UHU 1 , x2)) :S €(U 1). Therefor e,

R;

(8)

4-2

77

Numerical invariants of Cantor sets

Then , by (7) and (8) we get ~)diam(U1

x U2(U1,x2))d

Um

= Lf(U2(U1,x2)/ Um

Um Um

Since the diameter of Um may be taken arbitrarily small (by taking m large), we have md(A) :S b-d,a~a~µ(A) and so our argument is complete. □

Note that O < md(K) < oo ((a) in the proposition) is related to the fact that since K 1 and K2 are dynamically defined, H D(K 1 x K2) = H D(K 1) + H D(K 2). This also follows from the previous theorem stating that d(K;) = H D(K;), i = 1, 2, and the general product formulas H D(K 1 x K2) ;::: H D(K1) + H D(K2) and d(K1 x K2) :S d(K1) + d(K 2) together with the inequality d(K) ;:::H D(K). We now establish an interesting relation between Hausdorff dimension and thickness for Cantor sets in the line. In particular, if the thickness is large then the Hausdorff dimension is close to 1. PROPOSITION 5. If K C 1Ris a Cantor with thickness (log 2/ log(2 + 1/T)).

T

>

then H D(K)

PROOF: Let /3 = (log2/log(2 + 1/T)). We show that H13(U);:=:(diamK)t3 for every finite open covering U of K, which clearly implies the proposition. The key ingredient in this proof is the following elementary fact:

min{x 13+z 13Jx;:::0,z;::: O,x+z :S l,x;::: T(l-x-z),z;:::

T(l-x-z)}

=

1.

(9) We assume from now on that U is a covering with disjoint intervals. This is no restriction because whenever two elements of U have nonempty intersection we can replace them by their union, getting in this way a new covering V such that H/J(V) :S H{J(U). Note that, since U is an open covering of K, it covers all but a finite number of gaps of K. Let U, a gap of K, have minimal length among the gaps of K which are not covered by U. Let and be the bridges of K at the boundary points of U. (See Figure 4.8).

er

ce

78

4 Cantor Sets in Dynamics and Fractal Dimensions

~ ct~ ◄•--([

► +--

u

]()

(

er~

)[

)

Ar---•

Figure 4.8 By construction there are Ae, Ar E U such that ce Take the convex hull A of Ae U Ar. Then

c Ae and er c Ar.

and

f(Ar) 2 f(Cr) 2 T · f(U) 2 T(f(A) - f(At) - f(Ar)) and so, by (9), (f(Ae)).B+ (f(Ar)).6 2 (f(A)).6. This means that the covering U1 of K obtained by replacing Ae and N by A in U is such that H.a(U1 ) ::; Hp(U). Repeating the argument we eventually construct Uk, a covering of th~ convex hull of K with H.a(Uk) :S Hf3(U). Since we must have H (Uk) ;::: 13 (dim K).6, this ends the proof. r,

'-

Note that in general there can be no nontrivial upper estimates for the Hausdorff dimension in terms of the thickness, even in the dynamicalh· defined case. To see this, recall the earlier example of an affine Canto·r set K with Markov partition {K 1 , ... , Kk} with components all of length /3 · diamK, 0 < /3 < ¾, and gaps between K; and K;+i all of length (1-/3- k)-diam(K)/(k-1). As we saw, we have HD(K) = logk/log /3- 1. The thickness can easily be shown to be T(K) = f3(k - 1)/(1 - f3. k). Now consider a sequence of such Cantor sets characterized by k and f3k such that }~~ k · f3k = a E (0, 1). Then, ask-> oa, the Hausdorff dimension tends to 1 while the thickness converges to a/(1 - a). This fact is not really surprising since the thickness was defined as an infimum and so having T(K) small gives very little information concernin g the Cantor set. As mentioned before, this was our main motivation for introducing a variation of the thickness which we called denseness. We shall prove that Cantor sets with small denseness have small Hausdorff dimension . Let us first observe that if K is an affine Cantor set as above and 2f - 1 < k :S 2f, then 0(K) = (e - 1) + f/3(k -1)/(1 - /3k). This follows from th 0 and m(W 5 (p) n A) > 0. In this particular example, W 8 (p) n A is invariant under an expanding C 1 map but the bounded distortion property no longer holds. Given a sequence {f3n} of positive real numbers satisfying

L f3n< 2

and

n~O

we construct in J = [-1, 1], by the standard procedure, a Cantor set K 1 in such way that at the n th step we remove 2n intervals, Jn,k, k E {1, ... , 2n}, f3nis positive. For each of length ~;:. It is then clear that m(K1) = 2 -

L

n 2::1 and k E {2n-l

+ 1, ...

n~O

, 2n}, define

as follows: (i) gn,k is a C 1 orientation preserving homeomorphism; (ii) 9~,k(an,k) = 9~,k(bn.k) = 2, where Jn,k = [an,k, bn,k];

4

82

(iii)

sup xEln,k

Cantor Sets in Dynamics and Fractal Dimensions

12- 9~,k(x)j n-+oo -->

0.

The choice of the f3n's guarantees that (i)-(iii) coexist and makes possibl e this construction. Now, from the above conditions, we can continuously extend all th e 9n,k's to a homeomorphism g of class ci on [/30 /2, l], so that g' IKJ= 2. Finally, let Q be J x J and cp:Q --+ Q be a diffeomorphism given by

cp(x, y) = (g(x) , 9-i(y)), cp(x,y)

i

cp(x,y)

= (g(-x),

if

Q,

if

-g-i(y)),

if

(See Figure 4.10) +I

-------•

), I

-_-_-__---l

/11

],'

-_-__

,, ' /o I

/

/

o

I

I

I

I

I

-7: _

/11

1

/

IJ

I I

It

P0 l2

graph of g, partially defined

Figure 4.10 The reader may easily verify that cpis of class C1, p = (1, -1) is a hypercpn(Q) = K 1 x K 1 is a hyperbolic hors eshoe bolic fixed point of cp, A=

n

nEZ

and that Wi~c(P)nA greater than zero.

§3

= K1.

Local invariants

Besides, both A and K 1 have Lebesgue measure

and continuity

We conclude this chapter with some relevant facts on localized ver sions of the numerical invariants for Cantor sets introduced so far, and on the (continuous) dependence of these invariants on the Cantor set, at leas t for dynamically defined Cantor sets.

4-3 Local invariants and continuity

83

We give the definition of local thickness; local denseness, local Hausdorff dimension and local limit capacity are similarly defined. Let K C 1R be a Cantor set and k EK . The local thickness 7ioc(K, k) of Kat k is defined as

1ioc(K , k) = ,:limsup{ r(K) -,Q

IK

is the intersection of K with an interval

contained in an c:-neighbourhood of k }. For dynamically defined Cantor sets these notions have some additional properties. Let K be a dynamically defined Cantor set with expanding map llt. Th en for every U c K, U open, there is some n so that wn(U) = K. From this and the bounded distortion property it follows that the local invariants 1ioc(K,k), 010 c(K,k), HD1oc(K,k) , and dioc(K,k) are, in the dynamically defined case , all independent of k. Also , since the limit capacity and the Hausdorff dimension are invariant under diffeomorphisms , one has in t his case H D10 c(K, k) = H D(K) = d(K) = d10 c(K , k). The thickness and the denseness are not invariant under diffeomorphisms, and we may have r( K ) < 1ioc(K, k) or 0(K) < 010 c(K, k). For a discussion of the continuous dependence of the invariants on the Cant or set, we restrict ourselves to the dynamically defined case. Bearing in mind the dynamics of basic sets of surface diffeomorphisms , we define when two Cantor sets are near each other as follows. Let K be a Cantor set with expanding map W and Markov partition {Ki, ... , Kt}. Suppos e that 0 llt is c 1+ with Holder constant C, i.e. with l'11'(p)- w'(q)I ::; C IP- qj" for all p , q in a neighbourhood of K. We say that the Cantor set K is near K if K has expanding map 1l1and Markov partition Ki, ... , Kesuch that

- 1l1is

ci+ t and is ci near w, its derivative 'l!' has Holder constant such that (i , C) is near (c:,C),

C

- (Ki , ... , Ke)is near (Ki , .. . , Ke) in the sense that corresponding endpoints are near. An important consequence of this definition is the existence, for nearby Cantor sets K and K as above, of a homeomorphism h : K --+ K, 0close to the identity, such that 1l1 o h = h o W. We construct h as follows. Notice first that, because of the proximity assumptions in the definition, llt(K i) intersects (and then contains) Kj if and only if the same happens with 'l!(K i) and Kj. It follows that, given x E K, there is i; E K such that ii,n(x) E Ki {::}wn(x) E Ki, for all n 2: 0. Since 1l1is expanding, i; must be unique; we define h(x) = x. Clearly 'l!(h(x)) = h(w(x)). On the oth er hand we can obtain h-i by a symmetrical construction, so h is really a bijection. Checking that h is close to the identity presents no particular and Rn for K and K as in difficulty. Just construct Markov partitions the previous section taking connected components of the inverse images of

c

nn

84

4 Cantor Sets in Dynamics and Fractal Dimensions

the Kj, respectively Kj, by wn-i, respectively ii,n-l_ Then, one observes that x and h(x) belong to corresponding intervals of nn and i?,n for all n and that corresponding intervals are uniformly (meaning independentlv of n) close, due to the closeness of k to K, iJ! to \JIand to the bounded distortion property. We are left to show that h is continuous. We do more than that: we prove that it is Holder continuous. Take 8 > O such that d(K;,Kj) > 38 and d(K;,K 1) > 38 for all i =/-j. Now, for x,y EK with Ix - YI :::;8 we let n = n(x, y) 2: 0 be such that

and

1wn(x) - wn(y)I ::::28. By the definition of 8, the interval [wi(x), iJ!i(y)] is contained in some element of the Markov partition, for every O :::;i :::;n - l. On the other hand we may assume that ii,i(x) - W\y) 38 for O :::;i :::;n - l.

I

I :::;

To have this we just take _k close enough to K, in order to have lh(x) - xi :::; 8/2 for all x (note tha~ wi(x) = h(wi(x))). Then again [1J!i(x),ii,i(y)] must be contained in some K 1 , for all O :::;i :::;n - l. By the mean value theorem there are ~i E [wi(x), wi(y)], ~; E [1J!i(x),ii,i(y)] such that n-1

1wn(x) - wn(y)I = Ix - YI

II llll'(~;)I, 0

n-1

lwn(x) - wn(y)I = Ix -yl

II lw'(~;)I. 0

Clearly, we can take O k such that h and h- 1 both are C 7 implies that , · H D(K) :::;H D(K) :::;,- 1 · H D(K) (and analogously for limit capacity). This, in its turn, is a direct consequence of the definitions. This proof is similar to that in [PV,1988]. Now we state and prove the corresponding result for thickness and denseness. 2. The thickness and the denseness of a dynamically defined Cantor set K depend continuously on K. The same holds for local thickness and local denseness. THEOREM

Heuristically, the theorem is proved as follows. The global strategy is to show that the values T(K, U, u), with U = {Un} a presentation of K and u in the boundary of some bounded gap U = Un, depend equicontinuously on K in the sense that if k is close to K then T(K, h(U), h(u)) is close to r(K,U, u) for all U and u. Here h: K --> k is the conjugacy from W to Ill described above (assume k close enough to K to ensure that h exists) and h(U) is the presentation of k given by h(U) = {h(Un)}, where h(Un) is defined by 8h(Un) = h(8Un)- Observe that h, as we constructed it, is monotonic. For any given u and U we can, just by forcing h to be close enough to the identity, make T(K, h(U), h(u)) arbitrarily close to T(K,U, u). We can even make this happen simultaneously for all u (and U) for which the corresponding gap U is big, say with length bigger than some fixed a > 0. However, such a simple argument is insufficient to obtain the uniform closeness that we need. To deal with the small gaps we must use the bounded distortion property. The idea is to iterate the gap U and the U-component C of u until they become big. To be precise we fix /3 > 0 and take k = k(U, C) 2: 0 minimal such that f(iJ!k(U UC)) 2: /3. From the bounded

.

.

d1stort10n property we conclude that T(K,U,u)

:~::~i~~:

f(C) .

= f(U) 1s almost equal to

/3 > 0 and K taking /3 small enough. for k, \JIwe obtain that

their ratio admits a bound depending only on

and which can be made arbitrarily close to 1 by Analogously, from the bounded distortion property

f(h(C)) r(K, h(U), h(u)) = f(h(U))

.

1s almost equal to

f(\J!n(h(C))) _ . Moreover, f(wn(h(U)))

and this is a key point, the bound for the ratio of these last two values may be taken to be independent of k in a neighbourhood of K. This is a consequence of the fact that bounds for the distortion may be taken

86

4 Cantor Sets in Dynamics and Fractal Dimensions

to be uniform in a neighbourhood of any Cantor set. To explain this, let us first observe that the positive numbers c(8) constructed in the proof of the bounded distortion property vary continuously with the dynamically defined Cantor set. In fact, these c( 8) depend only on the Holder constants of the derivative of the expanding map and, by definition, nearby Cantor sets have expanding maps whose derivatives have nearby Holder constants. In particular, it follows that we can take (new) upper bounds c(8) as in the statement of the bounded distortion property which are uniform, i.e. independent of the Cantor set in a neighbourhood of K. We assume in what follows that k belongs to this neighbourhood. Now, if .e(IJ.!k(U))is big, that is larger than a, we can argue as before , -k

i.e. use the proximity of h to the identity to conclude that .e(~ (h(C)))

=

.e(iJ!k(h(C)))

.e(h(IJ.!k(C))). .e(IJ.!k(C)) . This, together with the estimates obf(h(IJ.!k(U))) is close to .e(IJ.!k(U)) tained above with the aid of the bounded distortion property, proves that T(K, h(U), h(u)) is close to T(K,U, u), as we wanted to show. Of course, we still have the problem that IJ.!k(U)may be small. Iterating further is no solution: it may not be possible to do it, if iJ,i ( C) gets out of the domain of IJ! before IJ.!i(U) gets large. Even if this does not happen , as we iterate the length of IJ.!i( U U C) gets bigger and so the bounds given by the bounded distortion property get rougher. Clearly, for the preceding argument we needed these bounds to be close to 1. Instead, what we do is to show that for our purposes this situation doesn't need to be taken into consideration. First, we observe that since .e(wk(U)) ::; a and .e(IJ.!k(U u C)) 2:

/3, if

we have chosen from the beginning

/3 >> a,

then

:~::~~~~

must be very big. The conjugacy h being close to the identity, the same

f(IJ.!k(h(C))) . . Usmg the bounded distortion property as above we .e(IJ.!k(h(U))) conclude that T(K,U, u) and T(K, h(U), h(u)) are very big. Since in the calculation of both the thickness and the denseness one must at some point take an infimum, these values are irrelevant for this calculation and so may be disregarded when proving the continuity of 0(K) and T(K). We now come to a formal proof. holds for

PROOF: Let A= sup JIJl'J and B = 20(K) + 8. Let c > 0, 8 > 0 and a> 0. Suppose that k is close enough to K so that Jh(x) - xi ::; a8 for all x EK. We prove that if a> 0 and 8 > 0 are chosen appropriately small (the precis e

4.3 Local invariants and continuity

87

condit ions are given below) then this implies 0(K)::; (1 + c) 2 0(K)

(a) (b) (c) (d)

+ c(l + c),

+ c)- 1, T(K)::; (1 + c) 2 T(K) + c:(1+ c), T(K) 2: (1 + c)- 2T(K) - c:(1+ c)- 1 . 0(K) 2: (1 + c)- 0(K) - c(l 2

This proves the first part of the theorem. Then we show that the second part. is an easy consequence of the first one. First we take a > O small enough so that the 2ABa-neighbourhood of K is contained in the domain of IJ!. Clearly, we may assume that the same holds fork and IJ.!.For U = {Un} a presentation of K, u a boundary point of a bounded gap U = Un and C the U component of K at u, take k 2: 0 minimal such that .e(IJ.!k(UUC)) 2: Ba. Then .e(wk(UUC)) ::; ABa (because f(wk- 1 (U u C)) ::; Ba) and so .e(IJ.!k(h(U) U h(C))) ::; ABa + 2a8 ::; 2ABa (as long as 8 ::; '\8 ). By the bounded distortion property we have

(1) and -c(2ABa)

e

< [(.e(IJ.!k(h(C)))) ;.e(h(C))l -

.e(IJ.!k(h(U))) ,


lik(C))·

I

t(h(>lik(U)))-t(>lik(U)) t(>li

< ABa-2a8+ABa-2a8 a • (a - 208)

+t(>llk (U))•t(h(>li

(V)>• lt(h(>lik

(C)))-t(>llk

(C))

(U)))

= 8 4AB

1 - 28

(3)

88

4 Cantor Sets in Dynamics and Fractal Dimensions

If 8 > 0 is sufficiently small this implies

I

e(wk(C)) £(\Jlk(h(C))) l'(IJ!k(U)) - £(\Jlk(h(U))) $

l

(3a )

€.

From (la), (2a) and (3a) it immediately follows that

T(K, h(U), h(u)) $ (1 + e) • ((1 + e) • T(K ,U,u) and

+ e)

T(K, h(U), h(u)) 2: (1 + e)- 1 ((1 + e)- 1T(K,U , u) - e).

(4a )

(4b )

!,et now l'(wk(U)) $ a . Then we must have l'(wk(C)) 2: Ba-a. Moreover l'(wk(h(U))) $ a+ 2a8 and £(\Jlk(h(C))) 2: Ba - a - 2a8 . This together with (1) and (2) implies l'(C) > l'(U) -

e- c(ABo).

Ba - a _ (B _ l) -c(ABo a e

)

(5)

and

l'(h(C)) > e- c(2AB o) . Ba - 2a8 - a l'(h(U)) a+2a8

=B-

28 - 1 . -c( 2AB o) 1+28 e

(6 )

Since we have chosen B = 20(K) + 8, we can suppose a and 8 small enough so that these relations imply

T(K,U, u) 2: (B(K)

+ 3)

(5a )

and

T(K, h(U), h(u)) 2: (B(K)

+ 3).

(6a )

Now we proceed to prove the affirmatives (a) through (d) stated near th e beginning of the proof. Recall that by definition

T(K)

= sup inf T(K,U, u),

B(K)

= inf

u

u

u

sup T(K,U, u). u

To prove (a) we must find for any given U a presentation U of k such that s\lp T(K,U, u) $ (1 + e}2sup T(K,U, u) + e(l + e) (al. ) u

u

There is no loss of generality if we assume that

supT(K,U, u) $ B(K) u

+ l.

(a2 )

4-3 Local invariants and continuity

89

Take U = h(U). From (a2) it follows that T(K,U, u) $ B(K) + 1 for all u and so (5a) never holds . Then , by the previous discussion we must have

(as well as (4b)) for all u. This immediately implies (al) and so (a) is proved . The proof of (b) is almost dual to the preceding one so we don 't writ e it down in detail. The only asymmetry comes from the fact that (6a) involves 0(K ) and not B(K). This is bypassed as follows. First , we may as above suppose that (b2) sup T(K,U, u) $ 0(K) + l. u

Now, from (a) (which we have already proved) we get that if k is sufficiently near K then B(K) $ B(K) + l. Then (b2) implies sup T(K ,U, u) $ B(K) + 2

(b3)

u

and now th e argument proceeds as before. To prove (r) we take , for each U, U = h- 1 (U) and show that (cl)

To do this we must to each u associate

u such

that (c2)

Again, it is sufficient to consider the points u for which

T(f(,U,u) :S inf T(K,U,u) + l. u

(c3)

Take u = h(u) and observe that if (c3) holds then T(K ,U,u) $ T(K) + 1 :S 0(K) + 1 and so (5a) doesn't hold. Therefore (4a) is true , and this is just (c2). T he proof of (c) is complete. The proof of (d) is dual to the one of (c) (recall also the remark in the proof of (b)) so we are done with proving the continuity of (global) thickness and denseness . Finally, recall that the local thickness of a Cantor set K at a point k E K is defined by Ti oc

(K, k)

= lim(sup{ T(K1)\K1 CK n B 0 (k) a Cantor set }). o-o

~et e > O be small. Given 8 > 0, take 8 > 0 such that h(K n B 0 (k)) c Kn B1;(h(k)). Let K 1 be a Cantor set in Kn B 0 (k) and let k 1 = h(K 1). If

90

4

Cantor Sets in Dynamics and Fractal Dimensions

h is close enough to the identity (i.e. if k is close enough to K ) th en the arguments above imply r(K 1) ;:::r(Ki) - c:. Since K1 is arbitrary it follows that

sup{r(K1)IK1 CK n B6(k) a Cantor set} ::; sup{r(K1)!K1 Ck n B 6(h(k)) a Cantor set}+ By making

8-+ 0

(and sob-+ TJ oc

c:.

0) we get

(K , k) ::; TJoc (K, h(k))

+€.

In the same way one shows TJ oc

(K, k);:::TJoc (K, h(k)) -

€.

This shows the continuity of local thickness. For local denseness the argument is the same. The proof of the theorem is now complete. □ REMARK 1: Consider a C 3 diffeomorphism r.pof a surface, with a basic set A and a saddle point p E A. For (p a C 3 nearby diffeomorphism th ere are A, a basic set, and p E A, a saddle point (near A and p, respectivel y), and the dynamically defined Cantor sets W"(p) n A and W"(p) n A are near in the above sense (if we take nearby parametrizations for W "(p) and wu(p) as in Section 1 of this present chapter). This follows from the cont inuous dependence on the diffeomorphism of basic sets and .their ci+ e:-stabl e and unstable foliations; see Appendix 1 and Remark 2 in Appendix 2, concern ing continuous dependence of Markov partitions. From this and the propo sitions that we just proved, we deduce the continuous dependence , with respect to the diffeomorphism in the C 3 -topology , of all the invariants of w u(p) n A that we have discussed , namely Hausdorff dimension, limit capacit y, thickness and denseness. To show this , one uses the arguments above together with the observation that C 3 diffeomorphisms and C 3 closeness are used only to obtain c 1+< expanding maps with nearby Holder constant s for the derivatives. This in turn provides bounds for the distortion of distances which are uniform in neighbourhoods of the diffeomorphism and the Cantor set. But, as we remarked before, at the end of Section 1, C 2 diffeomorphi sms induce Cantor sets satisfying the bounded distortion property (and the resulting expanding maps are indeed ci+e: for some c: > 0). The ar gument that we used there also yields uniform estimates for the distortion in a C 2 neigbourhood of the original diffeomorphism. Thus , all the above in variants of W"(p) n A depend continuously on r.p in the C 2 topology.

4-3

Local invariants and continuity

91

For Hausdorff dimension and limit capacity, one can go even further: in [PV ,1988] it is proved that the Hausdorff dimension and limit capacity of W"(p) n A depend continuously on the diffeomorphism in th e C 1 topology. This is done by using, as above, conjugacies with Holder constants near 1. We observe that this result had been obtained in [MM ,1983] as a consequence of a variational principle of the thermodynamical formalism.

CHAPTER

5

HOMOCLINIC BIFURCATIONS: FRACTAL DIMENSIONS AND MEASURE OF BIFURCATION SETS

In this chapter we bring together the Theory of Fractal Dimension s and Bifurcation Theory in Dynamical Systems. A number of (mostly recent) results, leading to further questions and conjectures as laid out in Chapter 7, shows that the first theory is of fundamental importance to the second, at least in the context of homoclini c bifurcations of nonconservative (say dissipative) systems. The result s ca n be stated in great generality if we focus our attention on the maxi mal invariant set of the restriction of the dynamics to a neighbourhood of the orbit of homoclinic tangency and an associated basic set. These results become of a global nature when global assumptions are made concerning filtrat ions and hyperbolicity of the positive or negative limit set. We now explain further the results, but leave the formal statements for t he sections following this introduction. So, in this chapter, we will consid er oneparameter families 'Pµ of surface diffeomorphisms which, as the param ete r varies, go through a homoclinic tangency say at µ = 0 which we assum e to be parabolic and to unfold generically; see Chapter 3. We want to know how big in the parameter space, near this bifurcating point, is the set of values that correspond to diff eomorphisms with a hyperbolic limit set. A main resul t here states that this set has a relatively large Lebesgue measure if we assu me small limit capacities (or Hausdorff dimensions) of the stable and un stab le sets of A, where A is the basic set associated with the homoclinic tan ge ncy. In this chapter we discuss the main ideas of the proof, leaving a compl ete presentation of it to Appendix V. We recall that the stable set W 8 (A) of a basic set A is the uni on of the stable leaves through points of A. In the present context of surfa ce diffcomorphisms, which we now assume to be at least C 2 , this stable foliati on is C 1. So the essential structure of W 8 (A) appears in its intersection with a curve e transverse to the stable leaves- since the foliation is C 1, Hau sd or ff dimension and limit capacity of W"'(A) n fl, are independent of fl, (and t hey are equal to each other as we saw in the previous chapter). In parti cul ar we may take, for a saddle point p E A, e = wu(p). We then define the stable limit capacity or Hausdorff dimension d5 (A) as d(W 5 (A) n w u(p)): the unstable limit capacity is similarly defined. Similar results for heteroclinic cycles as well as corresponding resul ts in higher dimensions are Htated in the last section. We also state a par tial

5.1 Construction of bifurcating families of diffeomorphisms

93

converse concerning nonhyperbolicity of maximal invariant sets when the ab ove Hausdorff dimensions are large (sum bigger than 1). As mentioned before, the result when the Hausdorff dimension is small (sum less than 1) becomes more of a global na~u:e when th~ ho1:1o~linic t angency occurs as first bifurcation and the pos1t1ve or negativ~ limit set of t he bifurcating diffeomorphism is hyperbolic: for a set of relatively large me asure in the parameter space the corresponding diffeomorphisms have thei r global limit set hyperbolic. The first result concerning relative measure of bifurcation sets (but not dea ling with fractal dimensions) was obtained in [NP ,1976], where homoclinic bifurcations from Morse-Smale diffeomorphisms was treated. The res ult we present here when applied to that case is somewhat stronger.

§1 Construction phisms

of bifurcating

families

of diffeomor-

We begin this section by indicating an interesting example of a persistently hy perbolic diffeomorphism on the 2-sphere 8 2 with infinitely many periodic or bits: the diffeomorphism itself as well as every small perturbation of it has a hyperbolic limit set. It is in fact similar to the horseshoe example from 2 Ch apter 1 but it is constructed in 8 2 ~ JR2 U oo instead of in JR . Many ot her examples, and in fact a full discussion about constructing a homoclinic tan gency as a "first" dynamic bifurcation, can be found in Appendix V. We take in 8 2 the diffeomorphic image of a square Q with two semicir cular discs D 1 and D 2 attached as indicated in Figure 5.1.

Q

Figure 5.1

94

5 Homoclinic Bifurcations

We let cp map Q U D1 U D2 inside itself as indicated in Figure 5.2 , i.e. so that in Q we have the above horseshoe example and in D 1 we have one hyperbolic sink S1 , attracting all points in D 1 .

(/J (Q)

Q

Figure 5.2 We extend cp to the complement of Q U D 1 U D 2 in S 2 in such a way that there is only one hyperbolic source S0 and such that for each x E 2 S - (Q U D1 U D2), lim cp-n(x) = So. It is easy to verify that in this case n ...... oo the positive limit set of cpconsists of S1,So and the maximal invariant subset in Q. This last set can be analysed as in Chapter 2 and it is hyperbolic. We let p denote the fixed saddle point in Q as indicated in Figure 5.2 and let A denote the maximal invariant subset of Q. The Cantor sets W 8 (p) n A and wu(p) n A are clearly dynamically defined (if cp is C 3 and if we take a correct identification of W 8 (p) and wu(p) with IR); these are the Cantor sets to which the results of Chapter 4 will be applied. The first bifurcating family of diffeomorphisms that we construct in this section is based on the above example of a diffeomorphism cp of S2 with a horseshoe, a source, and a sink. For simplicity we assume cp to be of class C 00 , although most results in this chapter are true for C 2 diffeomorphism s. Let C be a curve from rs E W 8 (p) to r,, E wu(p) as indicated in Figur e 5.3. U denotes a small neighbourhood of c which is divided by the loc al components of W 5 (p) nu and wu(p) nu containing rs and r,, in the region s U1, Uu, and Uu1- We shall obtain our one-parameter family by modif ying the map cp in U, i.e. by composing cp with Wµ, Wµ a one-parameter familv · of diffeomorphisms which are, outside U, equal to the identity. Before we describe Wµ, we analyse the dynamic properties of orbits pas sing through U; we assume that this neighbourhood U of c is sufficiently small

5.1 Construction of bifurcating families of diffeomorphisms

95

p

Figure 5.3 so that the following considerations are valid; see Figure 5.4. If x E U1UU11 then cpn(x) tends, for n --+ +oo, to the sink S1 and if x E Uu U Uni then .pn(x) tends, for n --+ -oo, to the source So. For x E Uu1, the positive iterates cpn(x), n --+ +oo, will stay near wu(p), but apart from that they may go to the sink or may stay near A (notice that A= wu(p) n W 8 (p); see the construction of cp above); in any case there are points x E U111such that ;pn(x) E U1 for some positive n. Similarly for x E U1, the negative iterates .p-n(x), n --+ +oo, will stay near w•(p), but apart from that they may go to the source or may stay near A; in any case there are points x E U1 such that 0.

- - ....

.... /

/

Ui

I

I

''

\

~

I I

I ,c

Uu I I

~W'(p)

'

Um

- - - .... Figure 5.4

/

wu(p)

96

5 Homoclinic Bifurcations

Now we come to the description of the one-parameter the points in U. We take wµ so that - for µ :S -1,

wµ

family

wµ

moving

is the identity,

-1, Wµ pushes points down in U (in the direction of Uur) so that, for µ < 0, Ur is still mapped inside Ur U Uu,

- forµ>

8 - for µ = 0 there is a tangency of '1>" 0 (Wu(p)) and W (p), or more precisely of Wo(Ur n Un) and Un n Uru; this tangency has quadratic order of contact and unfolds generically for µ > O into two transversal intersections. In Figure 5.5 we indicate the stable and unstable manifolds of p for the diffeomorphism Wµ o .a< 1, where 0 < >.< l and a > l are the eigenva lues of (dt.p)p- We say that >.is the dominating eigenvalue. Exactly as in the first example, we only modify t.pin a small neighbourhood of the curve C joining the points ruin wu(p) and Ts in W 8 (p) as in the figure. We define Wµ as before and '{)µ= Wµ o t.pso that we obtain a homoclinic orbit of tangency 0 for 0 in the above definition of UµNot e that the choice of a larger constant K leads to a slower rate of convergence of the limit.

§3 Homoclinic tangencies with bifurcation relative measure - idea of proof

set of small

In this section we want to outline the proof of the first theor em in the previous section. The proof of the more general and even global th eor em is the same except for a careful analysis of the consequences of the requir em ent that 0, we show that for µo sufficiently small, m(B( 0 and small. Then, to begin with. we must assume the transversality condition for all stable and unstable manifolds of cpo except along the (unique) orbit of tangency occurring say in wu(A;,) n ws(A;J. Besides, one has also to consider the unstabl e limit capacities of the basic sets Ak such that wu(Ak) n W 5 (A;,) =I-¢. Each such Ak is said to be positively involved with the tangency since wu(Ak) contains

5.4 Heteroclinic cycles

109

(a)

s1, s2: fixed saddles r1, r2: fixed sources a 1 , a2: fixed sinks (b)

s 1 : fixed saddle in a horseshoe s2: fixed saddle r 1 : fixed source a 1, a2: fixed sinks Figure 5.12 the orbit of tangency in its closure. Similarly, A.- iH negatively involved with the tangency if W·'(Ak) internects W"(A;,). We define the unstable limit capacity of the tangency d;' as the maximum of the unHtable limit capacities of the basic Hcts positively involved with the tangency. Similarly, we can define

llO

5 H omoclinic Bifurcations

the stable limit capacity of the tangency df. For a periodic orbit which is a basic set we define these limit capacities as zero even when they are sources or sinks. On the other hand for an attractor which is not periodic the stable limit capacity is 1; similarly for a repeller which is not periodic. as above we have the following. So, for a family O µo where B 9 (T,-1 ~

which were again discussed in the previous section. There, for ji, near - 2 and n big, we denoted by Pn(ji,) the fixed point of 'Pn,µ which is close to (x,y) = (2, 2). Corresponding to this fixed point, there is a periodic point with period n + N of ,Pµ which we also denote by Pn(µ). (Whenµ, ji,, and n appear simultaneously, it is understood that they are related by µ = Mn (ji,), see the theorem in Section 4, Chapter 3). In this section, which is entirely dedicated to the proof of the next propostion, we analyse the stable and unsta ble separatrices of Pn(µ) for µ near Mn(-2) . PRO POSITION l. For ji, near -2, there are compact arcs a;(ji,) and a~(ji,) in W 8 (Pn(ji,)) and wu(Pn(ji,)) containing Pn(ji,) and converging, for n --; '.X), to an arc in W 5 (po), respectively in wu(Po), containing at least one fundamental domain (Po is the saddle point of 1 and argui ng as in the proofs of the corollaries in Section 1 of this chapter, we obtain after thi s last perturbation a diffeomorphism which is in the boundary of an open set 2 UC Diff (M) with persistent homoclinic tangencies , as requir ed. REMARK 1: As a consequence of the main result in this chapter , we haw that the diffeomorphisms with a hyperbolic limit set are not C 2-dense in the space of all C 2 surface diffeomorphisms . (Similarly, the £-stable or n-st able

6. 6 Sensitive chaotic orbits near a homoclinic tangency

131

diffeomorphisms are not C 2-dense) . This fact, however is still not known in the C 1 topo logy, so we pose the following 1: Are the diffeomorphisms with a hyperbolic limit set C 1-dense in the space of all surface diffeomorphisms? PRO BLEM

§6

Sensitive

chaotic orbits near a homoclinic

tangency

In this final section we want to discuss the results that we have obtained so far in terms of occurrence of sensitive orbits and strange attractors. First , a homoclinic tangency, and its unfolding, give rise to hyperbolic basic sets of saddle type; in these basic sets, most orbits are sensitive. On the ot her hand , for a diffeomorphism in the plane near a homoclinic tangency , associa ted to a dissipative (area contracting) saddle point, there can be no nontri vial hyperbolic attractor near the orbit of tangency: such an attractor conta ins "holes" in its basin where the map must be expanding, see Plykin [P ,1974], which is impossible in the dissipative case. So, as long as th e dynamic s is hyperbolic , the chaotic orbits can only occupy a set of Lebesgue measure zero. Secondly, there is the phenomenon of the coexistence of infinitely many periodic attractors (sinks). Generically, the number of periodic attractors, with p eriod smaller than some constant, is finite. Thus, if there are infinitely many attractors, most should have very big period. Of course, periodic attrac tors are not sensitive, but in numerical experiments, where one can analyse only a finite part of an orbit, these periodic attractors of very high period may look chaotic . Finally, although we cannot expect nontrivial hyperbolic attractors, we do expect non-hyperbolic strange attractors. This is based on the fact that in the unfo lding of a generic homoclinic tangency there are Henon-like families of diffeomorph isms and for such families Mora and Viana [MV ,1991] proved the exis tence of "persistent strange attractors" . See the next chapter.

CHAPTER

7

OVERVIEW, CONJECTURES AND PROBLEMS A THEORY OF HOMOCLINIC BIFURCATIONS STRANGE ATTRACTORS

-

Based on recent developments , conveyed in the previous chapters and further discussed here, we now present some perspective, and indeed a program me. concerning homoclinic bifurcations and their relations to chaotic dynami cs. Actually, we consider homoclinic bifurcations as a main mechanism to unleash a string of complicated changes in the dynamics of a diffeomorphi sm (or, more generally, an endomorphism). Indeed, as we have seen, the one-parameter unfolding of a homo clinic tangency yields a striking number of dynamical phenomena: - hyperbolic Cantor sets (Chapter 2), - cascades of homoclinic tangencies (Chapter 3), and for locally dissipative surface diffeomorphisms · - cascades of period doubling bifurcations

(Chapter 3),

- relative prevalence of hyperbolicity of the limit set in a significan t number of cases (Chapter 5, Appendix 5) and its converse (Chapter 7), - infinitely many sinks (Chapter 6, Appendix 4), - Henon-like strange attractors, as proved in [MV,1991] extending the work in [BC,1991], to be discussed in this chapter. Thus, homoclinic bifurcations embody most of the known bifurcati ons of a nonlocal character, at least in the setting of surface diffeomorphis ms or three-dimensional flows without singularities. On the other hand , th ere is some evidence, still quite limited, that a nonhyperbolic diffeomorphi sm exhibiting (one of) the above complicated phenomena might be approxim ated by one exhibiting a homoclinic bifurcation. These considerations lead us to propose a number of related questions that when put together poin t toward a theory of homoclinic bifurcations . Our programme, as it will be discussed in this chapter, consists of (1) to determine a dense subset rt of all dynamical topology such that if cp E rt, then either -

n

Recall that if Po, ... , Pn are partitions, then

V Pi is the

partition whose

j =O

elements are sets of the form Pon•· -nPn for P1 E P1 and µ(Pon•· -nPn) > 0. Finally, we define hµ(cp) the (metric or measure theoretic) entropy of cp with respect toµ, as the supremum of hµ( cp,P) over all finite partitions of supp µ. \"otice that hµ(cp- 1) = hµ(cp).

This concept is also based on the intuitive notion of entropy that we have first presented. Instead of pursuing this point, which was quite clear with respect to the topological entropy , we now state the following important fact: the topological entropy of cp is the supremum of the metric entropies of cp with respect to all invariant probability (ergodic) measures. As a consequence, if h(;p)> 0 then there are invariant probability measures, even ergodic ones,

with respect to which the metric entropy is also positive. We refer the reader [B,1975a], [M,1987], [S,1976], [W,1982] for more references, details and some history. We now present a theorem of Oseledec [0,1968], the multiplicative ergodic theorem, that together with subsequent work of Pesin ([P,1976], [P,1977]) has very much influenced a basic line of research that allows one to extend results valid for hyperbolic diffeomorphisms, e.g. the stable manifold theory, to other classes of maps. We say that x EM is a regular point for cp if there are numbers >-1(x)> · · · > >-e(x) and a decomposition to

such tha t

for v E E j , v -=I-0, and 1 :::;j :::;£. The numbers ,\1(x) are called Lyapunov exponents of cp at x. The multiplicative ergodic theorem states that for every invariant probability measure µ, the set of regular points A is a Borel subset and µ(A) = l. Moreover, the maps x -> Aj(x) and x -> E 1(x) ar e measurable . If µ is ergodic, then the A1 's as well as the dimensions of the corresponding E/s arc constant a.e. (almost everywhere). We say that cp is nonuniformly hyperbolic if ,\1 (x) -=I-0, x a.e. in A and all IS j S £.

146

7 Overview, Conjectures and Problems

For nonuniformly hyperbolic C 2 (or even Cl+ °', a> 0) diffeomor phis ms, Pesin developed a "filtered" hyperbolic theory for the set A of regular poin ts. That is, ther e are closed, but in general not invariant, hyperbolic sets A 1 C - - - C Ak C · · ·, k E N and Ak = A. The main difference with the

LJ k

usual hyperbolic structure on A is that , as k increases, it may tak e long er and longer to detect the hyperbolic behaviour of dr.pon Eu(x) an d E 8 (x) for x E Ak, where Eu(x) and E 8 (x) denote the subspaces generat ed by the positive and negative Lyapunov exponents at x . Assuming that the measu re is ergodic and denoting by .X+ = min .X1, .X1 > 0, and by .X- = max .X1 , .X1 < 0. we have

ll(dr.pn)xvll ~ k - Ie>.+-t: llvll :S k e>.- +t: llwll ll(dr.pn)xwll for x E Ak, v E Eu(x), w E E 8 (x), n E N and E > 0 small . Moreov er, the stable and unstable sets of x E A are injectively immersed subm anifo lds of M, denoted , as in the hyperbolic case, by W 5 (x) and wu(x). (Noti ce, however, that in this case the angl es between Eu (x) and E 8 ( x) , x E Ak, m ay go to zero ask----->oo.) Thus, in the nonuniformly hyp erbolic case, many vectors will grow ex ponentially when it erating r.p.Vectors can be interpreted as "two infini tely close points" and, hence , we get exponential growth of their distanc es, whi ch implies, informally speaking , exponential sensitive dependence on the initial conditions . It seems reasonable to even expect that full sensi tivity is formally implied just by the existence of a positive Lyapunov expon ent. Based on the work of Pesin, Katok [K,1980] proved that the m axim al invariant set in each Ak is hyp erbolic with a dense subset of hyp erbol ic periodic points all of them exhibiting transversal homoclinic orbi ts. T his pretty result corroborates our pr evious comments about th e sens itivit y of the orbits. Moreover, since in general we have

known as Ruelle's inequality [R,1978], we conclude that if dim M = 2 and h( 0 then r.pis non uniformly hyperbolic with respect to som e ergo dic an d probability measureµ. In fact, there isµ such that h1,(r.p)is close to h(r.p) thus h1,(r.p)> 0. This implies the existence of a strictly positiv e Lyapun ov exponent by Ruclle 's inequality. Similarly, since h1,(r.p-1) = hµ('P), we get a strictly negative Lyapunov exponent , proving the assertion. Notice that. for invariant ergodic measures that arc absolutely continuous wit h rcsp 1. As we saw , residually in some intervals in the /t-paramctcr line, there arc infinitely many sinks for the corresponding maps cp1,. For the Henon-likF

150

7 Overview, Conjectures and Problems

strange attractors we can ask if it is possible that infinitely many of them occur simultaneously when a homoclinic tangency is unfolded. In any case. we have the following conjecture stating that the simultaneous exist ence of infinitely many sinks or Henon-like strange attractors might be a "rare'· phenomenon. CONJECTURE 1. The set ofµ 's for which 1, to ensure the existence of tangencies between leaves of F 5 (Aµ) and ?(Aµ) for every µ E (-£, 0) orµ E (0, c:), one can ask if the condition is necessary to achieve this persistence of tangencies. In work in development, Yoccoz and the first author of this book show that this is not the case for an open set of C2 diffeomorphisms. This leads to the following very interesting question . PROBLEM 3: Is there a necessary and sufficient condition, say involving fractal dimensions of the basic set A0 , to obtain an interval of the form (-c:,0) or (0,c) such that for eachµ E (-c:,0) orµ E (0, c:) th ere is a tangency between leaves of the stable and unstable foliations of Ao? Related to Problem 1 there is a relevant recent result of Diaz, Rocha and Viana [DRV,1991], stating that a saddle-node critical cycle correspond s to a point of positive density of Henon-like strange attractors for generi c oneparameter families of surface diffeomorphisms. That is, if SA( 0.

It is interesting to note that, by a remark of Mora, saddl e-node critical cycles are always present when generically unfolding a homoclinic tangc nc~·This can be seen as follows . As first pointed out in the last sect ion of Chapter 3 and much used in Chapter 6, we can study properties of cpl'that are "robust" under perturbations of the interval map y --+ l - ay 2 , since for some fixed integer N this is the limiting map as n--+ oo of 0. The key hypothesis in the results above is the hyperbolicity of th e fixed point p, i.e. the existence of a d,p-invariant splitting TpM = E• EBEu such that d 0 small enough in the construction of V (recall the outline of this construction in Theorem 5), then Vis We use this to justify our claims about the differentiability of the invariant sub manifolds introduced above. It is not difficult to check that V is rnor mally hyperbolic at p if and only if

er_

(1) when ever )q, >.2are eigenvalues of (dcp)p with l>. Let us 1 1 < a < i>21 consi der first the centre-stable manifold. Clearly, in this case we may take r

162

Appendix 1 Hyperbolicity: Stable Manifolds and Foliations

arbitrarily large satisfying (1). Hence, if cp is Ck, k finite, then w cs(p) may be chosen Ck, as we claimed . We point out that this is not true for k = oo: in general C 00 diffeomorphisms do not have C 00 centre-stable manifold s [S,1979], although they do have them of class Ck for any k ~ l. Thi s is related with the fact that wcs (p) is in general not unique. Analogo us comments hold for the centre-unstable manifold and the centre manifol d. Consider now the strong stable manifold. Again, we may take r as in (1) with arbitrarily large value, so W 88 (p) is Ck if cp is Ck. But now , due in part to the uniqueness of W 88 (p), this is still true for k = oo. The same holds for the strong unstable manifold, the stable manifold and the unstabl e manifold. We now want to use the theory of normal hyperbolicity to study the stable and unstable foliations, defined in a neighbourhood of a basic set of a surface diffeomorphism cp:M -+ M. Here, we are mostly concerned with the differentiability of these foliations. We first treat in some detail the fixed point case and then show how the conclusions can be extended to general basic sets. For diffeomorphisms cp: M -+ M as above, we construct a lift L(M)

1

M

.E£. L(M) _.£...

1

M

where L(M) = {(x, L) Ix EM and Lis a one-dimensional linear subsp ace of TxM} and Dcp is the diffeomorphism induced on L(M) by the deriv ative of cp. Clearly, if cp is Ck then Dcp is ck-l _ Let :F be a foliation of an open subset U of M by differentiable curves. Then U = {(x, L) Ix EU and Li s the tangent space of the :F-leaf through x} is a surface in L(M) and :F is a cp- invariant foliation if and only if U is a Dcp-invariant manifold . If U is then :Fis (i.e. there are local coordinates which trivialize :F) and we even have in this case that the tangent spaces Tx:F(x) to the leaves of :F depend on X. Let now cp be a C 2 diffeomorphism of M = IR2 and O E IR2 be a fixed

er

er

er

er

point of cp, with (dcp)0

= ( ~ ~),

0 < jaj < 1 < j>.j. If Eu denot es the

unstable eigenspace of (dcp0 ) then ou = (0, Eu) E L(IR2 ) is a fixed point of Dcp. The three eigenvalues of d(Dcp)ou are a>.-1 ..Hence , there is a two-dimensional locally invariant C 1 manifold U containing ou, such that TouU is the eigenspace of d(Dcp)ou corresponding to the eigenvalues a and >.. Note that [J projects diffeomorphically onto a neighbourhood U of 0 E IR2 . Then , there is a (unique) foliation :,:U such that TxP(x) = L for all (x, L) EU. Clearly, :,:U is a cp-invariant C 1 foliation and P(0) c wu(o)

Appendix 1 Hyperbolicity: Stable Manifolds and Foliations

163

(because :,:U(0)is an invariant submanifold tangent to Eu at the origin), so of 0 E IR2 • Moreover, one easily checks that U is r-normally hyperbolic at ou for all r < 1-log >./log a. Hence, if cp is C 3 and 0 < £ < - log>./ log a, £ :=:;1 then U is Cl+c, and so :,:U is a C 1+E foliation. In the same way we construct a C 1 stable foliation :,:snear zero and we show that :,:scan be taken of class c1+0 if cp is C3 . Before proceeding to the case of a general basic set, let us derive a corollary of the discussion above, which was much used in the text . :,:U is an unstable foliation on a neighbourhood

T HEOREM

7 [H,1964]. Let cp be a C 2 diffeomorphism of a surface Mand

p E M be a saddle point for cp. Then cp admits C 1 -linearizing coordinates near p. If cp is C 3 such coordinates can even be· taken to be C 1+c for some € > 0.

We detail the proof of the first part of the theorem; the C 3 case follows from the same argument. First we construct C 1-linearizing coordinates for cp j W 8 (p). Let a: IR -+ W 8 (p) be any C 2 parametrization of W 5 (p) with a( 0) = p, and define f: IR-+ JRby

Th en OE JRis an attracting fixed point for f with f'(0) = a, the contracting eigenvalue of (dcp)p- It is not difficult to check that h(x) = limn-+oo1:~x) is a well defined C 1 function in a neighbourhood of O ([S,1957]) . Clearly, we have h(f(x)) = a· h(x) for all x near 0. It follows that ~ = ho a - 1 (x) is a C 1-linearizing change of coordinates for cp I W 8 (p) near p, as we wished. Replacing cp by cp-1 , we get in the same way a C 1 coordinate, say 'r/, defined on wu(p) (near p), relative to which cp I wu(p)(ry) = >.rt,>.being the expanding eigenvalue of (dcp)p- We now define coordinates (~, rt) in a neighbo urhood of p as follows. Let :,:s,:F" be the stable and unstable foliations constructed above. For~ E W 5 (p) and 'r/ E wu(p) we take (~, ry) to be the int ersection point of P(~) and :F5 (ry). Since :,:sand :,:U are C 1 and transversal to each other this really defines C 1 coordinates near p. Moreover, due to t he invariance of the foliations we have

so the proof is complete. Now we indicate how to obtain stable and unstable foliations in a neighbo urhood of a basic set A of a two-dimensional diffeomorphism cp. One way that this can be done is by the limiting process described in Chapter 2 for the case of a horseshoe. A different but related proof involving the >.-lemma, wh ich we outlined above for the case of a hyperbolic fixed point, can also

164

Appendix 1 Hyperbolicity: Stable Manifolds and Foliations

be used to construct these foliations for any basic set of a surface diffeomorphism. Using this approach, we sketch the construction of the unstabl e foliation P and refer the reader to [M,1973a] for more details; of course , th e stable foliation is obtained in a similar way. After this is done, we discus s the differentiability of the foliations when the diffeomorphism is of class C 2 or C 3 . We observe that if the diffeomorphism is only C 1 , as considered in [M,1973a], then the foliations are in general not differentiable. To begin with define P in a fundamental neighbourhood of w s(A) that is in an open neighbourhood N of Wt (A) - 0, such that N n A = 0. That such a neighbourhood exists results from the fact that A is isolated [HPPS,1970] (see also [N,1980], [S,1978]). The definitio n of P on N must be done in such a way that 0, n big, cpn(x) is near the boundar y of cpn(U) and, hence, near wu(p) U W 8 (p). Since the part of the boundar y

Appendix 3

179

formed by W 8 (p) has decreasing length, cpn(x)is close to wu(p). This means □ that w(x) c wu(p) and so the proposition is proved. Note that this proposition explains that certain attractors are contained in wu(p), but not that they fill out all of it. In fact, there could be only one periodic attractor in wu(p). This is the reason why one is interested in nonexistence results for periodic attractors proved by Benedicks and Carleson [BC,1991). It is not hard to see that the above proposition applies to the Henon R.2 given by H(x, y) = (1 - tx 2 + y, fox); the existence of a map H: R 2 ------> homoclinic intersection was proved in [MS,1980]. In fact both Szewc and Tangerman proved more, namely that one can, in the case of the Henon map, choose U to be a neighbourhood of wu(p). Another interesting case can be obtained from the second example in Chapter 1, the pendulum, through small perturbations . With a first perturbation, we make the diffeomorphism, or even the differential equation attracting towards the {E = 1} level without perturbing the dynamics inside this level. With a second perturbation, we make transverse homoclinic intersections in all the branches of the separatrices of the saddle point (1r, 0) (and hence destroy {E = 1} as an invariant set). Although this example is defined not on the plane but on an annulus, the arguments of the proof of the proposition above still work.

APPENDIX

4

INFINITELY MANY SINKS IN ONE-PARAMETER FAMILIES OF DIFFEOMORPHISMS

In this appendix we shall prove a refinement of the main results of Chap ter 6. In that chapter we considered one diffeomorphism cp with a homo clinic tangency, and then proved that near this diffeomorphism there are man y diffeomorphisms with infinitely many sinks. Here, we consider a one-param eter family of diffeomorphisms cpµwith a homoclinic tangency (forµ= 0) , which satisfies certain generic properties , and then show that within this oneparameter family there are many diffeomorphisms cpµ, µ near zero, with infinitely many sinks (or sources). Such a result, based on Newhouse 's original work, was obtained in [R,1983) . We first point out that we can get this result by a rather formal argum ent in the following way. Let D = Difl'2( M) be the space of C 2 diffeomorphs ms on a 2-manifold M. Let :EC V be a codimension-1 submanifold consisting of diffeomorphisms with homoclinic tangency. We assume the usual generic properties (quadratic tangency and ldet dcpl at the saddle point different from 1) to be satisfied for all diffeomorphisms cp E :E. We shall consider one-parameter families µ 1---7 cpµ which intersect :E transversally for µ = 0. First, however, we reformulate the main result of Chapter 6. Let U C D be the (open) set of those diffeomorphisms cp which have a neighbourhood V such that, for each integer m, there is an open and dense subset V(m) c V so that each cp' E V(m) has at least m periodic sinks (or sources). From the results in Chapter 6, it follows that :E c U. Thi s means that, for any integer n, the following property is generic for one-param eter < 1/n , which families 0, the diffeomorphism .

( :::: means: equal up to a multiplicative factor which is bounded and bounded away from zero, uniformly in d ). Since O < >..< 1 < a and >..- 1 > a, the distance from cp- io(dl(R) to W 1:.C,(p)is much bigger than d (ford small). For some integer j 0 , independent of d, r E q:,- i0 (W 1:,.,(p}}; here W 1:,.,(p) is the interval in W '( p) consisting of points at distance less than one from p . Then the distance from cp- i( dol- i 0 ( R) to q:,- i 0 ( W1:,.,(p)) is also much bigger than d (for d small). So we have the following situation

~ c_

W ,::_,( pl

• ""'' 0 for h E (0, 1), M(h , A.(0)) attains its maximum for h = e- 2. Then Loge - 2 M{e - 2, A.(0)) = Log A.(0) .

(

e -2 ) 1/ 2

A.(0)

112 (A.(0)) . Log A.(0)

- 1 0, we have to choose r(e) so that O < r(e) < r and so that E > L · 0 113 ((K + I)· r(e)) . For such a value of r(e) the conclusion of the proposition holds. 3. Diffeomorphic images of scaled sets and the final result . PROPOSITION . Let A(µ) be a continuous I-parameter family of scaled sets and let \Jrµ:R --+ R be a continuous I-parameter family of C 1-diffeomorphisms with '1',,(0) = 0 and 'lr:(0) = I. Let K > 0 be a constant. Then, for every E > 0 there is an r( e) > 0 such that

A ,. ,(µ):) forallr,

('1',,(A{µ))

n [- (K +I) · r,(K + 1) · rl)

µ E [O,r(e)] .

This is a simple consequence of the fact that forµ

'lr,,(x)-x

---for

lxl --+ 0,

lxl

E

[O, r(e)] ,

-->O

uniformly in µ; see also [11].

*

In the case that 'lr;(O) I, one has to replace, in the conclusion, A, .,(µ) by the (r · e)-neighborhood of 'lr;(o) · A(µ) . In Section 1, we investigated the set of values of t such that A has an almost intersection with B, shifted over a distance t. In Section 2 we analyzed I-parameter families of scaled sets depending continuously on a parameterµ . In the1 present section we gave an estimate on the deformation of a scaled set by a C -diffeomorphism depending on µ. Combining all this we obtain the following theorem. THEOREM . Let A(µ) and B(µ) be two continuous I-parameter famili es of R--+R be two continuous scaled sets so that d(A) + d(B) < I. Let 'lr,,,'1>,,:

Appendi:x 5

211

1

I- parameter families of C -diffeomo,phisms such that '1',,(0) = 0 and cl>,,(O) = µ. Let K > O be a constant and let M,,, be the set of those µ E [O,r] such that the d istance between 'lr,,(A(µ)) n [- K · r, K · r] and cl>µ(B(µ))n [- K · r, K · r] is smaller than E • r. Then, for each 8 > 0, there is an E(8) such that

m(M, , , M with homoclinic ~xplosion which we consider are persistently hyperbolic for µ < 0. We first w ant to derive some consequences from this last assumption . For a diffeomorp hism M, M a closed surface, we know that if Q = Q( oo. In the case where M is a compact surface and A is a basic set , of a diffeomorphism M, M a closed 2-manifold, is that cp cannot have cycles and hence is a.stable [16]. To prove this, assume that cp has a cycl,e, i.e. that there are basic sets A 1, A 2, ... , Ak = A 1 , k;?: 2, such that W"(A;) n W'(Ai + 1 ) 0 for i = 1, .. . , k - I. From [10] we know that, with an arbitrarily small perturbation of cp we obtain a new diffeomorphism ip having one basic set A which contains basic (sub) sets Ai s; A with Ai close to Ai. This means that in a I-parameter family of diffeomorphisms, connecting cp and ip, new parts of basic sets and hence new periodic points are created. This contradicts the assumption that cp is persistently hyperbolic and hence that we may assume that ip as well as the diffeomorphisms between cp and ip are hyperbolic. Returning now to the I-parameter family cp,.: M-> M, we recall that for µ. = 0 we have: - cp = 'Po is hyperbolic on lim,. ,,.0 U( cp,,)= fi('Po); - cp has exactly one orbit of tangency between a stable and an unstable separatrix of periodic points, and this orbit of tangency is in fact an orbit of homoclinic tangency, i.e., of a tangency of the stable and unstable separatrix of the same saddle point p of cp = 'PQ; - the Q..set of cp is the union of fi('Po) and the orbit (!) of tangency. Since, for µ. < 0, cp,,is persistently hyperbolic, there are basic sets A;(µ.) depending continuously on µ. such that U( cp,.)is the disjoint union of these basic sets. Since there are no cycles we may assume that

*

(•)

W"(A;(µ.))

n

W'(A 1(µ.)) = 0

for all

i < l and

µ. < 0.

We define Ai(0) = lim,.,,. 0 A;(µ), so that fi(cp) = UA;(0). We want to prove that also for µ. = 0, ( *) holds. Suppose not. Then there are i < l so that W"(A;{0))

n

W'(Ac(0))

* 0.

We may assume that A; and A 1 are of saddle type. Since this intersection is empty for µ. < 0 and since the stable and unstable sets of these basic sets are fenced by stable and unstable separatrices of periodic points, there must be a tangency of the unstable separatrix of a periodic point in A ;(0) with the stable separatrix of a periodic point in Ac(0). But this extra tangency for µ = 0 is against our assumptions . In particular, all basic sets A;(µ.) are open in U( cp,.)for µ. .:o;0, except for Ai 0(0). But, it is clear from the above that even Ai 0(0) is the maximal %-invariant set in some neighborhood of it. That is, there exists V :::>A ;.(0) such that nnEZcp~(V) = A;(0). Then, from [3,§7], we conclude that %IA;(0) is conjugate 0 to cp,.IA;(µ.) for all i andµ.< 0. Therefore, as we mentioned in the introduction , the basic sets we consider here are open in the nonwandering set or they are homeomorphic to the ones having this property . Moreover, we can even consider

213

Appendix 5

basic sets A;(µ.) forµ positive and small as (canonical) continuation of A;(0) in the following way. If we choose neighborhoods Y; :::>Ai(0) in which A;(0) is the maximal 'Po-invariant set, then we take A;(µ.),µ. positive and small, as the maximal invariant set for cp,. in ¼- Again by [3, §7], we have that A;(µ.) is hyperbolic and cp,.jA;(µ.) is conjugate to 'PolA;(0). Let i 0 be the index of the basic set containing the saddle point P,. of cp,. whose stable and unstable separatrices make a tangency for µ = 0 (we do not exclude that A io(µ.) = p,.; in that case we call Ai.(µ.) a trivial basic set; otherwise we call it a non-trivial basic set). Observe that w·(Ai.(o))

= w•(Ai.(O) u

(!))

and W"(Aio(O)) = W"(Aio(O)

u

(!)).

So if we replace Aio(0) by A; 0(0) U (!), then(*) still holds. By [14] there is a filtration M 1 c M 2 c M such that (i) Mi is closed; (ii) M 1 c int(M 2); (iii) cp(M;) c int(M;); (iv) nnEZ cpn(M2- M1) = Aio(0) U (!)_ This means that for any (small) neighborhood U of A;.(0) U (!) there is an n( U) such that for each point x $. U, cpn< u >(x) or cp- n(u >(x) is in the complement of M 2 - lnt(M 1). Hence, forµ. near zero one still has this same property for cp,.instead of cp.This means that for any neighborhood U of A;.(0) U (!), there is a µ.0(U) > 0 such that for anyµ E [0, µ 0(U)], the nonwandering set U(cp,.) of cp,., intersected with M 2 - M1 , is contained in U. So for the rest of this chapter we may restrict our attention to such a neighborhood U, restricting µ. to [O, µ.0(U)]. 2. Bounds on the region where new nonwandering points can appear. Let again cp,,:M-> M be a I-parameter family of diffeomorphisms with a homoclinic For il-explosion. As before, U( cp,.)denotes the set of nonwandering points of cp,,. µ < 0, U( cp,.) is a hyperbolic set; the continuation of these hyperbolic sets, for µ ;?: 0, are denoted by fi(cp,.). So U( 'Po) = fi(%) U (!), where (!) is the orbit of tangency; U( 'Po) and fi('Po) will also be denoted by U( cp) and fi(cp).

Let r be a point in (!), the orbit of tangency of cp. Then for some constant K and µ > 0, µ. small, every orbit of fl( cp,.)\ fi(cp,,)has a point in a ( K · µ.)-neighborhoodof W"( p,.) near r. Again, P,. denotes the sadd[,epoint of cp,,with (!) c W"(p 0 ) n W'(p 0 ). PROPOSITION.

Proof We shall only give the proof in the case where the saddle point p has two positive eigenvalues, where the sides of tangencies and connections are as in

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the figure below

q

p

and where the expanding eigenvalue is dominating; see Chapter 2. We also assume that p is not part of a non-trivial basic set. The other cases can be treated similarly; see also the remark at the end of this section. As we have seen in the last section we can take a small neighborhood U, of r, U, being the component of r of a neighborhood U of Q(q>), and then the "new n-orbits" , i.e., the orbits of Q( q>I') \ Q( q>I'), will all pass through U, and stay in U for O ~ µ ~ µo(U) .

In the following considerations we assume that we have linearizing coordinates for q>I' in a neighborhood of pl'. We assume that in these coordinates Wi'~ ( pl') and W 1:,.,( pl') are independent of µ and that the coordinates of r and q are ( - I, 0) and (0, I); as usual r and q are points in the orbit of tangency in W1~(pl'), w 1:,.,(pl'). We assume that the tangency unfolds generically; to be more precise we assume the following situation near r for µ > 0:

W 'l p, I

the maximum distance of W'(pl') above W1~(pl') is µ (here one only considers the part of W ' ( pl') containing r ) . The proposition is a consequence of the following claim . For U, and µ 0(U) as above and sufficiently small, and for any x E U,, 0 ~ µ ~ µo(U), there is a positive no(x, µ) such that , whenever n ' > no(x , µ) and %- "'(x) E U,, then q:il' _",(x) has height less than 2µ above wu(pl' ). To prove this claim, take a point x E U, at distance d above W1~ (pl'). If %- "(x) is near q, then the distance from q:il' - "(x) to W 1:,.,(pl') is of the order d - Loga.fLogA., where Al' and al' are the contracting and expanding eigenvalues of / LogA,is much smaller than d. We (d%)A . Due to the dominance of a, d - Logo, - "(x) is in a small neighborhood of q; otherwise it would may a,;sume that q:>l' never return to U, under negative iterations of q:il'(in which case the claim would be trivially true for this (x, µ)) . Let m 0 be such that q:i"'o(r)= q. Then 'Pµ.-n-mo(x) E U,, and the distance from q:il' - n-mo(x) to W ' (pl') is of the order of

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d-Loga./ LogA•;i.e., there is a constant C, independent of d, such that the - n-mo(x) to W'(pl') is smaller or equal to C · d - Loga ,/ LogA,. For distance from q>l' d sufficientlysmall andµ near zero we have C · d - Loga , /LogA,< ½d. We forced and µ to be so small that this last inequality holds by choosing U, small. So if a point x E U, has a negative iterate x' in U, then p(x ', W ' (pl')) < 1/ 2 · p(x, W1~(pl')), where p is the distance function corresponding to the linearizing coordinates. If x' is below W1~( pl') then no further negative iterate of x will be in U,. The maximum distance of W'(pl') above W 1~(pl') is µ so that p(x' , W1~(pl')) < µ + 1/2 · p(x, W1~(pl')). This implies that after returning sufficientlyoften to U, under negative iteration we will finally be at a distance of less than 2µ from W1":,.,(pl') . This completes the proof of the proposition in this case. Remark . We observe that the shape of the region where the orbits of Q(cpl')\ Q(q>I') pass near r, forµ > 0, is as indicated in the figure below:

W '( p, l

It is not hard to see that if pl' is part of a non-trivial basic set for µ ~ 0, then the points of U, can also return (under negative iteration) between W 1~(pl') and W1:,.,( pl'), µ > 0, as in the case of the formation of a cycle by heteroclinic tangency; see [11].

3. Invariant foliations and further restrictions on the l.ocus of the nonwandering set. We continue with the same I-parameter family q:il':M - M with homoclinic ~xplosion. As before, pl' denotes the saddle point whose stable and unstable separatrices have a tangency forµ= 0. For µ ~ 0, A(µ) denotes the basic set of Q( q:il')to which pl' belongs. Forµ ~ 0, A(µ) is the basic set which is the continuation of the corresponding set for µ ~ 0. As mentioned in the introduction, there are C 1-stable and unstable foliations ~ • and ~u, defined on a small neighborhood of A(µ). They are C 1 in the sense that the tangent directions depend C 1 on x E M, andµ ER. For x in the domain of definition of these foliations, E,:(x) and E;(x) denote these tangent and ffe;,u through x. Since pI' E A(µ), ffe'' and ffe'" spaces of the leaves of .fF...' r r I' I' are both defined in a neighborhood of pl'. As usual r and q are points in the along orbit of tangency in W1~(p) and W 1:,.,(p). We extend the domain of ~u Wu(pl') until it includes a neighborhood of r; this extension is obtained by applying some positive iterates of cpl'to ~u. We also extend the domain of ffe'' I'

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until it includes r, but here we use negative iterates of