Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris
1706
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore
Tokyo
Sergei Yu. Pilyugin
Shadowing in Dynamical Systems
Springer
Author Sergei Yu. Pilyugin Faculty of Mathematics and Mechanics St. Petersburg State University Bibliotechnaya pl., 2, Petrodvorets 198904 St. Petersburg, Russia E-mail:
[email protected] Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme
PiUugin, Sergej Ju.: Shadowing in dynamical systems / Sergei Yu. Pilyug/n. - Berlin ; Heidelberg ; N e w York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 (Lecture notes in mathematics ; 1706) ISBN 3-540-66299-5
Mathematics Subject Classification (1991): 58Fxx, 34Cxx, 65Lxx, 65Mxx ISSN 0075-8434 ISBN 3-540-66299-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10650213 41/3143-543210 - Printed on acid-free paper
To m y sons Sergei and t(irill
Preface Let (X, r) be a metric space and let r be a homeomorphism mapping X onto itself. A d-pseudotrajectory of the dynamical system r is a sequence of points ={xkEX:kET/}or~={xkeX:kET/+} such that
r((~(Xk),Xk+l) < d. Usually, a pseudotrajectory is considered as a result of application of a numerical method to our dynamical system r In this case, the value d measures one-step errors of the method and round-off errors. The notion of a pseudotrajectory plays an important role in the general qualitative theory of dynamical systems. It is used to define some types of invariant sets (such as the chain-recurrent set [Con] or chain prolongations [Pi2]). We say that a point x (e, r a pseudotrajectory ~ = {xk} if the inequalities
r(r
0 there exists d > 0 such that for any d-pseudotrajectory ~ there is a point x that (e, r ~. From the numerical point of view, if r has the POTP, then numerically obtained trajectories (on arbitrarily long time intervals) reflect the real behavior of trajectories of r If r is a dynamical system C~ to r then obviously any trajectory of r is a d-pseudotrajectory of r with small d. Thus, if r has the POTP, then any trajectory of the "perturbed" system r is close to a trajectory of r Hence, we may consider the POTP as a weak form of stability of r with respect to C~ perturbations. Theory of shadowing was developed intensively in recent years and became a significant part of the qualitative theory of dynamical systems containing a lot of interesting and deep results. This book is an introduction to the main methods of shadowing. The book is addressed to the following three main groups of readers. The main expected group of readers are specialists in the qualitative theory of dynamical systems and its applications. For them, the author tried to describe a unified approach based on shadowing results for sequences of mappings of Banach spaces. It is shown that this approach can be applied to establish the classical shadowing property and limit shadowing properties in a neighborhood of a hyperbolic set, shadowing properties of structurally stable dynamical systems (both diffeomorphisms and flows), and some other classes of shadowing properties. In addition, we present a systematic treatment of connections
VIII
between the shadowing theory and the classical fields of the global qualitative theory of dynamical systems (such as the theories of topological stability and of structural stability). Next, some parts of the book (Sects. 1.1, 1.2.1, 1.2.2, 1.2.3, 1.3.1, 1.3.2, 3.1, 3.2, and 4.1) can be included into courses or used for the first acquaintance with the theory of shadowing by advanced students with basic training in dynamical systems. For this purpose, main definitions and results are illustrated by a lot of simple (maybe, too simple for specialists) examples. Proofs of basic results of the theory and description of some important general constructions contained in the sections mentioned above are given with all details and with necessary background from functional analysis. Finally, the book is addressed to specialists in numerical methods for dynamical systems. Some recent conferences (for example, the Conference on Dynamical Numerical Analysis, Georgia Tech, December 1995) showed that the idea of shadowing plays now an important role in this field and that "numerical dynamics" specialists need a detailed survey of results and methods of the shadowing theory. It was an intention of the author to describe two "numerically oriented" shadowing approaches. The first one is based on methods for verification of numerically obtained data. These methods allow to establish the existence of a real trajectory near a computed one and to give the corresponding error bounds (see [Cho2, Cho3, Cool-Coo6, Gr, Ham, Sau2] and others). The second approach establishes shadowing properties of dynamical systems generated by numerical methods (for example, discretizations of a parabolic PDE are realized as finitedimensional diffeomorphisms [Eil], see Sect. 4.4). These results allow us to study the influence of errors in application of numerical methods on unbounded time intervals. The book consists of 4 chapters. Chapter 1 is devoted to "local shadowing", i.e., shadowing in a neighborhood of an invariant set. We introduce the main shadowing properties in Sect. 1.1 and discuss relations between these properties. Section 1.2 is devoted to the classical shadowing result - the Shadowing Cemma by Anosov [Ano2] and Bowen [Bo2]. This result states that a diffeomorphism has the POTP in a neighborhood of its hyperbolic set. It is shown that this shadowing property is Lipschitz, i.e., if A is a hyperbolic set of a diffeomorphism r then there exist constants do, L > 0 and a neighborhood U of A such that for any sequence {xk} C U with
r(r
0} belongs to a neighborhood U of a hyperbolic set of a diffeomorphism r and, for some p > 1, t h e / : p norm
is small, then there is a point x such that
ll{hk}llp < LIl{dk}LIp with a constant L depending only on the neighborhood U. In Subsect. 1.4.3, we pass from the spaces s to their weighted analogs, the spaces s with norms
II{dk}ll~,,
=
rk
, ~ = {rk}.
We show that it is possible to establish the corresponding "s property" in a neighborhood of a compact invariant set, not necessarily hyperbolic, under appropriate conditions on the weight sequence ~. These conditions are formulated in terms of the so-called Sacker-Sell spectrum [Sac]. Another possibility to establish the "s is to assume that the weight sequence f grows "fast enough" (Theorems 1.4.6 and 1.4.8).
Xl
Hirsch studied in [Hirs4] asymptotic pseudotrajectories, i.e., sequences {zk} such that l/k
hmk_+oodk
< +k, where +k C (0, 1).
The main result of Hirsch [Hirs4] on shadowing of asymptotic pseudotrajectories is described in Subsect. 1.4.4. Section 1.5 is devoted to shadowing in flows generated by autonomous systems of ordinary differential equations. Technically, the shadowing problem for flows is significantly more complicated than the one for discrete dynamical systems. We prove in this section that if A is a hyperbolic set of a flow containing no rest points, then in a neighborhood of A pseudotrajectories are shadowed by real trajectories, and the shadowing is Lipschitz with respect to the "errors". In Chap. 2, we study connections between shadowing properties and the classical "global" properties of dynamical systems, such as topological stability and structural stability. We consider dynamical systems on a closed smooth manifold. Walters [Wa2] and Morimoto [Moriml] showed that a topologically stable homeomorphism has the POTP. We prove this statement in Sect. 2.1. It is also shown in this section that an expansive homeomorphism having the P O T P is topologically stable. Section 2.2, the main part of Chap. 2, is devoted to shadowing properties of structurally stable dynamical systems. In Subsect. 2.2.1, we prove that a structurally stable flow has a Lipschitz shadowing property [Pi4]. We begin to work with a flow since this case is technically more difficult than the case of a diffeomorphism (mostly due to the possible coexistence of rest points and of nonwandering trajectories that are not rest points). The main statement (Theorem 2.2.3) is reduced to shadowing results for sequences of mappings of Banach spaces with noninvertible "linear parts" (see Sect. 1.3). It was an intention of the author to make the presentation of Theorem 2.2.3 maximally "self-contained". Due to this reason, we give a detailed proof of the existence of Robinson's "compatible extensions of stable and unstable bundles" [Robil] (see Lemma 2.2.9). Shadowing for structurally stable diffeomorphisms is studied in Subsect. 2.2.2. It is shown that a structurally stable diffeomorphism has the Lipschitz shadowing property. Sakai noted that the POTP is "uniform" in a C 1neighborhood of a structurally stable diffeomorphism [Sakl] and that the C 1 interior of the set of diffeomorphisms with the POTP consists of structurally stable diffeomorphisms [Sak2]. We prove that the Lipschitz shadowing property is also uniform [Beg] and that a diffeomorphism in the C 1 interior of diffeomorphisms with this property is structurally stable. Without additional assumptions on the dynamical system, no necessary conditions for the P O T P are known. In Sect. 2.3, we study necessary and sufficient conditions of shadowing for two-dimensional diffeomorphisms satisfying Axiom A. We show that in this case the POTP is equivalent to the so-called C o
XII
transversality condition [Sak3], and the Lipschitz shadowing property is equivalent to the strong transversality condition (and hence to structural stability). In the same section, we describe results of Plamenevskaya [Pla2] on weak shadowing for two-dimensional Axiom A diffeomorphisms. A diffeomorphism r is said to have the weak shadowing property if given e > 0 there exists d > 0 such that for any d-pseudotrajectory ~ there is a trajectory O(x) of r with the property
C U~(O(x)) (here N~(O(x)) is the e-neighborhood of O(x)). An example of an Axiom A diffeomorphism of the two-torus T 2 with finite nonwandering set shows that necessary and sufficient conditions for weak shadowing have complicated structure, they are connected with arithmetical properties of eigenvalues of the derivative De at saddle fixed points. Section 2.4 is devoted to the long-standing problem of genericity of the shadowing property when the dimension of the manifold is arbitrary. We formulate (without a proof) a theorem from [Pi5] stating that a C~ homeomorphism has the POTP. In Chap. 3, we study the shadowing problem for some special classes of dynamical systems. Section 3.1 is devoted to one-dimensional systems. We prove two theorems of Plamenevskaya [Plal]. The first one gives necessary and sufficient conditions under which a homeomorphism of the circle has the POTP. In the second theorem, sufficient conditions for the limit shadowing property are obtained. As a corollary, it is shown that if a homeomorphism of the circle has the POTP, then it has the limit shadowing property. In Sect. 3.2, we consider linear and linearly induced systems. Following Morimoto [Morim3], we show that a linear mapping r = Az has the P O T P if and only if the matrix A is hyperbolic. Conditions of the POTP are known for a wide class of linearly induced systems, we treat in detail the so-called spherical linear transformations of the unit sphere S ~ C IR~+1 defined by the formula
Ax
r
IAxl"
We prove the following theorem of Sasaki [Sas]: r has the P O T P if and only if the eigenvalues of the matrix A have different absolute values. The second part of Chap. 3 is devoted to two special classes of infinitedimensional dynamical systems. In both cases, the shadowing problem is reduced to the same problem for auxiliary finite-dimensional systems. The first class, lattice systems, is studied in Sect. 3.3. Usually, the following three types of solutions for lattice systems are investigated: steady-state solutions, travelling waves, and spatially-homogeneous solutions. Under appropriate conditions, these solutions are governed by finite-dimensional diffeomorphisms. We describe conditions [Af3] under which an approximately static (approximately travelling or approximately spatially-homogeneous) pseudotrajectory is shadowed by a steady-state solution (correspondingly, by a travelling wave or by a spatially-homogeneous solution).
Xlll
Section 3.4 is devoted to shadowing in nonlinear evolution systems on Hilbert spaces. It is assumed that the evolution system S generated by a parabolic PDE ut = u~:,: 4- f ( u ) (0.7) has Morse-Smale structure on its global attractor ,4. We show that $ has a type of Lipschitz shadowing property in a neighborhood of the global attractor ,4 [Lar4]. In the last chapter, we describe some applications of the shadowing theory to numerical investigation of dynamical systems. Section 4.1 is devoted to methods of verification of numerically obtained data. We prove two theorems of Chow and Palmer [Chol, Cho2] on finite shadowing in one-dimensional and multidimensional systems. Coomes, Koqak, and Palmer [Cool-Coo6] developed a theory of "practical" shadowing for ordinary differential equations. Their methods allow to establish the existence of a real trajectory near a computed one. Section 4.2 is devoted to one of their methods, the method of periodic shadowing [Coo2]. This method gives computable error bounds for the distance between a computed closed trajectory and a real one. In Sect. 4.3, we consider connections between pseudotrajectories of a dynamical system and its "spectral" characteristics such as Lyapunov exponents and the Morse spectrum. In Subsect. 4.3.1, we study the problem of approximate evaluation of Lyapunov exponents. It is shown that in the evaluation of the upper Lyapunov exponent on a hyperbolic set, the resulting errors are Lipschitz with respect to the errors of the method and to round-off errors [Corl]. We show in Subsect. 4.3.2 that symbolic images of a dynamical system generated by partitions of its phase space [Osl] can be applied to approximate its Morse spectrum [Os2]. In Sect. 4.4, we investigate qualitative properties of semi-implicit discretizations of (0.7). In Subsect. 4.4.1, we study finite-dimensional diffeomorphisms generated by discretizations and the global attractors for these diffeomorphisms [Ell]. It is shown that, for a generic nonlinearity f ( u ) , these global attractors have Morse-Smale structure, hence we can apply the theory of shadowing for structurally stable systems (Chap. 2). In Subsect. 4.4.2, we apply shadowing results obtained in Sect. 3.4 to give explicit estimates (in terms of time and space steps) for the differences between approximate and exact solutions on unbounded time intervals [Lar4]. Cooperation with colleagues was very important during the preparation of this book. Special thanks are to V. Afraimovich, R. Corless, T. Eirola, S. Larsson, O. Nevanlinna, G. Osipenko, V. Pliss, O. Plamenevskaya, and G. Sell. The author is grateful to S.-N. Chow, M. Hurley, A. Katok, H. Koqak, K. Odani, C. Robinson, and K. Sakai for the attention they paid to the manuscript of this book during its preparation. The author's research was partially supported by INTAS grant 96-1158. The manuscript of the book was completed during the author's visit to the University of Alberta, Edmonton, Canada, supported by the Distinguished
•
Visitors Fund of the University of Alberta and by the Natural Science and Engineering Research Council of Canada. The author expresses his deep gratitude to Applied Mathematics Institute and Department of Mathematical Sciences of the University of Alberta and especially to Professor H. Freedman.
Co~e~s
Chapter
1. S h a d o w i n g N e a r a n I n v a r i a n t S e t . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism . . . . . . . . . . . . . . . 1.2.1 Hyperbolic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The Classical Shadowing Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Shadowing for a Family of A p p r o x i m a t e Trajectories . . . . . . . . . . . . . 1.2.4 The Method of Bowen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Shadowing for Mappings of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Shadowing for a Sequence of Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Conditions of Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Application to the Classical Shadowing Lemma . . . . . . . . . . . . . . . . . . 1.3.4 Theorems of Chow-Lin-Palmer and Steinlein-Walther . . . . . . . . . . . . 1.3.5 Finite-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Limit Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Limit Shadowing P r o p e r t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 s ................................................... 1.4.3 The Sacker-Sell Spectrum and Weighted Shadowing . . . . . . . . . . . . . . 1.4.4 A s y m p t o t i c Pseudotrajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Shadowing for Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C h a p t e r 2. T o p o l o g i c a l l y S t a b l e , S t r u c t u r a l l y S t a b l e , a n d Systems ................................................................
1 1 9 10 14 22 27 34 35 40 43 45 53 63 64 68 71 84 88
Generic 103
2.1 Shadowing and Topological Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Shadowing in Structurally Stable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Case of a Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Case of a Diffeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Shadowing in Two-Dimensional Diffeomorphisms . . . . . . . . . . . . . . . . . . . 2.4 C~ of Shadowing for Homeomorphisms . . . . . . . . . . . . . . . . . .
103 109 109 145 158 172
C h a p t e r 3. S y s t e m s w i t h S p e c i a l S t r u c t u r e . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 One-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear and Linearly Induced Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Lattice Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Global A t t r a c t o r s for Evolution Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
173 173 182 189 202
C h a p t e r 4. N u m e r i c a l A p p l i c a t i o n s o f S h a d o w i n g . . . . . . . . . . . . . . . . . . 4.1 Finite Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Periodic Shadowing for Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Approximation of Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Evaluation of Upper Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . .
219 219 223 233 233
•
4.3.2 A p p r o x i m a t i o n of t h e M o r s e S p e c t r u m . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 D i s c r e t i z a t i o n s of P D E s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 S h a d o w i n g in D i s c r e t i z a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 D i s c r e t i z a t i o n E r r o r s o n U n b o u n d e d T i m e I n t e r v a l s . . . . . . . . . . . . . References
.............................................................
Index ...................................................................
238 244 245 255 259 269
List of Main
Symbols
IR - the set of real numbers; IRn - the Euclidean n-space; 7I the set of integer numbers;
71+ = {k ~ 71 : k_> 0}; IN - the set of natural numbers; C - the set of complex numbers; GL(n, IR) (GL(n, C)) - the group of invertible linear transformations of 1Rn (respectively, of C'~); I is the identity operator (or the unit matrix); For a set A in a topological space, A is the closure of A, IntA is the interior of A, and OA is the boundary of A; For sets A, B in a metric space (X, r), N~(A) is the a-heighborhood of A, diamA = sup r ( x , y ) x,yEA
and dist(A,B) --
inf
xEA,yEB
r(x,y);
For a linear mapping A, flAIl = sup M=I
IAvl
is the operator norm of A; For a Banach space B, B(r) is the closed ball of radius r centered at 0; For a smooth manifold M, T,:M is the tangent space of M at x and T M is the tangent bundle of M; For a smooth mapping f , Df(z) is the derivative of f at x; := means "equal by definition".
1. S h a d o w i n g
Near
an Invariant
Set
1.1 Basic D e f i n i t i o n s Let (X, r) be a metric space. A homeomorphism r mapping X onto itself generates a dynamical system O:~_• by the formula 9 ( m , x ) = era(x), m E 77, x E X. The trajectory O(x) of a point x E X in the dynamical system 9 is the set O(x) = { ~ ( m , x ) : m e ~}. Usually, we identify the homeomorphism r with the dynamical system 45 it generates (and call r a dynamical system). A continuous mapping f : X --* X generates a semi-dynamical system by the formula 9 ( m , x ) -- fro(x), m E 71+, x E X . The trajectory O(x) of a point x E X in the semi-dynamical system ~ is the set
o+(x) =
x):m e 7]+).
We also identify f and $. The main objects of investigation in this book are dynamical systems and their approximate trajectories (pseudotrajectories) defined below. Let either K = 7] or K = 77+. Fix d > 0. D e f i n i t i o n 1.1 We say that a sequence ~ = {xk E X : k E K } is a "dpseudotrajectory" (or a "d-pseudoorbit") of a dynamical system r on K if the inequalities r(r z~+l) < d, k E K, hold. Fix e > 0. D e f i n i t i o n 1.2 We say that a point x E X "(~, r a d-pseudotrajectory ~ = {Xk} on K if the inequalities
(or "(~, r
2
1. Shadowing Near an Invariant Set r(r
< e, k e K,
(1.1)
hold. Below, if only one dynamical system r is considered, we will usually write simply "x (@shadows ~". This will lead to no confusion. In this book, we mostly study the shadowing properties of dynamical systems defined below. Let Y be a subset of X. D e f i n i t i o n 1.3 We say that the dynamical system r has the " P O T P " (the "pseudoorbit tracing property") on Y if given e > 0 there exists d > 0 such that for any d-pseudotrajectory ~ on 77 with ~ C Y there is a point z that (e, r shadows ~ on 77. If this property holds with Y = X , we say that r has the POTP. Remark. [1] Our usual term in this book is "shadowing" (not "tracing"), but we preserve the term " P O T P " , since it became standard. Sometimes a d-pseudotrajectory of a dynamical system r is considered as a result of a small random perturbation of r In this case, the P O T P means that near trajectories of a "randomly perturbed" r there are real trajectories of r Due to this reason, some authors use the term stochastic stability instead of the P O T P [Moriml]. Note that shadowing properties of random dynamical systems were studied, for example, in [Bll, Cho2]. We do not consider random dynamical systems in this book. Remark. [2] An analogous property connected with pseudotrajectories on 77+ is called the P O T P + . It is easy to show that there exist systems which do not have the POTP. E x a m p l e 1.4 Consider the circle S 1 with coordinate x C [0, 1) and a homeomorphism r of S 1 generated by the mapping f ( x ) - x. Fix d > 0 and take a sequence of points ~ = {xk : k E 77} C 5'1 such that x0 -- 0 and d xk+l = xk + ~(mod 1), k E 7/. Obviously, ~ is a d-pseudotrajectory for r Any trajectory of r is a fixed point. It is easy to see that, for d < 2/3 and for any trajectory p of r the set ~ is not contained in the (1/3)-neighborhood of p. This means that for e = 1/3 there exists no d with the property described in Definition 1.3. Let us describe some simple relations between the introduced notions. L e m m a 1.1.1.
1.1 Basic Definitions
3
(a) Assume that X is compact and that r has the following 'finite shadowing property" on Y C X : given e > 0 there exists d > 0 such that if for a set { X o , . . . , x m } C Y the inequalities r(Xk+l,r < d hold for 0 < k O. I f r has the P O T P on Y , then r has the POTP+ on Y . Proof. (a) Let e > 0 be given. Find a corresponding d > 0 given by the "finite shadowing property". Let ~ -- {xk : k C 7]} C Y be a d-pseudotrajectory for r Fix m > 0 and set x~ --- xk-m. By our assumption, there is a point ym E X such that r (r k(ym), x I < e, 0 < k < 2 m . Set w m = r
Then
r(Ck(wm),x)
0 be fixed. Find a corresponding d given by Definition 1.3. Let ( = {xk : k >_ 0} C Y be a d-pseudotrajectory of r on 7/+. Consider the sequence ~' = {x~ : k E 7/} such that x~ = xk for k > 0 and x~ = ek(x0) for k < 0. Obviously, ( ' is a d-pseudotrajectory of r belonging to Y. By the choice of d, there is a point x that (@shadows ~1 on 7/. It follows immediately that x (e)-shadows ~ on 77+. [] This l e m m a shows that if X is compact, then r has the P O T P if and only if r has the P O T P + . D e f i n i t i o n 1.5 We say that the dynamical system r has the "LpSP" (the "Lipschitz shadowing property") on Y if there exist positive constants L, do such that for any sequence {Xk E Y : k E 7/} with r(r
Xk+l) < d < do, k e 7/,
there is a point x such that the inequalities r(r
xk) < n d , k E 7/,
(1.2)
hold. If this property holds with Y = X , we say that r has the LpSP. Remark. An analogous property connected with pseudotrajectories on 7/+ is called the LpSP+.
4
1. Shadowing Near an Invariant Set
Of course, a statement analogous to L e m m a 1.1.1 is true for the LpSP and LpSP+. Let us give a simple example of a dynamical system that has the P O T P on some set but does not have the LpSP on this set. E x a m p l e 1.6 Consider a diffeomorphism r : ~ --, ]R such that r x2sgn(z), where sgn(x) is the sign of x. Set w =
9
= x +
Ixl < 1}.
Let us show that r has the P O T P on W. Fix arbitrary e > 0. Denote b = e2/2. Take a d-pseudotrajectory ~ = {xk : k 9 7]} C W with d < b. We claim that I~kl < ~ for k 9 7]. (1.3) To obtain a contradiction, assume that there exists Ixkl > e. If xk > e, we obtain the inequality Xk+l > r -- d > xk + b, and similar inequalities Xk+m > Xk + m b for m > 0
which imply that the sequence x,,~ leaves W as m grows. The case xk _< - r is considered similarly. It follows from (1.3) that any d-pseudotrajectory ~ is (~)-shadowed by the point x = 0. Let us show that r does not have the LpSP in W. Take a small d > 0 and consider xk = v/-d, k E 7]. Then we have -
x
+ll =
=
d.
Assume that r has the LpSP in W with constants L, do. For small d, the inequalities ICk(x) - xk] 0 such that any dpseudotrajectory ~ = {xk : k 9 7/} C Y is (e)-shadowed by not more than one point x. I f Y = X , we say that r has the SUP. The following statement (Lemma 1.1.2) shows that this property is almost equivalent to the well-known expansivity property. First let us define this property. D e f i n i t i o n 1.8 The system r is called "expansive" on a set Y C X if there exists A > 0 such that if for two points x, y the inclusions
Ck(x), Ck(y) e Y, k 9 7/, and the inequalities
r(r
Ck(y)) < ~, k 9 27,
hold, then x = y. If Y = X , the system r is called expansive. We say that A is an "expansivity constant".
L e m m a 1.1.2. (a) Assume that r has the S U P on a set Y C X with constant e. Then r is expansive on Y , and any A E (0, e) is an expansivity constant of r on Y . (b) Assume that for a set Y there exists a number A > 0 and a set Y1 such that r is expansive on Y with expansivity constant A , and the A-neighborhood of Y1 is a subset of Y . Then r has the S U P on II1 with c -= A / 2 .
Proof. (a) Assume that for x, y we have
Ck(x), Ck(y) E Y, k E 77, and
r(r
Ck(y))
0. Obviously, it is (@shadowed by x. It follows from the relations above that ~ C Y and that it is (e)-shadowed by y. This implies that x = y. (b) Now assume that for a &pseudotrajectory ~ = {xk: k e 7/} C 1/1 there exist points x, y such that
6
1. Shadowing Near an Invariant Set r(r
It follows that {r
< c and r(r {r r(r
< r k E 7/.
C Y, and r
< 2~ = ~a, k c 7/. []
This implies that x = y.
Remark. Thus, if a homeomorphism r has the SUP, then r is expansive. This means that the class of homeomorphisms having both the P O T P and SUP coincides with the class of expansive homeomorphisms with the POTP. This last class was studied by many authors (see a detailed survey [Aol] and the books [Ak, Ao2]). An expansive homeomorphism having the P O T P is often called topologically Anosov [Ao2]. It was shown by Hiraide [Hiral, Hira2] that if a manifold X admits a topologically Anosov homeomorphism, then strong restrictions are imposed on the structure of X. For example, the only compact surface with the mentioned property is a 2-torus. We devote the last part of this introductory section to the proof of a simple technical result. Sometimes it is easier to establish the P O T P (or the LpSP) not for the given dynamical system r but for r with some u E IN. It is easy to show that r has the P O T P (or it is Lipschitz and has the LpSP) if and only if r has the same property (with other constants). This approach was applied by Newhouse [Gu2] and others. We prove here a slightly more general statement (for sequences of mappings of Banach spaces and their finite superpositions) concerning Lipschitz shadowing. Let Ilk, k E 7/, be a sequence of Banach spaces, we denote by I.I norms in Ilk. Consider a sequence of mappings ~)k : Hk ""+ Hk+l.
For a natural number v and k E 7 / w e denote ~k,~ = r
o ... o ek.
Take zm E Hm~, m E 7], and construct a sequence xk E Hk so that x , ~ = zm and x~ = r
o . . . o e m ~ + , o e m ~ ( z m ) for k = , ~
+ m~, 0 < m l < ~ - 1.
(1.4)
L e m m a 1.1.3. Assume that there exist sets Vk C V/: C Ilk and a number s > 0 such that ek+, o r o... o r c v;+i+l for k E 7/, 1 < i < v - 1, and any r s
is Lipschitz on V~ with Lipschitz constant
1.1 Basic Definitions
7
Assume, in addition, that, for sequences Yk E Vk,~m = ym~ E Vm~, and zm E Vm~, the inequalities -- Yk+ll ~ d, k E 77,
[r
(1.5)
]Zm -- ~mi ~_ e, m E 7 7 , hold with some d, e > O. Then (1) for m E 77 we have
]~Pm~,~((m) --~m+l ]_~ Lid, m E 77,
(1.6)
where L1 = 1 + s + . . . + . . . s (2) for the sequence xk, k E 77, constructed according to formula (1.~), we have the inequalities [xk-- yk[ [2-1
(i~..o#_JlDr
k
+
~u+l
By construction, the obtained norm [.[p is continuous on the set A (we recall that the families S, U are continuous). Extend S, U to continuous (but not Dr families S I, U' on a closed neighborhood (i.e., the closure of a neighborhood) W0 of A so that statement (1) of our lemma holds. Now we extend [.[~ to W0 (decreasing W0, if necessary) and fix a constant N ' with property (1.12) for x 9 W0. For points x 9 A,y = r the mapping II~Dr (II~Dr is an isomorphism between S(x) and S(y) (respectively, between U(x) and V(y)), and the relations
llll~Dr IIlI~Dr
_ 1/,, II;DC(x)[u(~) = 0
hold (the operator norms are taken with respect to I.l*). Since//~, H~, r and D e ( x ) are uniformly continuous, given arbitrary e > 0 we obviously can find a neighborhood W(c, A) C W0 and a number 5 with the desired properties. To complete the proof of our lemma, it remains to approximate I.[* by a C ~ norm (and to decrease the neighborhood W, if necessary) so that all the estimates remain true. [] Note that by our construction the neighborhood W is bounded. Now we take a bounded neighborhood V of the hyperbolic set A for our diffeomorphism r a diffeomorphism r of class C 1 in ]I:U~, and define the number pl,y(r r
= sup Ir xEV
-- r
+ sup
xEV
IIDr
- DO(x)l[.
The proof of Lemma 1.2.1 shows that the following statement holds.
The neighborhood W, the norm 1.1~, and the extensions S', U' of the hyperbolic structure on A have the following property. There exists 51 > 0 such that if for a diffeomorphism r the inequality
L e m m a 1.2.2.
p l , w ( r 1 6 2 < 51
holds, then for x,y e W with l Y - r
ILr;Dr
< Alvin, I~Dr 1
(1.14)
< 51 we have
< ~lvl~, v 9 s'(x);
)II~PC(x)vly > XIvl=, [II~DC(x)vly 0, u E (0, 1) with the following properties. For any point x E U such that ek(x) E U for kl < k < k2 with kl < 0, k2 > 0 (infinite values of kl, k2 are admissible), there exist C 1 A-disks W~(x) and W~(x) centered at x and such that
(1) r163
C W~(r
if k2 > 1, r
C W~(r
if kl < - 1 ;
(2) if Yl, Y2 e W~(x), then Ir
- d(y~)l _< ~'lyi - y:l for I >_ o while l < ks;
if ~ , y ~ 9 W~,(x), then Ie/(Yl) -- r
--~ V-I[yl -- Y2[ for l ~_ 0 while l > kl and ~,-t[y I - Y21 < A;
(3) if k2 = cr and for a point y the inequalities Iek(x) - ek(y)l < A
hold for k >_ O, then y 9 W~(x); if kl = - o o , and for a point y the inequalities Iek(x) - ek(u)l < A
hold for k < O, then y 9 W~(x); (4) if for x, y 9 V we have Ix - Yl 0 the inequalities 1 ul = Ul--L--~ < 1 and
IIA~'II , IIA~'"II ~ . hold, then the operator I - A is invertible, and
I1(I- A)-lll _< R(,~, u)
1 =
(1 - A ) ( 1 - u l ) "
Proof. To prove statement (1), consider the operator C defined by oo
C = ~ _ , A k. k=O
Obviously, the series on the right converges, and the estimate IICrl < ~ IIAIIk = ~'~ Ak 1 k=o k=o 1- A holds. Let us write
oo
oo
( I - A)C = • Ak - E Ak= I. k=O
k=l
Similarly one shows that C ( I - A) = 1. This proves (1). To prove (2), we first consider the operator A0 given by [ A 8~ 0 0 A ~'~' ] " Let us show that I - Ao is invertible and that (I -
where Co is represented as
with
Ao) -1 = Co,
[ oo
(1.19)
c~0
oo
C~ = y~(A88) k, C~ = - Y~(A==) -k. k=O
k=l
Obviously, it follows from (1.18) that the series defining C~, C~ converge, and that 1
IlCgll _< -1- - ~ ' IIC~lt _
1 - A'
16
1. S h a d o w i n g N e a r an I n v a r i a n t Set
hence, 1 IIC011 ~ i - A'
(1.20)
Analogously to (1), we obtain the equalities (I -
A 8 8 )C• 8 = ~ _ , ( A " ) k - ~_,(AS") k = I k=0
k=l
and oo
(I -
A")C~
=
oo
)--~(A~") -k + ~ ( A ~ )
-
k=l
-k = I.
k=0
Similarly one shows that C ~ ( I - A 8s) = I and C ~ ( I - A ~') = I , hence (1.19) holds. Set A' = A o - A , A " = I - A0. It follows from our conditions and from (1.20) that IIA'II 0 we denote t~(T) = {~ e ~ : I1~11 < r}.
18
1. Shadowing Near an Invariant Set
Since our neighborhood W is bounded, there exist constants C1, dl > 0, and a function b(s) -* 0 for s --~ 0 such that IIDr
< C~ for z 9 W
and Ihk(vk)
-
hk(v~)l ~ b(max([[0[], IIr
- vll for I1~11, I1r
~ dl.
For ~ E B we set F(O) = ~', where Vk = r
1 "-~ V k _ l ) -- X k.
Since Ir - xk+,l < d, it follows from (1.23) and from the estimates above that for ~ E B(dl) we have
Ir
+ vk) - ~k+~l
1
19
u
-fMI.
if 9 9 = { v G ~ " = {w~}, then obviously I1~'11 -< ~119"11 and ~o on. Hence, DF(O) is represented with respect to decomposition (1.24) in the form A~ A ~
,
where
IIAssll, II(A~)-lll ~ A, IIA'~II, IIA~Sll ~ ~. Now (by our choice of ~, e) Lemma 1.2.3 implies that the operator I - DF(O) in B is invertible, and
II(I - DF(0))-'II _< n.
(1.25)
Set
G(~) = (I - DF(O))-I(F(~) - DF(O)9). Clearly, the equation F(~) = 9 is equivalent to G(9) = ~. Set H(9) = F(9) - DF(O)G i.e., H(~)k = [r - xk] + hk-l(vk-i). Then
IIH(0)II _< d.
(1.26)
It follows from properties of hk that for ~, 9' E/~(dl) we have
IIH(o) - H(r
_< b(max(llolh 11r
- 9'11.
(1.27)
We deduce from (1.25)-(1.27) that if d < 6, then
IIG(O)I[ < Rd and IIG(~) - G(~')II -< Rb(max(l[911, II~'ll))ll 9 - 9'11 for 9,9' E B(dl). Take d2 E (0, min(5, dl)) such that Rb(d2) < 1/2. It follows that 1
IIG(9) - a ( r
-< 2119- r
for 9, 9' E B(d2), hence G contracts on the ball B(d2). Set d2 d0=m, L=2R. Take d < do (note that R > 1, hence do < d2). For 9 e 13(Ld) we have IIG(9)[I _< IIG(0)I I + [IG(~) - G(0)[[ < R d + 22Rd = Ld. We is a x= the
see that G maps the ball B(Ld) into itself and contracts on it, hence there unique fixed point 9" of G (and also of F) in this ball. Obviously, the point x0 + v~ satisfies inequalities (1.2). The uniqueness of the fixed point implies uniqueness of the shadowing trajectory. []
Remark. It is easy to see that, for d < do, the mapping G has a unique fixed point not only in the ball B(Ld) but also in the larger ball B(Ldo). Take
20
1. Shadowing Near an Invariant Set
e > 0 and a neighborhood W0 of A such that 2e < Ldo and N~(Wo) C W. We claim that r has the SUP on W0 with constant e. Indeed, assume that for a d-pseudotrajectory ( = {x~} C W0 there exist points z, y such that -
0 there exists a natural number m such that qE r
f3 r
The sets W1 = Cm(W~(p)) and W2 = r are diffeomorphic to C 1 disks, we say that q is a transverse homoclinic point if the point q is a point of transverse intersection of the disks W1 and W2. We introduce the set
F = O(q). Obviously, F is a compact invariant set of r It is well known that if q is a transverse homoclinic point, then F is a hyperbolic set of r (it is a useful exercise for the reader to prove this statement). Poincar~ was the first who showed that the existence of a homoclinic point implies a very complicated structure of trajectories in a neighborhood of the set F. We establish here an important property of neighborhoods of transverse homoclinic trajectories.
1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism
21
1.2.4. If q is a transverse homoclinic point, then any neighborhood of the set F contains a countable family of different periodic points of r
Theorem
Proof. Take an arbitrary neighborhood U of the set F. We take the neighborhood U bounded and so small that r has the P O T P and the SUP on U (this is possible by Theorem 1.2.2). Then r has the P O T P and the SUP on F. Take A > 0 such that any d-pseudotrajectory ~ C F is (A)-shadowed by not more than one point. Find a number e E (0, A) such that N~(F) C U. We assume, in addition, that 2e < IP - ql. (1.29) Fix a number d corresponding to e according to Definition 1.3 (with Y = F). Let s be a Lipschitz constant of r in U. It follows from (1.28) that there exist numbers l < 0 , m > 0 such t h a t for the points q0 = Ct(q) and el = era(q) the inequalities d Iq0 - P l < d, Iql - P l < ~ , and the = {xk 0 < k < periodic
(1.30)
inclusion ql E U hold. Set N = m - 1 + 2 and define a sequence : k E 7]} as follows. Represent k E 7 / i n the form k = ko + kiN with N, and set xk = p if k0 = 0, and xk = r176 if k0 # 0. We obtain a sequence with period N, 99 9
Xo
=
p,
xl
=
qo,
99 9
2gN-1
~
ql,
XN
=
p,
XN+I
=
qo,
9 9 9
It follows from (1.30) that Ir
- Xl[ = [p - qo] < d and [r
-- XN[ < Llql -- Pl < d,
hence ~ is a d-pseudotrajectory of r By the choice of d, there is a point x that (e)-shadows ~. Set y = CN(x). Since zk = xk+N for all k, we have ICk(y) - xkl = ICk+N(x) --
k+NI < ', k e Z ,
i.e., the point y also (e)-shadows ~. The inclusion ~ C F and the SUP on F (with constant A > e) imply that x = y, hence x is a periodic point of r There exists k such that [r - q[ < e. It follows from (1.29) that x # p. Relations (1.28) imply that the set F contains no periodic points of r different from p, hence for the finite set O(x) we have O(x) f3 F = 0. Thus, we can find a neighborhood U' of F such that O(x) N U' = 0. The same reasons as above show that U' contains a periodic point x' ~ p, and so on. Hence, there is a countable family of different periodic points of r in U. Note that we proved that the trajectories of these points are in U. []
Remark. Palmer [Palml, Palm2] applied the shadowing approach to describe the structure of the set of all trajectories belonging to a small neighborhood
22
1. Shadowing Near an Invariant Set
of the trajectory of a transverse homoclinic point for a diffeomorphism. Later, he did the same for flows [Palm3]. Stoffer [Sto] proved a special shadowing lemma to study transverse homoclinic orbits for sequences of mappings of the plane. Steinlein and Walther [Stel, Ste2] gave a nonstandard definition of a hyperbolic set for a mapping in a Banach space and proved the corresponding shadowing result. This result can be applied to study homoclinic trajectories for infinite-dimensional semi-dynamical system. We prove a variant of this theorem of Steinlein-Walther in Subsect. 1.3.4 (see Theorem 1.3.4). Let A be an invariant set of a dynamical system r We say that A is locally maximal (or isolated) if there exists a neighborhood V of A such that the inclusion O(p) C V implies that p E A. It is easy to understand that if A is locally maximal, and r has the P O T P in a neighborhood of A, then pseudotrajectories belonging to a small neighborhood of A are shadowed by points of A. The set F defined before Theorem 1.2.4 is an example of a hyperbolic set which is not locally maximal (any its neighborhood contains a periodic trajectory not belonging to F). 1.2.3 S h a d o w i n g for a F a m i l y o f A p p r o x i m a t e T r a j e c t o r i e s It was mentioned that Anosov established the Shadowing Lemma in [Ano2] as a particular case of a more general statement, the so-called "theorem on a family of e-trajectories". This statement says that if A is a hyperbolic set for a diffemorphism r and if a small neighborhood of A contains a family of approximate trajectories of a diffeomorphism r CLclose to r then it is possible to shadow all this family by trajectories of r Let us formulate the theorem. We assume that A is a hyperbolic set for a C 1diffeomorphism r of IR% Consider a topological space X and a homeomorphism r of X. For a continuous mapping f : X --* ]R~ and for a diffeomorphism r of IR'~, set P ( f o r , r o f ) = sup If o r ( x ) - r o f(x)l. xEX
Obviously, the equality P ( f o r, r o f ) = 0 means that f maps trajectories of T to trajectories of r If the number P ( f o T, r o f) is small, f maps trajectories of r to "approximate" trajectories of r T h e o r e m 1.2.5. There exist neigborhoods Wo, W of the set A and positive numbers L, do with the following property. For any diffeomorphism ~ of IR~ with Pl,W0(r162 < do (1.31)
and for any continuous mapping f : X ---+W such that P(f oT,r
f ) < d < do,
there exists a continuous mapping g : X ---+IRn such that
(1.32)
1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism
23
(1) g o "r = C o g ; (2) sup=ex If(x) - g(x)] < Ld. In addition, if for a continuous mapping h : X ---+]R n analogs of statements (1), (2) (with h instead of g) hold, then h = g. Proof. We repeat the beginning of the proof of Theorem 1.2.3. For a hyperbolic set A of a diffeomorphism ff with hyperbolicity constants C, ~o, we fix numbers E ()~o, 1) and ~ > 0 such that the inequality s -- 0 and let Wo be the dl-neighborhood of W. Fix 81 given by Lemma 1.2.2. We consider below a diffeomorphism r such that pl,wo (r r < For p E W and w E IR'~ denote
hCv(w) = r
+ w) - r
- Dr
We take d2 E (0, dl) so small that ifp E W and [wl, [w' I < d2, then the segment joining the points p + w and p + w' belongs to W0. Since the neighborhood W is bounded, De is uniformly continuous on W0. Hence, it easily follows from the standard formula
hCv(w) - hC(w ') = ~o~ Dr
- w')ds - Dr
- w')
(1.33)
(where O(s), s E [0, 1], parametrizes the segment joining the points p + w and p+w') that we can find d2 having the following property. Ifp E W and [w[, Iw'l _< d2, then -
0. It follows from (1.34), (1.37) that for v ~ B(d~) we have
IIF(.)II < d +
(c~ + g-~1) Ilvll,
hence F is an operator from B(d2) to 13. We deduce from (1.38) that F is differentiable at 0, and
(DF(O)v)(f(x)) = Dr Now we represent B in the form/3 s @ B y, where B s = {v e B : v(f(x)) 9
S(f(x)),x9
X},
13" = {v 9 13: v ( f ( x ) ) 9 V ( f ( x ) ) , x 9 X } . Take x 9 X, z = r - l ( x ) , and denote y = f ( x ) . It follows from L e m m a 1.2.2, from our choice of r and from (1.37) (with d < 51) that, for w 9 S ( f ( z ) ) , the inequalities
III~Dr
No and CN(v) n v # O.
T h e o r e m 1.2.6. Let A be a hyperbolic set of a diffeomorphism r Assume that x is a nonwandering point of the restriction r Then any neighborhood of x
contains a periodic point of r
1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism
27
Proof. Take an arbitrary neighborhood U of x. Find a > 0 such that the ball of radius a centered at x belongs both to U and to the neighborhood W of A given by Theorem 1.2.5. Take d < do (from the theorem) such that Ld < a/2, and denote by V the intersection of the d/2-neighborhood of x with the set A. Since V is a neighborhood of x in A, we deduce from definition that there exists a natural number N such that c N ( v ) fq V 7~ O, hence Vo = r Take a point {0, 1 , . . . , N X --~ X , T(k) Define f : W). Then
f-I V r O.
y E V0, it follows that CN(y) E V. Consider the space X = 1} (with discrete topology) and the shift homeomorphism T : = k + l ( m o d m). X --* IR~ by f ( k ) = Ck(y) (note that y E A, hence f ( k ) E A C
f o T(k) = Ck+l(y) = r o f ( k ) for k = 0 , . . . , N
- 2,
f o r ( N -- 1) = f(0) = y, r o f ( N - 1) = CN(y), hence
P(f or,r
d < ~.
f)=lCN(y)--yl
By Theorem 1.2.5, there exists a mapping g : X ~ 1R'~ such that Ld < a/2 and g o r = r o g. Let Pk = g(k), then the equalities g o ~(k) = r o g(k),
k = 0,...,
Ig(x)-f(x)l _
0, u C (0, 1) such t h a t r y]) = [r r (1.45) when both sides are defined, and
1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism r(r
r
r(r
r
< u~r(y, z) if y, z E Vs'(x), n > 0, < u'~r(y, z) if y, z Z V6=(x), n > 0.
31 (1.46) (1.47)
D e f i n i t i o n 1.11 We define a "Smale space" to be a compact metric space (X, r)
with a map [, ] and a homeomorphism r satisfying (SS1) and (SS2) for suitable 71,6, u. Below (X, r denotes this space. The introduced notion is an abstract model of the behavior of a diffeomorphism near its hyperbolic set described in Theorem 1.2.1. A small neighborhood of a hyperbolic set cannot be invariant, thus it is impossible to apply the model directly (the only possible direct application is the case of an Anosov diffeomorphism, i.e., of a diffeomorphism r on a smooth closed manifold M such that M is a hyperbolic set for r [Anol]). Nevertheless, the model reflects the main properties used in Bowen's proof of the Shadowing Lemma. The sets V~(x) and V~(x) are analogs of W~(x) and W~(x) in Theorem 1.2.1, and Ix, Y] corresponds to the unique point of intersection of W~(x) and W~(y). Inequalities (1.46) and (1.47) are analogs of the corresponding properties (see statement (2) of Theorem 1.2.1). T h e o r e m 1.2.7. If (X, r
is a Smale space, then r has the POTP.
Proof. Let 7/, 6, u be the defining constants for (X, r and such that
Fix e > 0. Take 60 E (0, 3)
Since the functions r(x, [ x , y ] ) a n d r(y, [x,yl) are continuous for x, y with small r(x, y) and vanish for x = y, there exists 61 E (0, 7/) such that
r(x,[x,y]),r(y,[x,y]) < 6o for r(x,y) < 61. Find a natural number m such that
v'~6o < 61. Now we fix a number d > 0 such that
r(r
<min(61-um6o, 2 )
(1.49)
for any d-pseudotrajectory ~ = {xk : k E 7/} of r any natural s, and any 0
1 - u). Finally we see (taking (1.48) into account) that
r(r
~ + , ) < r(r +~(r
r
0, this implies that r(x, [y, z]) = 0, hence x = [y, x]. By (1.45), for proper x,y we have
r162
r
= [~,y],
this means that
r
' y] = [r
r
Thus, we can repeat the reasoning above (with n < 0) to show that x = [x, y]. Since our condition is symmetric, the same arguments show that
y = [~, y] = [y, ~], hence x = y.
[]
Let X be a compact metrizable topological space and let r be a homeomorphism of X. It is easy to see that if r is a topologically Anosov homeomorphism with respect to some metric compatible with the topology of X, then it is topologically Anosov with respect to any such metric. The following result obtained independently by Ombach [Oml], Sakai [ Sak4], and by Park, Lee, and Koo
34
1. S h a d o w i n g Near an Invariant
Set
[Par] shows that any topologically Anosov homeomorphism generates (in some sense) a Smale space. T h e o r e m 1.2.9. Let r be a homeomorphism of a compact metrizable space X . The following statements are equivalent. (1) r is topologically Anosov (with respect to some metric); (2) there exists a metric r such that ( X , r is a Smale space with respect to r. We refer the reader to [Oml, Sak4, Par] for the proof of this statement.
1.3
Shadowing
for Mappings
of Banach
Spaces
In this section, we describe some "abstract" shadowing results for sequences of mappings of Banach spaces. In Subsect. 1.3.1, we prove a general "local" shadowing statement (Theorem 1.3.1) [Pi4]. The main difference between the situation described in Subsect. 1.3.1 and the one in the classical Shadowing Lemma is the possible nonuniqueness of a shadowing trajectory in the first case. Let us note that the nonuniqueness of a shadowing trajectory is not a "rare" property; for example, in the case of a structurally stable dynamical system a shadowing trajectory is, in general, not unique. On the other hand, it was mentioned in Sect. 1.1 (see the remark after Lemma 1.1.2) that "global shadowing with uniqueness" is not a common phenomenon. Theorem 1.3.1 is one of the basic tools applied in this book to establish various shadowing properties (/;p-shadowing, weighted shadowing, and shadowing for flows in Chap. 1, shadowing for structurally stable systems in Chap. 2, shadowing near the global attractor for a parabolic PDE in Chap. 3). Of course, it is important to have a possibility to establish the uniqueness of a shadowing trajectory in the considered case of a sequence of mappings of Banach spaces. We give such a condition in Subsect. 1.3.2. In Subsect. 1.3.3, we describe a general scheme of application of Theorem 1.3.1. Subsection 1.3.4 is devoted to two "abstract" theorems on shadowing proved by Chow, Lin, and Palmer [Chol] and by Steinlein and Walther [Stel]. We show that one can prove the first mentioned theorem applying results of Subsects. 1.3.1 and 1.3.2. It is also shown how to modify the method of proof of Theorem 1.3.1 to establish the LpSP near a "Steinlein-Walther hyperbolic set". In [Pli2, PLi3], Pliss obtained necessary and sufficient conditions under which a family of linear systems of differential equations has uniformly bounded solutions. We "translate" these results into the shadowing language in Subsect. 1.3.5. It is shown (Theorem 1.3.6) that if a family {Ak} of invertible linear mappings in IR~ is piecewise hyperbolic, then the equations Ukq-1 =
Akuk Jr zk+l
1.3 Shadowing for Mappings of Banach Spaces
35
have a solution {uk} such that
II{uk}ll _< Mll{zk}ll, where the constant M depends on the characteristics of the piecewise hyperbolicity of {Ak}. It follows from the results of [Pli3] that a converse (in some sense) statement is also true. 1.3.1 S h a d o w i n g for a S e q u e n c e o f M a p p i n g s Below we show that a lot of shadowing problems can be reduced to the following "abstract" shadowing problem. Let Hk be a sequence of Banach spaces (k 9 7] or k 9 7]+), we denote by [.I norms in Hk and by I1.11the corresponding operator norms for linear operators. Let us emphasize that the spaces Hk are not assumed to be isomorphic (and in Chap. 2 we apply the main result of this section in the case when Hk are Euclidean spaces of different dimensions). Consider a sequence of mappings Ck : I l k ~
Hk+l
of the form Ck(v) = Akv + wk+l(v),
(1.51)
where Ak are linear mappings. It is assumed that the values ICk(0)l are uniformly small, say, Ir _< d. We are looking for a sequence vk E Ilk such that Ck(vk) ----vk+l and the values IVkl are uniformly small, for example, the inequalities sup Ivkl _< Ld k
hold with a constant L independent of d. First we prove an auxiliary statement. Consider two Banach spaces of sequences ~; = {Vk E Ilk}. Let s be the usual space with the norm
Ih~lloo = sup Lvkl (we use the same notation in both cases k E 7]+ and k C 7]), and let B be a Banach space such that its norm is monotonous, i.e., the inequalities Ivk] < lv~l imply IMIB 0.
(1.59)
For ~ E/3 set ~2(0) = {wk(Vk-a) : k >_ 0} with Wo = 0 (in the case of Hk, k E 77, set t0(0) = {wk(vk-1)}). It follows from condition (a2) that if ~ is a solution of the equation = g~(O),
(1.60)
then relations (1.59) hold, hence ~ satisfies equalities (1.58). Thus, it remains to solve Eq. (1.60) and to estimate its solution. Assume that inequality (1.56) holds. Let D be the ball {~ E/3 : ]]01113 0 such that inequalities (1.5~) and the inequality gNx < 1 (1.65) hold, where I+A
g~ = N1--:- X. Set L-
N1 1-tr
A do---L.
If for a sequence of mappings (1.51) we have
Ir
< d < do (k > 0 or k e 7Z),
then there exist points vk E Hk such that ek(Vk) = vk+l and [vk[ < Ld. Proof. We consider the case k > 0. Take B =/~oo, the norm of 13 is monotonous and condition (1.52) is trivially satisfied. We denote the norm of 0 E B simply by I[0[[. Define a linear operator G on 13 by
38
1. Shadowing Near an Invariant Set
where Un1 = P n z n ' Un2 = E
An-l...
un3 =
AkPkzk,
B . . . . BkQk+lzk+l.
k=n
k=0
Let us show that G maps B into B and estimate its norm. Take 2 C B. Obviously,
I1~111_ 0. Fix a number A ~ (0, 1) and set Akv
= .~-1"o
for k < 0, Akv = Av for k _> 0.
Obviously, conditions (1.63) and (1.64) are satisfied with Bkv = ~v, k 0 and set Wk+l(V) = d for all k, v. We can take a = 0 in (1.54), so that condition (1.65) is automatically satisfied. Fix arbitrary v0. We deduce from the equalities Ck(vk) = Vk+l that Vk+l = ~-lvk + d for k < 0, Vk+l = ,kvk + d for k >_ 0. It follows that vk=~lklv0--d(1+...+t vk=~kvo+d(l+,k+...+t
N) for k < 0 , k-l) for k_>0.
Hence, for any v0 with Iv0[ _< d we obtain a sequence 0 = {vk} satisfying the equalities Ck(vk) = Vk+l and such that ]lvll_< ( 1 + 1 _ - - ~ ) d . Note that the described construction is a model of behavior typical for structurally stable systems. As the simplest example, take a diffeomorphism r of the circle S 1 which has two hyperbolic fixed points, p with De(p) = ~ < 1, and q with De(q) = )~-1, and such that any trajectory of r tends to p U q. Obviously, the behavior of pseudotrajectories of r is similar to the one described in our example (we leave the details to the reader). In addition, there exist interesting infinite-dimensional examples in which Hk are the same for all k, but Sk "grow" and Uk "decrease" at every step. E x a m p l e 1.13 Consider Ilk = loo, k C 77, where loo is the usual Banach space of sequences x = {Xk : xk E IR, k C 2[}, with Llxll -- sup Ix l. k
For k E Z let Ak : Ilk ---* Hk+l be defined as follows: (Akx)m =
{ 1/2x.~, r e < k , 2x,~, m > k.
Obviously, conditions of Theorem 1.3.1 are satisfied in this example with ~ = 1/2, N = 1, { xm, r e < k , (Pkz)m =
O,
m > k.
40
1. Shadowing Near an Invariant Set
1.3.2 C o n d i t i o n s of U n i q u e n e s s
Let us again consider a sequence of mappings Ck : Hk ~ Hk+l
which have form (1.51), where Hk are Banach spaces. We give sufficient conditions for the uniqueness of a sequence {vk} obtained in Theorem 1.3.1. T h e o r e m 1.3.2. also that (b')
Assume that condition (a) of Theorem 1.3.1 holds. Assume
AkUk C Uk+l and
1
IIAkrv~tl- X;
(1.67)
(d) there exist numbers n0, A > 0 such that
Iwk+x(v) - wk+l(v')l _< ~01v - v' I for Ivl, Iv'l _< z~,
(1.6s)
and for ~ = Nno the inequalities
A+2~;7>1,
A 2~ -+-- ;Iz~l-
~lzk].
(1.75)
Assume that there exists 1 E 7/ such that zt r 0. Consider two possible cases. Case 1. There exists m _< I such that
I~1 > Iz~ml"
(1.76)
z~+ 1 ~ 0 and Iz~+, I > Iz~+l I.
(1.77)
Then obviously z,~ ~ 0. Let us show that in this case
It follows from (1.76) that
Izml < Iz;,I + Iz~l < 2lz~l. Now we deduce from (1.69) and (1.75) that 1 this proves the first inequality in (1.77). It follows from the obtained inequality and from (1.74) that IZ~+ll < ~lz;.I + 2~lz=l < ~ + 2~ -- 7lz~+kl hold for all k > 0. But these inequalities imply that
Iz~+kl >_ ~lz~,l ~ r162as k --, ~ , and this contradicts to (1.73). We proved that case 1 is impossible. Case 2. For all m _< l we have
Iz,~l < Iz~l.
(1.78)
42
1. Shadowing Near an Invariant Set
Then
Izml _< Iz~l + Iz~[ _< 21z~l, ~ < z. It follows from (1.74) t h a t IZ~+l[ < AIz~l + 2nIz~[ _< Al[Z~l, m _< I, where A, = A + 2n < 1. This gives the inequalities IzZI _< AllZf_~l _ 0 (we apply (1.73) here). Since k >_ 0 is a r b i t r a r y in (1.79), we see that z r -- 0. But then (1.78) implies that also z? = 0, hence zl = 0. T h e obtained contradiction completes the proof. []
Remark.
[1] An analogous s t a t e m e n t can be proved in the same way for a sequence of m a p p i n g s Xk : Hk ~ Hk-1 of the form
x~(v) = B~v + w~_l(v) under the conditions
BkUk C Uk-1, BkSk C Sk-1. We leave the details to the reader.
Remark. [2] We also can obtain conditions of uniqueness for a "one-sided" sequence of m a p p i n g s r : H k --* Hk+l, k _> 0. Assume t h a t Hk = Uk for k >__0, t h a t condition (b') of T h e o r e m 1.3.2 holds, that there exist n u m b e r s x0, A > 0 such that (1.68) is fulfilled, and, finally, t h a t 1
---~o>1. A Then the relations
Ck(vk) = vk+,, r
= U~+l, Iv~l, [u~l _< ~, k > O,
imply t h a t vk = uk, k > O. We again take zk = uk - vk. It follows that the inequalities 1
1
Izk§ >_ ~lz~[- ~01zkl _> 71z~l, k _> 0, where 7 = X - no, similar to (1.75), are true. If we assume that zz ~ 0 for some l E 7/+, this will lead to and the obtained contradiction will prove our statement.
1.3 Shadowing for Mappings of Banach Spaces
43
1.3.3 A p p l i c a t i o n t o t h e Classical S h a d o w i n g L e m m a We describe in this subsection a scheme of application of Theorem 1.3.1. The main application here is the Shadowing Lemma in the case k ~ 0 (Theorem 1.3.3 below). Similar arguments are applied in this book to study more complicated situations (such as shadowing for structurally stable systems in Chap. 2), so it is useful for the reader to see their application in the simplest case. This scheme is applied literally in the same way to establish s and weighted shadowing (see Theorems 1.4.2 and 1.4.5 below). Let us describe here the common first step of the scheme (below we denote by SL,/2p, and WS the cases of the Shadowing Lemma, of/2p-shadowing, and of weighted shadowing, respectively). The main idea of the proof is the following one. Let ~ = {xk} (where k C 71 or k E 71+) be a sequence in a neighborhood W of an invariant set A for a diffeomorphism r such that I~(Xk) -- Xk+ 1 [ ~ d.
We consider the sequence of Banach spaces Hk = IR~ and the sequence of mappings Ck(v) = r + v) - xk+l. The "extended" hyperbolic structure S, U given by Lemma 1.2.1 (for the SL and s cases) and a similar structure connected with the Sacker-Sell spectrum (for the WS case) allow us to show that we can represent Ck in form (1.51) and then apply Theorem 1.3.1 (in the SL case) or its modifications (in other cases). Thus, we consider a compact invariant set A for a diffeomorphism r of IRn. If A is a hyperbolic set of r let C, A0 be the hyperbolicity constants of A. We take e > 0, A E (A0, 1), and construct the corresponding Lyapunov norm and extended hyperbolic structure S ~, U' in a neighborghood W of A. In the conditions of Theorem 1.4.5, let C, A0 be given by Theorem 1.4.4. We again take e > 0, A E (A0, 1), and apply in this case Lemma 1.4.1 to construct a "Lyapunov" norm and an extended structure S t, U' in a neighborghood W of A. The same reasons as in the proof of Theorem 1.2.3 show that, without loss of generality, we may assume the Euclidean norm to be Lyapunov. We write below S, U instead of S ~, Uq Decreasing W, if necessary, we can find a constant N > 0 such that [IH~[I, [[Hi[ [ 0 such that for the number 1 t% -- 4N1' where N1 =
NI(N, A) is
defined in Theorem 1.3.1, the inequality 2Ne < no
(1.81)
44
1. Shadowing Near an Invariant Set
holds. Now we take da > 0 such that, for x , y E W with Ix - y[ < da, inequalities (1.10) and (1.11) of Lemma 1.2.1 (for the SL and Z;p cases), and similar inequalities of Lemma 1.4.1 (for the WS case) are satisfied. Take a sequence ~ = {xk : k _> 0} C W with IXkTl --
~)(Xk)l < d ~ dl, k ~ O.
Set Hk = IR~, k > 0. Define Ck : Hk --~ Hk+l by Ck(V) ~- ~)(Xk '~ v) -- Xk+l.
Since the neighborghood W is bounded, we can find do E (0, dr) such that, for the representation Cdv) = Dr + Xk+~(v), the inequalities [Xk+I(V) -- Xk+I(V')[ ~ K01V --
Vtl for [vl, Iv'l _< do
(1.82)
hold. Denote Dk = DCk(0), Sk = S ( x k ) , P ~ = 11~k, Uk = U(xk), Qk = H2 k. Set, for k > 0, Ak = A~ + A~, where ASk = P k + l D k P k , A'~ = Qk+lDkQk. Let us estimate IIAk- Dkll = IIA~ + A~ - Dk(Pk + Qk)[] _
0} if
- x +11 < d ___ do. Thus, for the point x = x0 + Vo we have
Ir
- xk[ _ O. []
This completes the proof. 1.3.4 T h e o r e m s o f C h o w - L i n - P a l m e r a n d S t e i n l e i n - W a l t h e r
Many shadowing results were established for mappings of Banach spaces. In this subsection, we describe in detail a variant of "abstract" shadowing lernma proved by Chow, Lin, and Palmer in [Chol] and a theorem of Steinlein and Walther [Stel]. A detailed treatment of exponential dichotomies and shadowing for mappings of Banach spaces is given by Henry [He3]. Chow, Lin, and Palmer studied in [Chol] a sequence of C 1 mappings Ck : Hk --* Hk+l, k E 77, of Banach spaces H~ under the following conditions: (1) there exist subsets Tk C Hk such that Ck(Tk) C T~+I; (2) for any x E Tk there is a continuous (in x) splitting
Ilk = Sk(x) ~ Uk(x)
(1.86)
46
1. Shadowing Near an Invariant Set
such that
DCk(x)Sk(x) C Sk+l(r
DCk(x)Uk(x) = Uk+l(•k(X)),
and DCk(x) : Uk(x) ---+Uk+l(r is an isomorphism with a bounded inverse; (3) there is a constant g > 0 such that if Pk(x) (Qk(x)) are the projectors in Hk onto Sk(x) parallel to Uk(z) (respectively, onto Uk(x) parallel to Sk(x)), then IIPk(x)lh IIQk(x)ll -< N for all x E Tk, k E 71; (4) there exists A C (0, 1) such that, for any finite sequence xk, zk+l ---C k ( z k ) , . . . , x~+l = r with xk E Hk and any integers k _< n, the inequalities
I]DC~(z~)...
DCk(Xk)Pk(Xk)ll< g)~ n-k+1,
IlDCk(xk) -1 ...DG~(x,~)-aQn+l(Xn+,)l[ 0 such that Ck and D~bk are bounded and continuous in closed A-neighborghoods T~ of Tk uniformly with respect to x in T~ and k E 71. The main result of [Chol] states that there exists r > 0 such that given r C (0,r one can find d > 0 with the following property. If for a sequence {yk E Tk}, k C 71, the inequalities
Ir
- ~+11 < d, ~ ~ 71,
(1.89)
hold, then there exists a unique sequence xk E Hk, k C 77, such that Ck(xk) = xk+l and Ixk - Ykl -< r k E 77.
(1.90)
Let us show how to prove this statement applying Theorems 1.3.1 and 1.3.2. Fix a natural number l, such that N~ ~ < A
(1.91)
and define, for k E 7I, Set G m = H,~v, m E 7/. Take p E T,~ and denote Po = P, Pl = r
. . . , Pmu+u-1
=
~bm~,+t,-2(Pm~+~-2)"
Then we can write, for p E Tm~, ~m~,.(p + v) = ~.~.,~(p) + A,nv + w'~+x(V), where
Am = DCm~+~-l(pm~+~-l)... DCm~(po). It follows from the definition of Am and from (1.87), (1.88), and (1.91) that for
1.3 Shadowing for Mappings of Banach Spaces
47
w : = wm~(p), w = s, u, P, Q; p e T,~, and for A,~ :Gm ~ Gin, analogs of conditions (a) and (b) in Theorem 1.3.1 are satisfied. By condition (5) of [Chol], given ~ > 0 we can find A0 e (0, A) (A is used in the definition of T~) such that for w'~+~ an analog of (1.54) holds (with A0 instead of A). Take numbers do, L given by Theorem 1.3.1 for ~, N, A0. The same condition (5) implies the existence of a uniform Lipschitz constant s for ~bk on the sets T~. Take dl > 0 such that
diLl 0 such that 1 -Ke A o + K e < A, ~00
1 > ~.
(1.98)
Set N = K 3, introduce N2 = N2(N, A) (see T h e o r e m 1.3.5), and find ~ > 0 such that inequality (1.96) holds. We assume that the inequality 2eK 2 < ~
(1.99)
is fulfilled. Find A E (0, r) (the number r is from condition (1) in Definition 1.14) such that the following statements hold: (sl) if r + v) = r + Dr + X(x, v), then
[X(X,V ) _X(X,V,) I < a for x E T, Ivl, lv'l < Z~; --
2
(s2) for x, y E T, the inequality Ix - Yl -< A implies that (s2.1) lip(x) - P( )ll, IIQ(x) - Q(y)II < and (s2.2) if L is a linear subspace of H such that Q(x)L = U(x), then Q(y)L = T h e existence of A follows from our assumptions on r and T. Take a sequence ~ = {xk : k E 77} C T such that
52
1. S h a d o w i n g Near an I n v a r i a n t Set
Ir
- ~k+,l < d < z~.
Denote
Pk = P(xk), P~ -- P(C(xk)), Qk = Q(xk), Q~ -- Q(5(xk)), and set Sk = PkH, Uk = QkH. Define linear operators
Bk = Pk+lDC(xk)Pk + Qk+iDC(xk)Qk, Ck = Pk+lDC(xk)Qk. Take v E Sk. Since Qkv = 0 and Pk+IH = Sk+l, we deduce that Bkv E Sk+l. It follows from (2.3), (s2.1), and (1.98) that
IBkvr _< tP;nC(xk)vl + I(P~+l -
P;)DC(~)vl _ Q i kDr
_ Qk)Dr I ,
>_ _>
(~_1 K~),v,_>~-Ivl.
Hence, Bk satisfies condition (a3) of Theorem 1.3.5. Obviously, I[Ckl[ _< K 3 = N. Since K _> 1, the estimates [IPkll, [[Qkl[ < N are fulfilled. Define, for Iv[ < A, -- Xk+i] "Jr nC(x~)v
Ck(V) -~- [r
+ X(xk, v).
A sequence {vk} satisfies the relations r
"Jr Vk) = Xk.4-1 -t- Vk-t-1
if and only if Ck(vk) = Vk+l. Represent Ck(v) in form (1.51) with Ak = Bk + Ck and Wk+l(V) = [r -- Xk+l] + (Dr -- Ak)v + X(Xk, v). Let us estimate [l(Dr - Ak)[[. Note that since Q'kDr dition (2.1) of Definition 1.14, we can write
= {0} by con-
1.3 Shadowing for Mappings of Banach Spaces I + QkDr
ck = Pk+lDr It follows from Dr
53
= (Pk+l + Q k + l ) D r
+ Qk) that
II(DC(xk) -- Ak)ll = II(Qk+l - Q'k)Dr
0 defined for a C (0,7r/2) and having the following property. Assume that L1 and L2 are transverse linear subspaces of ]R"~ such that Ol t(L1, L2) >_ -~. If Mi = Li + xi and [x~[ _< c,i = 1,2, then there exists a vector y E 1141M M2 such that lYl-
a.
Then there exist positive constants R,'),, lo, ll,g (depending on N, a) such that for any d > 0 the following holds: if D is a (~Rd, lo, d)-ball with respect to (S, U), and $1 = S' + x', U~ = U' + x', where [x'[ _< 2d, then D contains a subset D1 being an (Rd, l~, gd)-ball with respect to ($1, U1). Remark. An analog of this lemma (for small disks in stable and unstable manifolds of trajectories in a small neighborhood of a hyperbolic set) was stated without proof in [Pi2] (see Lamina A.8). Proof. Let L be a linear subspace of IR'~ of the form
(1.110)
L = {x + F x : x E U},
where F U = S. Obviously, we can find a constant 10 (depending only on a) such that if IIFll < 10, then o~
Z(L, U) _< ~. Set
N
11 - sin(a/2)
'
(1.111)
9 = 2(p(a) + 1), 3' = 2N(ll + 1), R - 2p(a) 1 +la"
Let D be a (TRd, lo, d)-ball with respect to (S, U). Assume that D is given by a formula similar to (1.109). Consider the corresponding linear subspace L given by (1.110). Since dimL=dimU, it follows from the second condition of our lamina that there exists a linear subspace L' Q L such that S' @ L' = IR'L Inequality (1.111) implies that Ol
/(L', S') > ~.
(1.112)
Obviuosly, we can write
L ' = {x + F'x : x E U'}, where F'U' = S'. Take a vector v E L', let y = P'v E S' and x = Q'v E U', then obviously y = F'x. It follows from (1.112) that
1. Shadowing Near an Invariant Set
58
I~1 _> Ivl sin(a/2), hence lyl < NIvl NT.
(1.120)
and Take a family {Ak}, let it be piecewise hyperbolic with constants T, N, )~, a on segments I0 = < -(x:),tl >, . . . , Im =< tm,OO > . Fix a sequence 2 G s By Lemma 1.3.2, there exist vectors u~,n E I j , j -0 , . . . , m, such that for u j analogs of (1.107) and (1.108) hold on 5" Set
d = LII II. Let SJ(k), UJ(k), k E Ij, be the subspaces given by Definition 1.16. Denote
sJ(k) = sJ(k) + ut, uJ(k) = uJ(k) + ut. To simplify notation, denote tl = a, t2 = b. Obviously, the set Do = {x e U~ is a (TRd, 0, 0)-ball with respect to (S~
Ixl T and that
v~ = A k - 1 . . . A ~ v I for k 9 I1,
(1.122)
we deduce from (1.102) and (1.118) the estimate [v~[ < gAb-~gd < d.
(1.123)
' vk, v k' similarly to w~, v~. Now we take two vectors w~, w a' 9 D 1 and define wk, wk, Represent Vk = zk +yk, v'k = x'k +y~, where zk, z'k 9 U~(k), Yk,Y'k 9 S~(k) 9 Since the set D 1 - u~ is an (Rd, li,gd)-ball with respect to ( S l ( a ) , U l ( a ) ) , we have [Ya -- Vlal ~ ll[Xa], hence
lyol < gd + l Rd. Formulas similar to (1.122) show that [yk] _< NAk-~Iy~[ < N(l~R + g)d for k 9 11.
(1.124)
The same reasons show that - ykl < N
and
k-~
- yol
,~a-k Ix~ - xkl >_ - ~ - I x ' . - x.I
(1.125)
for k 9 I1. Since
[Y: - Ya[ 0 such that a M < 1, where M is given by Theorem 1.3.6, and assume that functions wk(v) satisfy the inequalities
Iwk(v') - wk(v)l _< ~ 1 , ' - vl. /f
(1.130)
Iw~(O)l ~ d, k e 77, with some d > O, then there is a sequence ft satisfying
(1.131)
?dk+l : Akuk + W k + l ( U k )
and such that I1~11 --- Ld, where Z = M(1 - a M ) -1 Proof. Fix d > 0 and assume that condition (1.130) is satisfied. Set e = a M and u = M d . Construct a sequence f~J E B , j >_ 0, as follows. Set u ~ - 0. Let ~1 be a solution of u k+l I Aku~ + wk+l(0) such that II1~111 ~ M[I~(O)I I < M d = u.
For j > 1 define inductively ~j+l as solutions of Ujk+l +l :
AkU~k+l + Wk+l kUkJ
with the following property: they satisfy the equations b j-l-1 " (U~"t-1 U~) -~- j k+l -- U~+I = Ak Yk+l,
where YJk+l = Wk+l (UJk) -- W k + l (~/,~-1) ,
and the estimates ii~J +1 _ ~2Jll _< MII~?3ll hold (this is possible by T h e o r e m Since ]1~1 -
~~ _< M d
1.3.6).
= u and Ila j+l -
~Jll -< aMlr ~j
- ?~j-1 [I
~--"
ell ~j
- ~J-lll
1.4 Limit Shadowing
63
forj >_ 1, we see that II~j - ~J-lll ~ {~j--1/], hence the sequence uJ converges to some ~. It easily follows that fi satisfies (1.131). Since I1~11 ~ II ~1 - ~~
+ I1~ ~ - ~111 + . . .
~ ~ + ~ +,..
-
~
1--g
-
Ld,
our theorem is proved. Consider the equations
uk+l = (Ak + Bk)uk + Zk+l,
(1.132)
where Bk are linear operators. Taking Wk+l (uk) = Bkuk + Zk+l, we represent the equations above in form (1.131). It follows from Theorem 1.3.7 that if the family {Ak} satisfies the conditions of this theorem, then there exists r = e(N, ~, a) such that if IIBkll, Izkl < e for k E 77, then there is a sequence ~ satisfying (1.132) and such that Ilfil[ < 1. Now we can formulate the necessity statement proved by Pliss in [Pli3]. T h e o r e m 1.3.8. There exist functions N : N(C, e), c~ = c~(C, e), O = O(C, ~), and )~ = A(C, e) E (0, 1) defined for C, e > 0 and having the following property. Let {Ak : k E 7/} be a family of invertible linear mappings of IRn such that IIAklI, IIA;lll < C. If, for some e > 0 and any Bk,zk with [IBk[I, Izk] < e, Eqs. (1.132) have a solution f, with Ilf,II < 1, then the family {dk} is piecewise hyperbolic on 71 with constants O, N, A, and c~. The proof of Theorem 1.3.8 is rather complicated. We do not give it here and refer the reader to the original paper [Pli3].
1.4 Limit Shadowing In the usual statement of the shadowing problem (see Sect. 1.1), the values r(xk+l, r are assumed to be uniformly small. We can impose another condition on these values, d, = r ( x k + l , r
~
0 as k ~
oo,
(1.133)
and look for a point x such that h, = ~(r
~k) --+ 0 as k ~
~.
We study the introduced shadowing property (we call it the limit shadowing property) in Subsect. 1.4.1. It is shown that in a neighborhood of a hyperbolic set a diffeomorphism has this property (Theorem 1.4.1). In Subsect. 1.4.2, we investigate the rate of convergence of the values hk in terms of dk. Theorem 1.4.2, the main result of this subsection, shows that if
64
1. Shadowing Near an Invariant Set
the sequence {dk} belongs to a Banach space s p _> 1, then the sequence {hk} belongs to the same space. Passing from the spaces s to their weighted analogs, the spaces s we obtain a possibility to establish the "s in a neighborhood of an arbitrary compact invariant set (not necessarily hyperbolic) under the corresponding conditions on the weight sequence ~ (see Theorem 1.4.5 in Subsect. 1.4.3). These conditions are formulated in terms of the so-called Sacker-Sell spectrum [Sac]. Another possibility to establish the "s is to assume that the weight sequence grows "fast enough" (see Theorems 1.4.6 and 1.4.8 in Subsect. 1.4.3). Hirsch studied in [Hirs4] asymptotic pseudotrajectories, i.e., sequences {xk) such that limk~odff k < )~, where 0 < s < 1. He found conditions under which an asymptotic pseudotrajectory is asymptotically shadowed. The main shadowing result of [Hirs4] is described in Suhsect. 1.4.4. The main results of Subsects. 1.4.1-1.4.3 were obtained in [Ei2].
1.4.1 L i m i t S h a d o w i n g P r o p e r t y Let r be a dynamical system on a metric space (X, r). D e f i n i t i o n 1.17 We say that r has the "LmSP" (the "limit shadowing property'? on Y C X if for any sequence ~ : (xk : k E 7/) C Y such that (1.133) holds there is a point x such that r(r
Xk) ---+0 as k ---* co.
If this property holds on Y = X , we say that r has the LmSP. From the numerical point of view, this property of a dynamical system r means the following: if we apply a numerical method that approximates r with "improving accuracy", so that one-step errors tend to zero as time goes to infinity, then the numerically obtained trajectories tend to real ones. Such situations arise, for example, when we are not so interested in the initial (transient) behavior of trajectories but want to get to areas where "interesting things" happen (e.g., neighborhoods of attractors), and then improve accuracy. It is easy to see that there exist systems which do not have the LmSP. E x a m p l e 1.18 Consider the system r on the circle S 1 with coordinate x E [0, 1) such that any point of S 1 i8 a fixed point of r It was shown in Sect. 1.1 that this system does not have the POTP. To show that it does not have the LmSP, consider the sequence {Xk : k E 7/} such that x0 = 0 and
1.4 Limit Shadowing
65
1 xk+l = xk + ~-7--:--7., - t l -(mod 1), k >_ O. Obviously, condition (1.133) is satisfied, but since the series 1
~-~k+l
k>O
diverges, the sequence xk does not tend to a fixed point of r Now we give an example of a system that has the LmSP but does not have the POTP. E x a m p l e 1.19 Consider again the circle S 1 with coordinate x 6 [0, 1). Let r be a dynamical system on S 1 generated by a mapping f : [0, 1] ~ [0, 1] with the following properties: f is continuous and increasing; the set { / ( x ) = x} coincides with {0, 1/3,2/3, 1}; f ( x ) > x for x E (0, 1/3) U (1/3, 2/3); - f ( x ) < x for x E (2/3, 1). Thus, r has three fixed points (p = 0, q = 1/3, r -- 2/3) on S 1, the point r is asymptotically stable, the point p is completely unstable (i.e., it is asymptotically stable for r and q is semi-stable. Take a natural m and denote by V~ the 1/m-neighborhoods of the points s -- p, q, r. Set -
-
-
WI~ = (0, 1/3) \ (WTM U V~), W~ = (1/3, 1) \ (WTM U Vqm U V~). Note that there exist positive numbers am suchthat f ( x ) >_ x + 2am for x E W • ,
If(x) - r] _< Ix - r[ - 2am for x E W~.
To show that r has the LmSP on S 1, take a sequence ~ = {xk : k >_ 0} C S 1 such that Obviously, there exist numbers m0 and bl > 0 such that for m > m0 the following holds. For any z 6 r the inequality dist(x, V~) _> bl for s, u 6 {p, q, r}, s ~ u, is fulfilled. Find k0 such that for k > k0 we have dk < bl. Below we take m > m0 and k > k0. It follows from our choice that if xk E V~, then Xk+~ cannot "jump" into V~ with u # s. Take an index m and find a number b2(m) which is less than any of the following three values:
(1_1)_
1)
66
1. Shadowing Near an Invariant Set
Now we claim that for any m there exists an index s C {p, q, r} and /c(m) such that xk G V~ for/C _>/c(m). (1.134) It follows from our choice of m0,/c0 that the fixed point s with the described property does not depend on m (since "jumps" from V~ into V~ with different s,u are impossible). Thus, if we prove this statement, this will establish the LmSP for r The same reasons show that if xk C Vpm 12 VqTM U V~ for /C >_ /Co, then we have nothing to prove. Thus, it remains to consider two possible cases. We fix m and take l(m) such that, for k >_ l(m), the inequalities dk < min(am, b2(m)) are fulfilled. Case 1. xz(m) E W~. It follows that for k >_ l(m) we have IXk+l - - r I < Ixk -- r I -- am while Xk E W~. By the choice of b2(m), there exists /c(rn) such that (1.134) holds with s = r. Case 2. xz(m) E W1m. It follows that for/C >_ l(m) we have Xk+l > X k "-~ a m
while xk C WlTM. Hence, either there exists k(m) such that (1.134) holds with s = q, or there exists /el such that Xkl E V2TM, and case 2 is reduced to case 1 (note that we cannot "return" from V~ into W1~ by the choice of b2(m)). It remains to show that r does not have the P O T P on S 1. Take arbitrary d > 0 and let 1 X 0 --
3
d
2'
1 Xl =
d
~ ~V 2 '
Xk =
r
/C < 0; X k -~- ( ~ k - l ( x l ) , ]g ~> 1.
Since f(xo) e (Xo, 1/3), we see that { = {xk} is a d-pseudotrajectory for r Any trajectory of r belongs to one of the sets [0, 1/3], (1/3, 2/3], (2/3, 1), while intersects small neighborhoods of the points 0, 1/3, and 2/3. This shows that r does not have the POTP. Let us formulate and prove the main statement of this subsection. T h e o r e m 1.4.1. Let A be a hyperbolic set for a diffeomorphism r oflR n. There
exists a neighbourhood W of A such that r has the L m S P on W . Remark. For a fixed point A, this result was proved in [Ak] (Theorem 13 of Chap. 9). One can prove the stated result applying Proposition 11 of Chap. 10 in [Ak], but we prefer to give here a simple direct proof. Proof. Take a neighborhood U of A and numbers u, A given by Theorem 1.2.1. Let W0 be a neighborhood of A on which r has the LpSP with constants L, d~
1.4 Limit Shadowing
67
(see Theorem 1.2.3). Set W1 = U VI W0. Decreasing 12, if necessary, we can find a neighborhood W of A such that the 12-neighborhood of W belongs to W1. For a sequence ~ -- {xk, k E 7/} C W, set dk = Ixk+l - r
(1.135)
and assume that dk --* 0 for k --* co. Take an integer j0 > 1 such that ,4 < Ldojo. For every j _> j0 find an index kj such that 12 dk < ~ < d o for k > kj. Since r has the LpSP on Wo, there exist points yj such that 12 12 Iek(yj) - xkl _< ~- < ~-, k > kj, j > jo.
(1.136)
It follows from the choice of j0 that ek(yj) E W1 C U for k > kj, j > j0. By (1.136),
Ir
- r
< 12 for k > kj, j > j0.
Theorem 1.2.1 implies that in this case ek~(yj) E W~(r
for k > kj, j > jo,
and hence
Iek(yjo) -- ek(yj) I ~ uk-k312 for k > ki, j > j0. (1.137) Take z = YJo. By (1.137), there exist numbers Ij > kj for j > j0 such that 1 < = for k > lj, j > j0. 3 We deduce from this inequality and from (1.136) that given ~ > 0 we can find j such that iek(z) _ r
Ixk - ek(x)l < Ix~ - r + Ir - r 12 1 < 2-)=+ j < ~for k > tj.
1. 1.4.3 T h e S a c k e r - S e l l S p e c t r u m a n d W e i g h t e d S h a d o w i n g We study in this subsection a shadowing property which is a "weighted" variant of the s described by Theorem 1.4.2. We fix a sequence ~ = {rk >__1 : k > 0}, a number p > 1, and consider instead of the space s (to be exact, instead of the space s the space s of sequences 9 = {vk C ]Rn : k > 0} with the norm
72
1. Shadowing Near an Invariant Set
II ll ,p =
r lvklP) lip
The shadowing problem is formulated similarly to the one in the previous subsection, i.e., given a dynamical system r in IRn and a sequence = {xk E IRa: k > 0} with small [[g(~)[[~,p, to find a point x such that [[h(x,~)t[e,p is small (see the definitions of g, h before Theorem 1.4.2). We will show that it is possible to establish this "s in a neighborhood of an invariant set A of a diffeomorphism r not assuming that A is hyperbolic. To do this, we need the concept of Sacker-Sell spectrum for an invariant set [Sac]. Note that in [Sac] the spectrum is defined for a very general class of dynamical systems, while we describe it here only for diffeomorhisms of IR~. Let A be a compact invariant set for a diffeomorphism r of ]R~. We identify the set of nonsingular n x n real matrices with the group G L ( n , IR) of invertible linear transformations of IRn. Fix p > 0 and consider the mapping 9 p : A x 7] ~ G L ( n , IR)
given by 9 p(x, k) = pkDCk(x).
D e f i n i t i o n 1.21 We say that ~p has "exponential dichotomy" over a point x E A if there exists a projector P = P ( x ) in IR '~ and numbers K > 0 , a E (0, 1) such that i l ~ ; ( x , m ) P t / l ( x , l)li < g a m.' for I < m, lir
< g ~ ' - ~ for m < I.
For a point x E A, the resolvent T~(x) is defined by 7~(x) = {p: ~p has exponential dichotomy over x}. Now we define the spectrum S ( x ) by
= (o,
\ R(x),
the Sacker-Sell spectrum Z ( A ) by Z(A) = U EAZ(X), and the resolvent 7~(A) by
n(A) = (0, +oo) \ We need one more definition.
1.4 Limit Shadowing
73
Definition 1.22 We say that the set A is "invariantly connected" if it cannot be represented as the union of two disjoint nonempty compact invariant sets. The main result about the spectrum S ( A ) of a compact invariant set we need is the following statement obtained by Sacker and Sell (it is a corollary of Lemmas 2, 4, 6, and Theorem 2 in [Sac]). T h e o r e m 1.4.4. Assume that A is invariantly connected. Then (a) the spectrum S ( A ) is the union of k < n compact intervals, Z ( A) = [al, bl] U . . . U [ak, bk]; (b) for any p E ~ ( A ) there exist constants C > O, Ao E (0, 1), and a continuous family of linear subspaces S(x), U(x) C IRn for x C A such that
(b.1) S(x) 9 U(x) = ~'~; (b.2) D r
= S(r
Dr
= U(r
(b.S)
pklnCk(x)vl _< C~oklvl for p-klDr
v 9 S(x), k > O,
< CAkolvl for v e U ( x ) , k >_ 0;
(1.140) (1.141)
(c) if p > bk, then S(x) = { 0 } , U ( x ) = IR n for x C A. Remark. In [Sac], the spectrum is defined in such a way that analogs of (1.140) and (1.141) hold for e x p ( - k # ) l n C k ( x ) v l (instead of pklDCk(x)vl) , hence S(A) and the original Sacker-Sell spectrum are related by the transformation p H e x p ( - # ) . Of course, this does not change the geometry of the set 2J(A).
The main result of this subsection is the following statement. T h e o r e m 1.4.5. Let A be an invariantly connected compact invariant set for a diffeomorphism r of ]Rn. Assume that for some r,p >_ 1 we have p = r lip C 7Z(A). Then there ezists a neighbourhood W of A and numbers L, do > 0 such that if a sequence of points ~ = {xk : k > O} C W satisfies the inequality
Ilgff)ll~,~ ~ d ~ do, then there is a point x such that
IIh(x,5)ll~,v _< Ld,
(1.142)
w h e r e , = { r k, k >_ O} .
/f [p, cr fq ~U(A) = 0, then we can find W and do such that a point x with property (1.132) is unique.
74
1. Shadowing Near an Invariant Set
In the proof, we use the l e m m a below. 1.4.1. Assume that p E ~(A). Let ~0 E (0,1) and C be given by Theorem 1.4.4. Then for any c > 0,~ E ()~o, 1) there exists a neighborghood W = W(c, ~) with the following properties. There exist positive constants N', 5, a C ~ norm I.Iz for x E W, and continuous (not necessarily Dr extensions S', U' of S, U to W such that (1) S'(x) @ U'(x) = IR~, x E W; (2) for x,y E W with lY - r < 5, the mapping / / ~ D r (llyDr is an isomorphism between S'(x) and S'(y) (respectively, between U'(x) and U'(y)), and the inequalities
Lemma
p111;DC(x)vl~ 1 and such that p = r lip E ~(A). Let B = s obviously the norm of B is monotonous and satisfies condition (1.52). It follows from Lemma 1.4.1 that
AkSk
=
Sk+l,
AkUk
= Uk+l.
Now we define a linear operator ~ on s in the same way as in Theorem 1.3.1, G~ = ~1 + ~. + ~3, where 1
u n =
PnZn,
2
t t n =-
~-1 ~_,
An-,...AkPkzk,
un3 = -
k=-oo
~2
B,~. .. BkQk+lZk+l 9
kmn
Let us show that ~ maps/:~,p into s By (1.80),
and estimate its norm. Take 2 E s
I@II~,,< NII21Kp. Let
us
write n--I
P
r"]u~F = Ip'~u~I' = k~__o(PA._l)... (pAk)Pkpkzk
9
Since Pkzk C Sk and (1.143) holds, we obtain the inequality
n--1
])P
We apply the Minkowski inequality (see the proof of Theorem 1.4.1) and obtain the estimate
Ilelk, 1. We claim that, for a sequence ~ = {xk : k > 0}, relation (1.142) cannot hold with d < d~ for two different points x, z'.
1.4
Limit Shadowing
77
To obtain a contradiction, assume that (1.142) holds for different x and x'. Let Ck(x) - xk = vk, Ck(z') - xk = wk.
Since IMIoo -< IMkp, we deduce from (1.142) that
I~1, Iwkl
Ld~o9
(1.149)
It follows from our assumptions that {Ok(x)}, {r Comparing the relations
C W. Set zk = vk - wk.
x~+l + Vk+l = r
q'[Zo[, and pk§
[Zk+ 1 =
pk p]zk+l I >_ p%lzkl,
we get by induction that
Pklzkl >_ "?lzol for k ___ O. If z -r z', then [Zol -r O, and it follows that for ~ = {zk} we have
Ipkzkl "
II~ll,,, =
k
0
>_
bkzoI p
k
= oo.
0
The contradiction with the inequality INk,
completes the proof.
-< Ilvll,,p + II~ll,,p _< 2Ld~o
[]
Now let us show that if the weights "increase fast enough", then we can directly establish a "weighted" shadowing property with the uniqueness of the shadowing trajectory. T h e o r e m 1.4.6. Let r be a Cl-diffeomorphism of ]R'~ and let U be a bounded set in IR'~. Assume that p > 1 and that the weight sequence ~ = {r k ~ 1 : k _> O} satisfies the conditions
78
1. Shadowing Near an Invariant Set
rk+l k pork for k >_ ko,
(1.150)
p = pl/, > M = max (1,supllDr " ~eu\
(1.151)
where
Then there exist L, do > 0 such that for a sequence ~ = {xk : k >_ O} C U satisfying the condition [[g(()lle,p _ 0} is a nonnegative sequence, and p, rI > 1, then
s
[
. ~ XlP
E aj o
rlJa
j>o
This implies (1.155). For p = 1, inequality (1.155) holds trivially with C(r/, 1) = 1.
1.4 Limit Shadowing
79
and consider the equation S~ - DF(O)~ -- @ which is
Let us fix ,~ E s equivalent to
Vk+, = n r
+ w~, k > O.
(1.156)
To solve this last equation, set Dk,k = I and Dj+l,k = D r 1 6 2
Dr
We claim that the sequence 0 = {vk : k > 0} defined by
Vk = -- ~ DjE~l,kwj, k >_ O, j=k
(1.157)
satisfies (1.156). To show the convergence of (1.157), apply (1.151) to find E (M p, P0) and set Y = MP" It follows from (1.155) that oo
Ivkl 0 such that rk < clrko for 0 < k < ko - 1. Now we estimate
k=O
k=0 rj
k=ko rj
The first sum does not exceed cl --rk
co
k~_:l ~j-k o ~
'
k=o
p~
cl < c2 = --korko, -
co
and the second one is estimated by
~
-~ ~_ m we have hence Ivkl _> M-1]Vk_ll > Mm-k]v,~l,
vk = D r
and
rklvkl" >- clvml'M ~" k-=m
\(~ - -po ;]
~k
=
~.
kmm
This shows that ~ does not belong to/:e,p. Hence, the operator S - DF(O) has a bounded inverse in/:e,p. We see that Eq. (1.154) is equivalent to = ( S - D F ( O ) ) - ~ ( F ( O ) - DF(O)~).
This last equation is solved exactly in the same way as in the proof of Theorem 1.2.3 (we take into account that IVkl < [If~lkp). [] To complete this subsection, we consider the following weighted analog of the s p a c e / : ~ . Fix again a sequence ~ = {rk >_ 1 : k > 0} and denote by/2~,oo the Banach space of sequences ~ = {vk E ]R~ : k > 0} with the norm
1.4 Limit Shadowing
81
II011~,oo = suprklvkl.
k>0
The same reasons as in the proofs of Theorems 1.4.5 and 1.4.6 show that their analogs are true for the spaces /:e,oo. Let us explain only how to obtain the main estimates. In the analog of Theorem 1.4.5, we take p E R(A) and consider the space /:~,~ with rk = p k . The basic step is to estimate IIGll. For example, to estimate the second term in the expression defining the operator ~, let us write
p~l,~l
= k~=o(PA~-I)'''(PAk)PkPkWk 1---~.
(1.161)
Then there is L > 0 with the following property: for any sequence {xk E IR '~ : k C 7]} such that the value
sup w-'kllr
-- xk+, I
kEZ
is finite, there exists a unique point x such that sup w -IkllCk(x) - xkl < n sup w -Ikllr kEZ
- Xk+l[.
kEZ
One can apply reasons similar to the ones in the proof of T h e o r e m 1.4.6 to obtain the f o r m u l a t e d result. Let us show t h a t for the introduced space s a similar result holds (under a condition weaker t h a n inequality (1.161)). We give here a short direct proof close to the one in [Fe]. T h e o r e m 1.4.8. Assume that for a mapping r given by (1.159) inequality (1.160) holds, and that 1 P > 1 ------K" (1.162) Consider ~ = {rk >_ 1 : k >_ 0} such that rk+l >_ prk, k > O. Then for any sequence ~ = {xk E IR'~ : k >_ 0} such that the value
IIg(5)ll , is finite, there is a unique point x with the property
IIh(x,OIl,,oo
LIIg(5)II,,
where
L
~
r
p(1 - K ) - 1
,
(i.163)
1.4 Limit Shadowing Proof.
83
Let us write the main equation r
~- Vk) = Xk+l ~-
Yk+l
in the form vk+l - vk = f ( x k + vk) + xk -- r
+ r
-- xk+l,
or
.A9 = CO + ~,
(1.164)
where ( A o ) k = vk+l - vk, (GO)k = f ( x k + vk) + xk - r
and ~k = r
-
x~+l
= -g~(0.
Set, for ~ E s co
(~)~
~,.
= _ ~
i=k
To estimate IIBII, take ~ E s162 and set ~' = n0. Since rk < pk-lr~ for i > k, we obtain the inequalities
,'klvs < ~klv,
I=
r, lv~l
I1~11~,~ >- I1~11,,~ -II~G~II~,~ >
->(1-K
p -P1
) 'l~''~'~176
that our solution is estimated by _&_
,-1 - p ( 1 - KP) - I II~ll~,oo_< 1_-~-__~11~11~,~
IIwll~,oo,
as was claimed.
[]
Remark. If the function f in (1.159) is of class C 1, then condition (1.160) implies that r is a diffeomorphism. Let us estimate [[(Dr Take x, 5' E IRn, let y = r y' = r It follows from the inequalities
l y ' - yl >- I x ' - 5 [ -
If(5')
-
f(5)l
>
(1
- K)Ix'- 51
that i
,,..~.1. .~.~.~j-, < 1 - g' hence in the considered case condition (1.162) of Theorem 1.4.8 is similar to condition (1.151) of Theorem 1.4.6. 1.4.4 A s y m p t o t i c P s e u d o t r a j e c t o r i e s Hirsch studied asymptotic shadowing of asymptotic pseudotrajectories in [Hirs4] (see also [Ben]). Before stating his result, we give some definitions. Consider a sequence a --- {ak E IR'~ : k > 0}. We denote 7~(a) = l i m k - ~ lak 1l/k, where, as usual, lim is the limsup. D e f i n i t i o n 1.23 A sequence ~ = {xk : k > 0} is called an "asymptotic pseudotrajectory" of exponent )t < 1 for a diffeomorphism r of IRn if
1.4 Limit Shadowing
85
ze(g(~))
0 and a neighborhood U of the set K such that g,p(r
C r
(1.168)
for x E K and p E (0, p0), and tr
- r
> ~lx - yl
for x,y E U with Ix - y [ < p0. We fix an asymptotic pseudotrajectory ~ = {Xk} C K of exponent A f o r r such that (1.167) holds. For d > 0 we denote by Bk(d) the closed ball of radius d k centered at Xk. Take a number d such that A < d < min(1, #). We claim that
(1.169)
86
1. Shadowing Near an Invariant Set
Bk+l(d) C r
(1.170)
for large k. Fix a number u such that AO
is not empty. Take a point x C r Since Ok(z) E Bk(d) for k _> m, we see that I h k ( x , g ) l - - ICk(x) - *kl _< d k, k _> m , (1.174) hence ~(h(~,~))
_< d.
Take two numbers dl,d2 satisfying inequalities analogous to (1.169). Let xl,x2 be the corresponding points for which analogs of (1.174) hold. Let us show that x~ = x~. Set d = max(d~,d2). Denote Zk = ICk(xl) -- Ck(x2)l, it follows that zk _< 2dk for large k. There exists m such that r
C U, i = 1,2, and Zk < Po
for k > m. Since zk+, = ICk+l(xl) - r
> ~z~
for k > m, we see that
2d k >_ Zk >_ I~k-rnzm, k > m. We obtain the inequalities
1.4 Limit Shadowing d) k
>
87
"m
-- 2#m
which are contradictory for large k if z,~ ~ 0. This shows t h a t xl = x2. Now we fix a sequence of numbers dt > ~, l > 0, such t h a t dr ~ )~ as l ~ oc. It follows from the previous a r g u m e n t s t h a t for large l there exists a point x independent of l and such that R ( h ( x , ~)) _< d,. Obviously, this point x has the desired property.
[]
Remark. Let us show t h a t the t h e o r e m above can be reduced to T h e o r e m 1.4.6. Introduce the n u m b e r Mo = m a x IlDr xEK
then obviously #oMo = 1. It follows from condition (1.167) t h a t /kMo < 1, hence we can find a n u m b e r M > Mo such t h a t )~M < 1 (below we assume that M > 1). Find a heighborhood U of K such that sup IlDC-l(x)l] < M. xEU
Let ~ = {xk : k ~ 0} be an a s y m p t o t i c p s e u d o t r a j e c t o r y of e x p o n e n t /k for r such that ~ C U. Take sequences ul > )~ and p~ such that uzpz < 1, ul ~ ~, and 1 pz ~ ~
(1.175)
Pl > M
(1.176)
as l --* oo. Since AM < 1, we have
for large I. Fix an index l such t h a t the last inequality holds (for simplicity, below we write u and p instead of ut and pl). For large k we have Igk(~)i -~ uk, and since pu < 1, the series converges. It follows from (1.176) t h a t we can apply T h e o r e m 1.4.6 (with p = 1 and rk = pk). Find m such that P~Ig~(r
< do.
k)'m
B y T h e o r e m 1.4.6, there exists a unique x such t h a t
< Ldo k)'m
(and the s a m e reasons as in T h e o r e m 1.4.9 show t h a t this x does not depend on
I).
Since pkIhk(x, ~)i --< 1 for large k, we obtain the inequalities Ihk(z, ~)1 0,
~l(s)=w+sv~+o(s), where, as usual,
0(4
- -
-~
0 as
s ~
0.
3
By statement (3) of L e m m a 1.5.1,
I//%=1- IlI~
0 be given. Then there exists a set of points {Y0,..., ym} such that
L e m m a 2.1.2.
106
2. Topologically Stable, Structurally Stable, and Generic Systems
(a) r(xk,Yk) < rl , O < k < m; (b) r ( r , Yk+x) < 2 d , 0 < k < m (c) yi # yj for O < i < j < m.
1;
Proof. We use induction on m. For m = 0 our lemma is obviously true. We assume that the lemma is true for m - 1, and prove it for m. Let r/ > 0 be given, we consider r / < d. Choose A E (0, 71) such that the inequality r(x, y) < A implies r(r r < d. By our assumption, we can choose {yo,... ,ym-1} so that r(yk, xk) < ~, 0 < k < m -
1; r(r
< 2d, 0 < k < m -
2,
and y~yjfor0 2 and let r be a dynamical system. Given a natural m and a positive A, there exists d > 0 with the following property. If ~ = {xk}
is a d-pseudotrajectory for r then there exists a system r E Z ( M ) and a point y E M such that p0(r r < Z~ and r(r
xk) < A for k = 0 , . . . , m.
(2.4)
Proof. Fix a system r a natural number m, and A > 0. Find dl E (0, A) such that the inequality r ( x , y ) < dl implies r ( r r < A. Take d = dl/4 and consider a d-pseudotrajectory ~ for r By L e m m a 2.1.2, there exists a set { y 0 , . . . , y,~} such that r(r
< dl/2, k = 0 , . . . , m d(yk,xk) < A, k = 0 , . . . , m ;
and
YI~Yj, O 0. Find A E (0, e) such that for any system r E Z(M) with po(r162 < A there exists a mapping h : M --~ M having the properties described in Definition 2.1. Find d > 0 such that the statement of Lemma 2.1.3 is true. Let ~ be a d-pseudotrajectory for r Fix a natural number m and apply Lemma 2.1.3 to find a dynamical system r and a point y such that p0(r r < A and r(r xk) < `4, k = 0 , . . . , m. Take x =
h(y),
then Ck(x) = h(r r(r
r
k 9 7I. Since < ~ for k 9 71,
we see that r(r
~ ) < ~ for k = 0 , . . . , m.
The number e > 0 is arbitrary. It follows from part (a) of Lemma 1.1.1 that r has the POTP. This completes the proof of Theorem 2.1.1.
Remark. Let us emphasize that topological stability is a really stronger property than the POTP. A generic continuous dynamical system has the P O T P (see Sect. 2.4), while topological stability is not a generic property [Hu]. Yano constructed a simple example of a dynamical system on the circle S 1 which has the P O T P but fails to be topologically stable [Y1]. An explanation of the example of Yano is given in Sect. 3.1, after the proof of Theorem 3.1.1.
108
2. Topologically Stable, Structurally Stable, and Generic Systems Now we prove Theorem 2.1.2.
L e m m a 2.1.4. Assume that a dynamical system r is expansive with expansivity constant ~. aiven ~ > o there ~xi~ts g > 1 such t h a t / f
r ( # ( x ) , # ( y ) ) < ~ for
all k with Ikl < N, then r(x,y) < .L Proof. Let )~ > 0 be given. If no N can be chosen with the property stated, then for each N _> 1 there exist points xN, YN such that r(r ek(yN)) < a for all k with Ikl < N, and d(xN, YN) >_ A. Choose subsequences Ni and points x, y with xN, ~ x, YN, ~ y as i ~ ~ . Then r(x, y) > ~, and r(r ek(y)) < a for all k. This contradicts the expansivity of r [] To prove Theorem 2.1.2, we fix e > 0 such that 3e < a and choose d > 0 corresponding to e by Definition 1.3. Take r E Z ( M ) such that P0(r r < d. Then any trajectory of r is a d-pseudotrajectory of r By part (b) of Lemma 1.1.2, r has the SUP with constant a/2. Hence, for any x E Mthere is a unique point y such that r(r
r
< e for k E 7/.
This defines a mapping h : M --* M, h(x) = y. Take k = 0 in the inequality above to show that r(x, h(x)) < e. Since
~(#(h(r
r
< ~, k ~ 7/,
and
r(r162
= r(r162
< e, k G 77,
we see that the d-pseudotrajectory {xk = ek+l(x) : k E 7l} is (e, r by the points h(r and r Since x is arbitrary, the SUP implies that hor162 Now let us show that h is continuous. Let )~ > 0 be given. Apply Lemma 2.1.4 to find N such that if r(r Ok(v)) < a for [k] < N, then r(u, v) < )~. Choose 7/> 0 such that r(x,y) < ~ implies r ( r 1 6 2 < a/3 for ]k] _< N. Then for x, y with r(x, y) < y we have
r(r
ek(h(y))) = r(h(r
i. Then there exists T > 0 such that 1 C ( S ( T , y)) > i + -~. Since G is continuous and tm --* oo as m --~ oo (this follows immediately from (2.13) and from the compactness of f2i), for large m we have 1 G ( S ( T , y,~)) = G ( S ( T - t,,, xm)) > i + -~. But since T - tm < 0 for large m, the last inequality contradicts to the relations
G(~.(T -t,,,,xm)) < G(xm) -~ i as m -* oo. This contradiction completes the proof.
[]
We will need also the following simple statement. L e m m a 2.2.6. For any neighborghood V of f2~ there exists a number 0 < d < 1
such that, for any d' E (0, d), the set G - l ( i - d') n
is a subset of V. Proof. Consider the set 1
HI = a - ' ( i - [ ) n W'(r~,). First we show that the set H1 is compact. Indeed, consider a sequence {Xm} C H1, and let x he a limit point of this sequence. Since G is continuous, we have
118
2. Topologically Stable, Structurally Stable, and Generic Systems
G(x) = i
1
2"
Since G increases along trajectories, the relation x ~ )4;S(~Q~) implies that there is an index l > i such that x E VYs(~21). Then we can find T > 0 such that 1
G(Z(T, x)) > i + ~. Since G and ~ are continuous, it follows that for large m we have 1
G(~.(T, xm)) > i + ~, and this contradicts to the inclusions xm E Ws(J2i). Hence, H1 is compact. Since any trajectory through H1 tends to J2i, standard reasons show that there is a number T > 0 such that ~(t, H1) C V for t > T. Obviously, we can find a number d > 0 with the property
a ( ~ ( t , x ) ) . In addition, if the basic set ~2i is a rest point, then S'(x) = S~ for x E )4;~(~2~). Since for x E ~2~ we have
U~
@ $(x) = T~M,
120
2. Topologically Stable, Structurally Stable, and Generic Systems
we can take the neighborghood V so small that
U'(x) @ S'(x) = T~M for x E V.
(2.18)
We denote a neighborhood of 12i with this property by Vii. By (2.18), U' and S ~ introduce "coordinates" in T,M, x E Vi. Now we prove L e m m a 2.2.7.
Proof. Since ]2 is continuous on Q, it follows from the construction of ~3 on O(Q) that it is continuous on O(Q). It remains to show that V is continuous on W~(12i). For this purpose, we apply a variant of argument used in [Palil] to prove the so-called )~-Lemma (see also [Pill). First let us assume that our basic set g2i is not a rest point. Since the set 121 is compact, there exists a positive constant #0 such that IX(x)l
~01X(y)l for x,y E hi.
(2.19)
Now we fix a constant N >_ 1 such that
II// lh ll// ll, II/1~
g for
x e
/2i,
where//~, / / ~~, / / ~o are the complementary projectors onto the spaces S~ < X ( x ) > of the hyperbolic structure on ~i. Set
(2.20) U~
/~ = N(1 + #o), obviously, # > 1. Find positive numbers T and A such that P- R ~
< 1, where R =
,~o T.
(2.21)
Take a vector v E T:M, x E Vi, and represent it in the form v = v" + v ~, where v s E S'(x), v ~ E U'(x) (here we refer to the "coordinates" corresponding to (2.18)). If v ~ # 0, we can define the inclination a of v by the formula O/--
IvS[
Iv=l
(in the original paper [Palil], inclinations were denoted by A, and due to this reason the result concerning them was called the A-Lemma; in this book, we denote by A characteristics of hyperbolicity, and we do not want to give other meanings to this symbol). Denote xl = ~ ( T , x) and consider the operator F ( x ) : TxM --+ Tx, M, F ( x ) v = D 3 ( T , x)v.
2.2 Shadowing in Structurally Stable Systems
121
If xl E Vii, then we can represent F(x) in the form
Fs~ F ~ according to (2.18) (i.e., Fs~v" E S'(xx) etc). First let us consider a point x E •i. Take v = v" + v" E T~M and represent v ~ = v ~ s + v ~ where v "~ E S~ and v ~ E < X ( x ) > . It follows from the definition of hyperbolic structure t h a t
F(x)v ~" E S~
and [F(x)v'Sl < AoTlv"l < Iv~ I,
F(z)v" ~ U~
and IF(x)v"l >__,~oTIv=l.
On the other hand, it is well known t h a t if v ~ = cX(x), then F(x)v ~ = c X ( x l )
(see [pill, Chap. 5), hence IF(z)v~ _< ~olv~ . Since Iv'l, Iv~ _< NIv% we see that IIF~[I _ RIv=l
(2.22)
hold. Denote :Dx = .~([0, T],:D). It follows from condition (2.17) t h a t there exists a constant K such that, for v E 12(x),x E 791, the inclination a of v does not exceed K . Take a point x0 E 91. Denote xk+~ = .E(T, xk) for k > 0. Since 14]*($2~) is ~ - i n v a r i a n t and G(xk+l) < G(zk), the choice of d implies t h a t zk E 14Js(~2~)MV~. Let ak be the inclination of vk = Fk(z)vo, represent vk = vZ + v~' according to (2.18). Let us e s t i m a t e Iv~l _< IF~.vgl + IF~v'~l IF=:v~l - IF:=v~l >_ R I v a l - ~lv~l. Combined with (2.23), this inequality gives
2t, lv~)l + ~lv~l (~1 ~-~
RI,31
-
~lv~,l -< p~o + ~
(we take into account the inequality a0 Now we decrease the neighborghood c Q'. In this case, for any z E -~(t,Q) inclination a of v does not exceed A. inequalities
< A
< A and the second condition on 5). Q chosen to define Y so that ~ ( m o T , Q) with t > m o t and any v E ])(x), the We also assume that, for x E Q, the d 2
G(x) < i - -
(2.25)
hold. Apply an analog of L e m m a 2.2.6 for )'Y~(J2~) to find a number dx > 0 such that
a - l ( i + d') n W : ( ~ , ) c V,~ for d' E (0, dl]. S e t / / 2 = G - l ( i + dl) M VVu(ai). To show that 1,' is continuous on )d;=([21), it is enough to check the continuity of Y on H2 and on [21 (this second case is treated similarly, and we leave it to the reader). Take a point y E H2 and consider a sequence of points yp E O(Q) such that Yv~yasp~. Since
i < G(~(t, y)) _ 0; D Z ( t , p ) U ( p ) C U(q), t 0 such that
L e m m a 2.2.11.
ItP(p)ll, IIQ(p)II -< N, p E M,
where P(p), Q(p) are the projectors in Z(p) onto S(p) parallel to U(p), and onto U(p) parallel to S(p), respectively; (e) IO.E(t,p)vl < ~lvl for v ~ S(p), t _> T - 1,
IO•(-t,p)vl
< ~lvl for v e f ( p ) , t > T - 1.
Take K _> N such that
[X(p)[ < K for p E M and iin~(t,p)[i < K for [t[ 0, p E M we set Be(p) = {v E T p i : Iv ] < c}.
2.2 Shadowing in Structurally Stable Systems
133
It is well known that, for some c > 0, any expv is a diffeomorphism of the ball E~c(p) onto its image, the first derivatives of expv , exp; 1 are uniformly (in p) bounded on E2c(p) and on expp(E2~(p)), and Dexpv(0 ) -- I.
(2.44)
We fix this c below. It is easy to see that a family of linear subspaces W(p) C TvM, p C M, is continuous at z E M if and only if q ~ z implies D e x p z l ( q ) W ( q ) ~ W(z). Now we formulate the uniformity of the "weak upper semicontinuity" described in statement (2) of Lemma 2.2.10. The subbundles S~, U~ are uniformly continuous on Vi. A point p E M t.s. from Vi according to formula (2.38) with to E [0, To]. We easily deduce from the proof of Lemma 2.2.10, from the compactness of M, and from the uniform boundedeness of derivatives of exp~-a that the following statement holds. L e m m a 2.2.12. Given l3 > 0 there exists a > 0 such that if z, p C M, tl,t2 C ]R, q = 3 ( t l , p), y : 3(t2, z), and the inequalities
r(z,q) < a, r(y,p) < a, I t x - T l _ < 1, It2 + Tl 0 there exists a2 > 0 such that if a _< a2, then the angle between v' and Z ( z ) is less than /32. Note that these al depend only on the corresponding/3i. Since derivatives of expp are uniformly bounded, given/3a > 0 there exist numbers aa > 0 and Sl > 0 such that for a < aa we have dist(Tl(S), Z(z)) 0 such that (1.65) holds. Now we find u E (0, vo) such that
4 K ( S K z + 6K 2 + 9K + 1)u < n/2.
(2.50)
Below we denote by d' positive constants that depend only on u, n, K, N. At each step of the proof, we consider (d, T)-pseudotrajectories such that d does not exceed the minimal d' previously chosen. Since we choose d I finitely many times, no generality is lost. We take d' < min(b, d2) (b and d2 are given by Lemma 2.2.14). We fix d' such that, for any points x , y C M with r ( x , y ) < d' and y' = exp~ l(y), the inequalities
IDexp-~'(y)vl IDexp~(y')vl _ O, then y E C. Proof. Take e < A, then it follows from (2.73) and from the inclusion x E W'(p) that
r(r
r
0 such that any point z E W~(r k' (p)) with r(z, x") < r belongs to C'. Take e = r then ek, (y) E C ~, and it follows that this r has the required property. []
Remark. A similar statement is true for a point x C W~(q), below for x E W~(q) we also refer to L e m m a 2.2.20.
2.2.21. If for a diffeomorphism r statement (2) of Theorem 2.2.8 holds, then r E ~ I ( M ) .
Lemma
Proof. Take a neighborhood W of r such that any diffeomorphism r E W has the P O T P , we claim that, for any r C W, all periodic points are hyperbolic. To obtain a contradiction, assume that a diffeomorphism ~b' E W has a nonhyperbolic periodic point p (to simplify notation, we assume that p is a fixed point, other cases are treated similarly). Hence, the linear mapping L' = D r
TpM ~ TpM
has an eigenvalue #1 with I#1] = 1. We consider the case #1 = 1 (it is useful for the reader to consider other possible cases). There exists a perturbation r E W of r such that p is a fixed point of r and the eigenvalues # 1 , . - - , #~ of the linear mapping L = Dr TpM ---* TpM satisfy the conditions /zl = 1; I/~j] ~ 1, j ~ 1. Obviously, we can find a neighborhood V of p, a diffeomorphism r E W, and local coordinates (Yl,... ,Yn) in V such that p = 0, and r coincides with L in V with respect to these coordinates. Let Q be the one-dimensional invariant subspace of L corresponding to/~1, we assume that Q is given by Y2 = ... -- yn -- 0. Then any point of the set Q' = {yl = 0 : i = 2 , . . . , n} M V is a fixed point of r while for any point z E V \ Q its trajectory {r cannot be a subset of V.
: k E 71}
2.2 Shadowing in Structurally Stable Systems
155
Let us show that r does not have the POTP. Take e > 0 such that the set Q~= Q'N
{lUll ~ ~}
is a subset of V, and dist(x, Q~) > e for x e M \ V. Take an arbitrary d > 0 and find a natural number N such that e < Nd. Consider a sequence ( = {xk : k E 77} defined as follows. Set x0 = (e, 0 , . . . , 0). Represent k E 77 in the form k = 2 m N + l with 0 < l < 2N - 1, and set xk = (zk, O , . . . , O ) , where zk+ ~ e Zk -- -~
Zk+l ~-
for m odd , formeven.
By construction, ~ C Q~. Since any point of Q~ is a fixed point of r it follows from the choice of N that ~ is a d-pseudotrajectory of ~b. For any point x of Q' there is an index k such that r(x, xk) k e. For any point x ~ Q', the trajectory Ck(x) contains a point x' E M \ V, hence dist(x', ~) k e. This shows that r does not have the POTP. [] Since (5) =v (2), it follows from Lemma 2.2.21 and from Theorem 2.2.9 that if for a diffeomorphism r condition (5) holds, then r satisfies Axiom A. By Theorem 2.2.4, to establish the implication (5) ~ (1), it remains to prove that r satisfies the strong transversality condition. Fix a number 1; such that any r E W has a Lipschitz constant not exceeding 1:. Assume that v is a point of nontransverse intersection for WS(p), W"(q), where p, q e f2(r Since the set ~ ( r is hyperbolic, we see that v ~ ~?(r Let us take a small neighborhood V of v, denote v' = r and V' = r We can take V so small that Ck(V) f'l V -- 0 for k • 0. (2.74) Find numbers k+, k_ such that ck+ (v') e IntW~(r k+ (p)), Ck-(v) E IntW~(r We consider open disks C + C IntW~(r
Co C IntW~(r
and the corresponding disks c + = r
c-
= r
containing v ~, v, respectively, and so small that C-cV,
C +CV',
then by (2.74) we have Ck(C+) N V = 0, k > 0.
(2.75)
156
2. Topologically Stable, Structurally Stable, and Generic Systems Apply L e m m a 2.2.20 to find e > 0 such that if
r(r
Ck(x)) < ~, ~ > 0,
r(r
r
then x 9 C +, and if
< ~, k < 0,
then x 9 C - . Since v is a point of nontransverse intersection of We(p) and W"(q), we have the inequality
T, We(p) + T,W"(q) # T,M. Assume that the stable manifold We(p) is m-dimensional, and the unstable manifold W~'(q) is/-dimensional (note that m, l < n). We consider two cases. Case 1. l + rn > n. In this case, we introduce local coordinates y = (Yl,...,Yn) in V so that v = 0,
T, We(p) + T,W"(q) C {yn = 0}, and C - C {y,, = 0}. Let C = r
(2.76)
c V. We choose coordinates so that
T,~C = {ym+l . . . . .
yn = 0}.
Assume that in a neighborhood of the point v the disk C is given by the equation = g(~), where
~- (Yl,...,Ym), with g 9 C 1, g(0) = 0, and
~
~ -~
(yrnT1,..-,Yn),
(0) = 0.
The spaces T~C = T~We(p) and T,~C- = T,,W"(q) have dimensions m and I with m + l _> n and belong to the (n - 1)-dimensional space {y~ = 0}. Hence, the dimension of their intersection is at least one. We assume that {Y2 . . . . .
Yn = 0} C T,,C gl T,,C-.
(2.77)
We can perturb r on V so that for the perturbed diffeomorphism r the following statements hold: (1) r 9 W, where W is the neighborhood from statement (5) of Theorem 2.2.8; (2) there exists a C 1 disk C' containing v such that
r
= c +,
and C' is given by the equation ~ = G(T/), where G 9 C 1, G(0) = 0,
2.2 Shadowing in Structurally Stable Systems
157
OG N (o) = o, and G~(~) ~ 0 for 0 < 171 < b
(2.78)
with some b > 0 (here Gn is the nth component of G). Let us note some properties of the disk C'. Relations (2.76) and (2.78) imply that if a point w ~ v belongs to C' M C - , then r(v,w) ~_ b. It follows from (2.75) that Ck+'(C') = c k ( c + ) for k > 0.
(2.79)
For small t > 0 we consider two smooth curves,
h(t) = (t,O,...,O, G m + , ( t , O , . . . , O ) , . . . , G , ( t , O , . . . , O ) )
C C',
and c(t) = ( e l ( t ) , . . . ,c~(t)) c C such that c(0) -- 0, ~ttc(0) = ( 1 , 0 , . . . , 0 ) . For small t > 0 we define a sequence of points ~(t) = {xk(t): k E 77} as follows: xo(t) = c(t), xl(t) ----r xk(t) = Ck(x0(t)), k < 0;
xk(t) = Ck-'(xl(t)), k > 1. The diffeomorphisms r and r coincide outside V, hence
r(xk+l(t), r
< d(t) := s
c(t)).
Since d(t) --+ 0 as t ~ 0, we have
d(t) < 2ut < do for small t > 0. By our assumption, there is a point x(t) such that
r(xk(t), Ck(x(t))) < Ld(t)
(2.80)
(do and L above are from statement (5) of our theorem). By our choice of e and the construction of ~(t), for small t the point x(t) belongs to C' M C - . As was noted, for any point w E C ' M C - such that w ~ v, we have r(v, w) > b. It follows from (2.80) that r(xo(~), x(t)) --~ 0 for t -~ 0. Since xo(t) ~ 0 as t ~ 0, we see that x(t) = v for small t > 0. Since
158
2. Topologically Stable, Structurally Stable, and Generic Systems dcl(0
=
d 1; ~ c i ( 0 ) -- 0, i # 1,
we have r(x(t), c(t)) > t + o(t) for small t > 0. On the other hand,
hence d(t) = o(t), and it follows that
r(x(t), c(t)) = r(x(t), x0(t)) = o(t). The obtained contradiction shows that case 1 is impossible. Case 2. 1 + m < n. In this case, we can apply the transversality theorem [Hirs2] to obtain a diffeomorphism ~b C W that coincides with r on M \ V, and a C 1 disk C' C V such that
r
+, C-nC'=~.
Consider the sequence ( = {xk : k E 7/} defined as follows: Xk = ~)k(l)), ~g ~ 0; Xk = c k - l ( v ' ) ,
~ :> 1.
Since r coincides with r outside V, (2.74) holds, and r
# v', we have
< d, where d = r ( r
v'). We can take r such that s
r(r
v')
0, the connected component of the intersection N~(x) n ~-k(W~,(~k(p))) containing x is an embedded open segment of class C 1, we denote this set by If x 9 WS(p), and a > 0 is small enough, we denote by Bi,~(z), i = 1,2, the open components of the set
No(x) \ D e f i n i t i o n 2.9 We say that 4) satisfies the '~g~ transversality condition" if for any point z C M such that x 9 WS(p) n WU(q), where p, q 9 ~(~) with dimWS(p) =dimW~(q) = 1, and for any small positive a we have C'~(x) N Bi,,(x) r O, i = 1,2. It is easy to see that if the condition above holds for some point x, then it holds for any point of the trajectory of x.
160
2. Topologically Stable, Structurally Stable, and Generic Systems The main result obtained by Sakai in [Sak3] can be stated as follows.
An (A,2)-diffeomorphism r has the POTP if and only if r satisfies the C O transversality condition.
T h e o r e m 2.3.1.
We will prove here the "only if" part of Theorem 2.3.1, the reader is referred to [Sak3] for the remaining part of the proof.
Proof. To obtain a contradiction, let us assume that r has the POTP, but the C O transversality condition is not satisfied. Hence, there exists a point z E WS(p) M W~(q) such that p,q C D(r dimWS(p) =dimW~(q) = 1, and for small a > 0 we have C2(z) n
Bl,~(z) =
0.
Denote Pk = Ck(p), k E 77. The stable and unstable A-disks
W~(x) and W~(x) through points x E g2(r described in Theorem 1.2.1 are constructed in the case of a manifold as follows. Let {S(p), U(p)} be the hyperbolic structure on Y2(r There exist mappings
f~: {v e s(x): Ivl ~ A} --+ s(~) and
f2: {v ~ g(x): Ivl ~ A} ~ s(x) Of class C 1 such t h a t
W~(x) = exp.({(% f~(v)) : v E S(x), Iv I _< A}) and
W~a(x) = exp~({(v, f~(v)) : v E U(x), Iv[ _< ZI}). It follows from the proof of Theorem 1.2.1 (in the case of a compact manifold) that the norms tlDRll, IIDf~II are uniformly (in x E f2(r bounded, and the angles between the spaces S(x), U(x) are uniformly separated from 0. Hence, there exist positive numbers b _> e0 with the following properties. If Ark = N2b(pk), then - the set
( Wi (pk) n Nk ) \ p~ is the union of two smooth open segments (they are denoted W~,1 and W~,2); the set -
Nk \ W~a(pk) is the union of two open two-dimensional disks (they are denoted D~, and D~, so that W~,~ C D~, i = 1,2);
2.3 Shadowing in Two-Dimensional Diffeomorphisms
-
161
if v E W~,2 and r(v,pk) = b, then dist(v, 01) >_ co.
Denote W [ ( p ~ ) = W~(p~) n Nb(pk), ~ = s, u.
We choose the indices of the segments W k,i ~ so that W~+I, ~ i C r it is easy to understand that in this case for any point y E D~, i = 1, 2, we have either r E D},+l or r ~ Nk+l. Obviously, the point z fixed above can be taken so that z E W~(p). It follows from our assumption that there exists an open segment C C C~(z) containing z and such that one of the sets CMDio,i = 1,2, is empty. Assume that CMD~ = 0. By Lemma 2.2.20, there exists el > 0 such that if r(r
r
< e~ for k < 0,
then x E C. Take e < min(eo, el). Since r is assumed to have the P O T P , for this e we can find the corresponding d. There exist natural numbers l, m with the following properties:
~(r
< d,
and for the point w E W l+rn,2 ~ with r(w, pt+m) = b the inequality ~(r
t.
Since r = Ct(z) and x, = r our choice of l and m implies that is a d-pseudotrajectory for r Assume that ~ i s ( e , r by a point x. Since e < el, we see that x E C, and hence x E D0x. If x E W ' ( p ) or Ck(x) E 01 for all k e [0, l + m], we have = ~(~, r
r(xz+~, r
> ~o.
If there exists k E [0,/-t-m] such that Ck-~(x) E D~_l but Ck(x) ~ D~, then Ck(x) ~ Nk, hence
~(~, r
> b > ~0
since by construction
for 0 < k < l + m. The obtained contradiction completes the proof.
[]
162
2. Topologically Stable, Structurally Stable, and Generic Systems Now we give conditions under which an (A,2)-diffeomorphism has the LpSP.
T h e o r e m 2.3.2. An (A,2)-diffeomorphism r has the LpSP if and only if r is
structurally stable. Proof. By Theorem 2.2.7, the structural stability of r implies the LpSP. It follows from Theorem 2.2.4 that to prove our theorem it remains to show that an (A,2)-diffeomorphism having the LpSP satisfies the geometric strong transversality condition. To obtain a contradiction, assume that an (A,2)-diffeomorphism r has the LpSP (with constants (L, do)) but does not satisfy the geometric STC. Hence, there exist points p,q 9 ~2(r having a point z of nontransverse intersection of WS(p) and W~(q). Obviously, in this case we have dimW~(p) = dimW"(q) = 1. Apply Lemma 2.2.20 to find open segments C + C WS(p) and C - C W"(q) containing z and a number r such that if
r(r
r
0,
r(r
r
< ,, k < 0,
then x 9 C +, and if then x 9 C - . Decreasing the segments C +, C - , if necessary, we can introduce coordinates (y,v) in a neighborhood of z so that z is the origin, the inclusion
C - c {y = 0} holds, and
c § = { ( y , g ( y ) ) : lyl < b} with some b > 0, where da g 9 C 1, g(0) -- 0, and ~-~y(0) = 0. If there exists bl > 0 such that g(y) - 0 for lYl < bt,
(2.81)
then r does not satisfy the C O transversality condition, and by Theorem 2.3.1 r does not have the POTP. Hence, in this case r does not have the LpSP, and our theorem is proved. If there exists no bl such that relation (2.81) is satisfied, then there is a sequence y~ such that y~ --~ 0 and g(y~) ~ O. For definiteness, we assume that y~ > 0. Obviuosly, in this case there is a sequence ym such that
g(ym) = 0 and g(y) ~ 0 for y 9 (ym, y~)
2.3 Shadowing in Two-Dimensional Diffeomorphisms
163
(of course, it is possible t h a t y,~ - 0). For the n u m b e r s hm = ~ ( y m ) we have lim hm = 0 as m -+ co. Let s be a Lipschitz constant of r Take h > 0 such t h a t
2hLs < 1
(2.82)
and find m such t h a t hm < h. T h e r e exists yo E (ym, y~) such that lYo - YmI < lYo - Ykl and Ig(Yo)I < 2hlyo - Y,~I. Fix points Xo = (yo,0), z' = (Yo,g(Yo)), and Xl = r sequence ~ = {xk : k E 7/} defined as follows:
Consider the
xk = Ck(xo), k _< 0; xk = C k - l ( x l ) , k > 1. Since r(xo, x') = ]g(Yo)], we have
r(Xl,r
< ~lg(Y0)],
it follows t h a t
r(XkTl,~)(Xk)) < d :~- ~]g(Y0)t. We can take m and yo such t h a t
Ig(y0)l
0 there exists d > 0 such that any d-pseudotrajectory of r is weakly (e)-shadowed by a trajectory of r It is easy to give an example of a system which does not have the WSP. E x a m p l e 2.12 Take M = S 1 (with coordinate a E [0, 1)), and consider r E Z(M) such that r - a. Fix an arbitrary d > 0, and define a system r E Z(M) by r
(mod 1).
Obviously, trajectories of r are d-pseudotrajectories of r (0, 1/2), x, y E M we have
{r
: k e g} r
and for any d E
gl/4(x)
Since {x} is the trajectory of x in r this proves that r does not have the WSP. It is also easy to show that there exist systems which have the WSP but do not have the POTP. E x a m p l e 2.13 Take again M = S 1 with coordinate a E [0, 1), and consider the system r E Z(M) generated by the mapping
f(a)=a+fl
(modl),
where fl is irrational. Every trajectory of r is dense in S 1, this obviously implies the WSP. Assume that r has the POTP, take e = 1/4, and find the corresponding d (see Definition 1.3). There exists a rational 3' = l/m such that 13' - fl[ < d. Consider the system r generated by
g(a)=a+3"
(mod 1),
obviously, trajectories of r are d-pseudotrajectories of r It follows from the structure of 3, that any x E S 1 is a fixed point of r Since for p E S 1 the sets {r are dense in S 1, we see that for any pair x,p there is k E 7/such that
2.3 Shadowing in Two-Dimensional Diffeomorphisms
165
1 eke(x)) >_ a.
r(r Hence, r does not have the POTP.
Plamenevskaya studied conditions under which an (A,2)-diffeomorphism has the WSP [Pla2]. She considered this problem for an Axiom A diffeomorphism of the two-dimensional torus T 2 with finite nonwandering set. She showed that, in contrast with the simple geometric condition for the P O T P given in Theorem 2.3.1, necessary and sufficient conditions for the WSP are very delicate. Let us describe the example of Plamenevskaya. Represent T 2 as the square [-2, 2] x [-2, 2] with identified opposite sides. Consider the metric r generated by the supnorm. We study a diffeomorphism r of T 2 with the following properties. The nonwandering set ~(r is the union of 4 hyperbolic fixed points,
a(r = {o,;1,p,, s}, where the point s is asymptotically stable, the point o is completely unstable (i. e., it is asymptotically stable for C-a), and Pl,P2 are saddles. It is assumed that with respect to coordinates (u, v) E [-2, 2] x [-2, 21 the following conditions hold: (cl) o = (1,2), Pl = ( - 1 , 0 ) , p~ = (1,0), s = ( - 1 , 2 ) ;
(c2) WS(pl) =
W~'(px) = {-1}
x
WU(m)=
[-2,21
x
{0};
( - 2 , 2), W*(p2) = {1}
x
( - 2 , 2);
(c3) there exist neighborhoods O1, 02 of pi,p2 such that
r
= Pl + Dr
- Pi) in Oi, i = 1,2;
(c4) there exists a neighborhood O of the point z = (0, 0) such that
r
c o,, r
c
and r is affine on 06(0); (c5) the eigenvalues of De(P1) are - # , u with ~ > 1, u E (0, 1), the eigenvalues of De(p2) are -A, ~ with A E (0, 1), ~ > 1. It follows from conditions (c2), (c3), and (c5) that r
= (u(u + 1) - 1 , - # v ) in O1
r
= (~(u - 1) + 1 , - $ v ) in 02.
and Note that by Theorem 2.3.1 the diffeomorphism r does not have the POTP since it does not satisfy the C o transversality condition.
166
2. Topologically Stable, Structurally Stable, and Generic Systems
T h e o r e m 2.3.4. The diffeomorphism r has the WSP if and only if the number log A log #
(2.84)
is irrational. Remark. It is worth noting that the value (2.84) appeared in the qualitative theory of dynamical systems as a functional modulus of local topological conjugacy in a neighborhood of a heteroclinic curve joining two saddles [DM]. Proof. First we show that if the value (2.84) is irrational, then r has the WSP. Note that in this part of the proof we do not apply condition (c4). Below = {Xk : k 9 7/} is a d-pseudotrajectory of r (we do not repeat this when we impose conditions on d). Fix e > 0. To simplify notation, we will find d corresponding to 2e instead of e in Definition 2.11. We take e so small that Nr
C 0;, i = 1, 2.
Consider the sets V1 = ( - l - e , - 1
+e) x
,e
and V2=(1-e,l+e)
x k 2 ,e
.
It is easy to see that there exists a number dl 9 (0, e) and neighborhoods
V(s), V(o) of the points s, o such that V(s) C N~(s), V(o) C N~(o), and
Nd,(r
C V(s), Ndl(r
C V(o).
It follows that if d < dl and xk E V(s), then x~ E V(s) for I > k (similarly, if xk 9 V(o), then x, 9 V(o) for l < k). The set 1/1 belongs to the basin of attraction of the point s, and the set V2 belongs to the basin of attraction of 0 for r There exist numbers T1 > 0 and d~ 9 (0, dl) such that
cTI(Nd2(V1)) C V(s), r
C V(s),
and if d < d2 and r(xo, y) < d2, then r(xk, Ck(y)) < dl for ]k[ < T1. We see that if d < d2, xk 9 V1, and r(y, xk) < d2, then
l >__k} c
2.3 Shadowing in Two-Dimensional Diffeomorphisms
167
(we recall that O+(X) and O - ( X ) are the positive and the negative Ctrajectories of a set X). Similarly, if d < d2, xk E V2, and r(y,xk) < d2, then
{x,: l _< k} C N2,(O-(g)). In addition, we take d2 such that
Now we fix the following subsets of V1 and V2: Wl
:
( - 1 - d2, - 1 -4- d2) x
,e
and
W2=(1-d2,1+d2)
x ~, 2 ,e .
Take d3 > 0 such that
(2.86)
ud2+d3 < d2 and 2 d z ( # + l ) < e.
We claim that this d3 has the following property: if d < d3, Xko (7_ Nd2(Pl), and xkl ~ N,(pl) for some kl > k0, then there exists k E (k0, ka) such that xk C W1. Indeed, let xk = (uk, vk). We can consider kl such that xk E N,(pa) for k < kl, it follows from the choice of e that for k0 _< k < kl we have
r
vk) = (u(uk + 1) - 1 , - # v k ) .
Hence, if Uk E (--1 -- d2, - 1 + d2) and r(r
Zk+l)
0 with the following property: if ( is a d-pseudotrajectory of r such that ~ C Mi, then r r E M/ for any k. Now it remains to repeat the proof of Theorem 2.2.7 to establish our claim. Let do, L be the corresponding constants (for both sets Mi). Take d E (0, min(d0, dh)) and such that dL < 2e. We want to show that for any d-pseudotrajectory ~ of r there is a point y such that ~ C N~(O(y)). It follows from our previous considerations that it is enough to consider a d-pseudotrajectory ~ such that
~nQ,r
i=1,2.
We assume that the d-pseudotrajectory ~ we work with is such that xk E N~(o) for negative k with large [kl, and x~ E N~(s) for large positive k (other possible cases ase treated similarly). Then it follows from the choice of d that ~ intersects the sets W, W1, W~. Set
2.3 Shadowing in Two-Dimensional Diffeomorphisms
169
kl = m a x { k : xk e W2}, k2 = m a x { k : xk E W}, k3 = m a x { k : xk E W1}, then we have kl < k~. < k3. By the choice of T2, the inclusion Zk~+T~ E Q2 holds. Our condition (c3) on r implies that {zk: kl o
J
The inclusions
$2 c w c Ndp~) hold, and it follows from the choice of d4 that for any point x C $2 we have {xk: k < k3} C N2~(O(x)) (here we take (2.87) into account). Thus, to prove our statement it remains to find a point x E $2 such that its trajectory intersects the set X1, since in this case we have {xk: k > k3} C X2,(O(x)). Set LJ_>O
J
If we show that $1 n r
# O,
this will complete the proof of the first part of Theorem 2.3.4.
(2.88)
170
2. Topologically Stable, Structurally Stable, and Generic Systems
Take the points yj = (u2,(-1)JA%l),j > O. Property (c3) implies that yj C $2 for large j. Set zj = cT2(yj), let zj = (u},v}). By condition (c2), the linear mapping DCT2(u2, 0) has an eigenvector (1,0), let the matrix of this mapping be 0
c
"
Then we can write
v} = c(-1)JASv, + a(j)A j,
(2.89)
where c 7~ 0 a n d a ( j ) ---* 0 as j ---* cx~. Obviously, for A > 0 small enough there exists m0 such that S, n [ ( - 1 - d~, - 1 + d~) x ( - A , A ) ] =
=
U ( - 1 - d2, - 1 + d2) x ((--1)m#-mv3 -- #-rod', (--1)ruff--my3 -[- #-md'). m ~ rrl,0
Since there exists j0 such that zj e ( - 1 - d2,-1 +d2) x ( - A , A) for j > j0, it is enough to find j > jo and m > mo such that
or
- val < d'.
(2.90)
Since the numbers log A and log # are incommensurable and have different signs, the set {(--1)m+J/~mAJ : m , j > 0} is dense in IR. Now relation (2.89) implies that there exists a solution of (2.90). This completes the proof of the first part of Theorem 2.3.4. Now we show that if the value (2.84) is rational, then r does not have the WSP. Assume that log A _ r log # s for some natural numbers r, s, i.e., A = 7 r, # = 3'-8 with 3' E (0, 1). Take a > 0 such that N2~(pl) C O1 and N2~(p2) C 02. Let e > 0 be so small that
N~(z) C O
(2.91)
e < 2-),1/2(1 - 3,1/~).
(2.92)
(recall that z = (0, 0)) and
2.3 Shadowing in Two-Dimensional Diffeomorphisms
171
To obtain a contradiction, we assume that there exists d > 0 such that any d-pseudotrajectory of r is weakly ~-shadowed. Let us construct a dpseudotrajectory as follows. Fix the points z, = (-1,'71/2a) and z2 = (1,a). We have
r
~ pl, Ck(z2) ~ p2, r
~ pl, and r
~ p2
as k ~ (x~. Hence, there exist numbers k~,i = 1,2,3,4, such that the set
= {~k(z1) : k __>kl} U {~k(z) : k2 ~_~k ~__k3} U {(~k(z2) : k ~ k4} is a d-pseudotrajectory of r Assume that ~ is weakly (0-shadowed by O(p). Then O(p) n N d z ) # 0, we assume that p E N~(z). In addition, O(p) N N~(z2) # 0. It follows from property (c5) of r and from (2.91) that the coordinate u decreases along the trajectory of p, hence O-(p) n N~(z2) # 0. Then p E N,(z) n [jUoCJ(N,(z2))] By (c3) and (c4), for x E N~(z2) M r
with j > 0 we have
CJ(z) = P2 + Dr hence
Ndz)n[jUor c U (-~, 0 x ((-lya'7"J - ~'J~, (-lya'7"J + '7"JO. j>o
It follows that
p e U (-~, ~) • ((-lya'7"J - '7"J~, ( - i y a ' 7 ~j + '7"J~). j>0
Similarly,
p e U (-~, ~) • ((-1)me'7 sm+'/5 - 'Tsm~, ( - l y a ' 7 "~+1/5 + '7~m~). m>O
Hence, for some j, m > 0 the inequality
la'7,j _ a'7~+,/51 < ('7,j + "7~m)e holds. We deduce from (2.92) that the last inequality is equivalent to the inequality
1 - "T. . . . j+1/5 < '71/5(1 _ '71/2) if srn > rj, and to the inequality
'7,/5(1
_
'7,j ....
1/5)
3 not every homeomorphism is C~ by diffeomorphisms [Mu], thus this method is not applicable if the dimension of M is arbitrary. Note also that the genericity of the weak shadowing property (see the previous section) in Z(M) for any dimM was established in [Cor2]. =
The proof of Theorem 2.4.1 is based on the theory of topological transversality [Ki, Q]. We do not give it in this book.
3. Systems with Special Structure
3.1
One-Dimensional
Systems
Consider the circle `81 with coordinate x E on `81 induced by the usual distance on the We fix a homeomorphism r of .81 and phisms [r = { r
[0, 1), we denote by r the distance real line. consider the family of homeomor~ IN}.
It was mentioned that a homeomorphism r has the POTP if and only if the homeomorphism Cm with some natural m has this property (see Sect. 1.1). Thus, if one homeomorphism of the family [r has the POTP, then all homeomorphisms of this family have it. Obviously, we may restrict our consideration to homeomorphisms preserving orientation. For a homeomorphism r we denote by Fix(C) the set of fixed points of r This set is closed. Let P : IR --+ S a be the mapping defined by the relations
P ( x ) E [O, 1), P ( x ) = x
(modl),
with respect to the considered coordinates on `81. To study the dynamical system generated by a homeomorphism r preserving orientation, we introduce the so-called lift q~ of r [Nil], i.e., a continuous increasing function such that (a) the function ~(t) - t is 1-periodic; (b) P o ~ = C o P . We fix a lift 9 of r such that ~(0) e [0, 1). It is well known (see Chap. 1 of the book [Nil]) that for any point x E ]R there exists the limit lim ~ ( x ) 7t'-* O0
n
and this limit does not depend on x. This number is called the rotation number of r we denote it #(r The main property of the rotation number is the following one: r has a periodic point if and only if the number #(r is rational [Nil].
174
3. Systems with Special Structure
For two points a, b E S 1 (we identify them with the corresponding points of we denote by (a, b) the open arc of S 1 corresponding to the set (a, b) C [0, 1) if a < b, and to the set (a, 1) U [0, b) C [0, 1) if b < a. Similar notation is applied for closed arcs. Let a, b be two fixed points of a homeomorphism r preserving orientation. Assume that (a, b)MFix(r = O. We say that (a, b) is an r-interval if
[0,1)),
~(t) - t > 0 for t E (a, b) or for (a, 1) U [0, b), correspondingly. Otherwise, we say that (a, b) is an l-interval. Obviously, if (a, b) is an r-interval, then for x E (a, b) we have Ck(x)
a, k
r
b,
--* or
and an/-interval has a similar property. Now we state necessary and sufficient conditions under which a homeomorphism of S 1 has the POTP [Plal]. The same problem was solved earlier by Yano [Y2], but we prefer to follow [Plal] here, since this allows us to apply similar methods to treat both shadowing and limit shadowing for homeomorphisms of S 1" T h e o r e m 3.1.1. A homeomorphism r of S1 has the P O T P if and only if the family [r contains a homeomorphism r such that (a) r preserves orientation; (b) the set Fix(C) is nowhere dense and contains at least two points; (c) for any two r-intervals (l-intervals) ( a, b) and (c, d) there exist l-intervals (correspondingly, r-intervals) (p, q) and (s, t) such that (p, q) C (b, c) and (s, t) C (d, a). We begin the proof of Theorem 3.1.1 by some auxiliary lemmas. One easily proves the following statement (compare with examples 1.4 and 1.18). L e m m a 3.1.1. If for a homeomorphism r the set Fix(C) contains a nondegencrate arc (i.e., an arc that is not a point), then r has neither the P O T P nor the LmSP. Below, in Lemmas 3.1.2 and 3.1.3, we assume that r has the POTP. By Lemma 3.1.1, it is enough to consider homeomorphisms r such that the set Fix(C) contains no arcs (i.e., this set is nowhere dense). L e m m a 3.1.2. If a homeomorphism r of S 1 preserves orientation and has the POTP, then r has periodic points.
Proof. To obtain a contradiction, assume that r has the P O T P but its set of periodic points is empty. Then Fix(C) = 0, and we can find e > 0 such that
3.1 One-Dimensional Systems
r(x, r
> 3e for x 9 S ~.
175 (3.1)
Take a n u m b e r d given for this e by Definition 1.3 and such t h a t d < e. Let us construct a d-pseudotrajectory ~ of r as follows. Take a point x 9 S 1 and consider the set O(x), the t r a j e c t o r y of x. By our assumption, Ck(z) # Ct(x) for k # l, hence the infinite set O(x) contains two points era(z) and Cn(x) such that r ( r Cn(x)) < d. Assume t h a t m < n and denote x0 = era(x), then Cn(x) = Cg(x0), where N = n - m > 0. It follows t h a t
r(xo, CU(x0)) < d.
(3.2)
Represent k 9 7] in the form k = lN-4- a, where l 9 7/,0 < s < N , and set zk = r T h e n ~ = {xk} is a d-pseudotrajectory of r Indeed, for s < N - 1 we have r and for s = N -
= r162
= r
=
1 we have Xk :
cN-I(x0)
, r
---- c N ( x 0 ) , X k + l :
XO,
hence it follows from (3.2) that r(r Xk+l) < d for all k. By our assumption, there exists a point x 9 S 1 t h a t (e)-shadows ~. Take points x~, x' 9 [0, 1) C IR such t h a t P(x'o) = Xo, P ( x ' ) = x. Since r(z, Xo) < e, the length of one of the arcs joining x0 with x is less t h a n e. We a s s u m e for definiteness t h a t 0 does not belong to this arc, then Ix0 - x' I < e.
(3.3)
Now we construct a sequence ~' = {x~ 9 IR: k 9 7]} such that P(x'k) = xk and !
!
z k<xk+ l<x~+l,
k9
(3.4)
T h e point x~ is fixed, it follows from xk # Xk+l t h a t the sequence is uniquely determined. T h e set Fix(C) is empty, hence ~ ( t ) - t ~t 0 for t 9 [0, 1). It follows t h a t 45(t) - t ~t 77 for t 9 lit, and now the inequality 0 < ~(0) < 1 implies t h a t 0 ~Pk+l(x') > dhk(x ') > x~ -- e.
(3.7)
176
3. Systems with Special Structure
Since P(qbk+i(x~)) = r and the point x (@shadows ~, the point ~k+l(x') belongs to the e-neighborhood of a point x~+ 1 + l, l E 77. We have to show that l -- 0. Assume that I~k+i(x ') - (x~+ 1 + l)l < e for some l r 0. Then it follows from (3.7) that I
I
I
x k + 1 - 2e > xk+ 1 + 1 > x k -- 2c. We deduce from
(3.4)
and from IX~+l+l-xk+ 1' I=lll- >1
that
hence either
X ; + 1 "4- l ~ [X~, X~ "t- 11,
Xk' or
!
xk+l
-2e
>
d>
The obtained contradiction establishes (3.6). It follows from (3.6) that lim ~k(x) - lim x~: k--*~ k k--*~ k The equalities P(x'o) = P(X'N) = Xo imply that X~r - x oI = m for some natural ! m, this gives X'kN = x o + k m , and we see that I
#(C) = lim x~ + k m k-~oo
kN
_ m N
Hence, the rotation number of C is rational, and C must have periodic points. The obtained contradiction proves our lemma. [] Thus, if C0 has the POTP, then the family It0] contains a homeomorphism preserving orientation and having periodic points. Hence, this family contains a homeomorphism C such that the set Fix(C) is not empty. For this diffeomorphism r the following statement holds. L e m m a 3.1.3. / f a , b EFix(r sign on (a, b).
then either (a, b)AFix(r
= O orq~(t)-t
changes
Proof. To obtain a contradiction, we assume that there exist points a, b, c E Fix(C) such that c 9 (a, b) and ~(t) - t _> 0 on (a, b). Let a < b.
3.1 One-Dimensional Systems
177
Take e > 0 such that 2e < min(c - a, b - c). Find d > 0 corresponding to this e by Definition 1.3 (recall that r has the P O T P by our assumption). Let us construct a d-pseudotrajectory of r as follows. Since the set Fix(C) is nowhere dense, there is a point Yo E (a, b) such that c-e>yo>a+e,
andyoq~Fix(r
Then we can find points do, bo EFix(r
such that
yo 9 (do, b0) C (a, b), and ~(t) - t ~ 0 on (ao, bo). It follows that (do, bo) is an r-interval, hence Ck(yo) --+ bo for k --* cr Take kl > 0 such that
r(r kl (Yo), bo) < Find Yl qWix(r
d
5"
such that d d b 0 + ~ < yl < bo+ 2 '
There exists an v-interval (hi, bl) containing YI, it follows that r
--* bl for k --* cr
Similarly we find a number k2 and a point y2, and so on. Obviously, in the course of this process we obtain a point bn such that bn - c > e. Set xk = Ck(x0) for k < O, xk = Ck(yl) for 1 < k < kl, Xk = c k - k l - a ( y 2 )
for kl + 1 < k < kl + k2 + l,
and so on, and xk,+k = C k ( y n ) for k > O, where k' = k l + . . . + k n - 1 + n - 1. It follows from our construction that ~ = {xk} is a d-pseudotrajectory of r Let x be a point that (@shadows ~. By the choice of x0, x E (a,c). Then r(r b=) > e for k > O, while xk --* b,~, and x cannot (@shadow ~. The obtained contradiction completes the proof. [] Remark. The same reasons show that if r is a preserving orientation homeomorphism of S 1 such that the set Fix(C) consists of one point, then r does not have the P O T P .
Obviously, Lemmas 3.1.1 - 3.1.3 imply the necessity of conditions of Theorem 3.1.1. Now we prove their sufficiency.
178
3. Systems with Special Structure
Fix e > O. It follows from conditions (b) and (c) that we can find fixed points al, bl,..., a2N, b2N of r with the following properties: 0 k0, we take ~(n) = ko. Otherwise there exists kl > k0 and an index m such that xkl E I , , We consider the case when 1,~ lies in an r-interval. Then there exists k~ > kl such that xkI E I " U Im+x. If I,~+1 lies in an /-interval, then xk E 1'~ for k > k~, thus we can take ~(n) = k~. If I,~+1 lies in an r-interval, there are two possibilities. Either xk E I " for k >__ k~ (and again we take ~(n) = k~) or there exists k2 > kl such that xk~ E 1,,+1. One of the segments 1 1 , . . . , 1 N
3.1 One-Dimensional Systems
181
lies in an l-interval, hence there exist ~ = g(n) and q such that x~ 9 Iq, Iq is an v-interval, and Iq+l is an /-interval. Obviously, this g(n) has the desired property. The theorem is proved. [] Now we show that for one-dimensional dynamical systems the P O T P implies the LmSP [Plal]. T h e o r e m 3.1.3. If a homeomorphism r of S 1 has the POTP, then r has the
LmSP. Proof. Assume that a homeomorphism r of S 1 has the POTP. By Theorem 3.1.1, there exists a natural m such that r = Cm satisfies the conditions of Theorem 3.1.2. Consider a sequence {xk} such that (3.15) holds. It follows from the inequality zn--1 rt----0
and from (3.15) that
r(xk+m, r
---* 0 as k ---* oc.
Hence, for Yk = xkm we have r(yk+l, r
--* 0 as k --* o~,
and it follows from Theorem 3.1.2 that there is a point x such that r(r
~k~) ~ 0 as k --* ~ .
= v(r
We claim that for any l, 0 < l < m - 1, we have v(r
~k~+,) -~ o as k ~ o~,
this will prove our theorem. Our claim follows from the inequalities
V((/)km+l(x) ' Xkm+l) ~ r(r +v(r
r
r
.~_r(r + . . . + r(r
r
(,Tkm+l))..[ -
xk~+~),
since every term on the right tends to zero as k tends to infinity.
D
Now let us mention some results on shadowing for semi-dynamical systems generated by continuous (but not necessarily invertible) mappings f : [0, 1] --* [0, 1]. In this case, it is natural to study an analog of the P O T P + on [0,1] (we also call it POTP+). Let f be a continuous function on [0,1]. Denote by F the set of all interior fixed points of f . Define the set
182
3. Systems with Special Structure H= {xEF:
for any e > 0 there exist y, z E ( x - c , x + e )
suchthat
f(y) < y and f(z) > z}. Pennings and Van Eeuwen [Pe] proved the following statement.
Theorem 3.1.4. Let f be a nondecreasing continuous function mapping [0, 1] to itself. The semi-dynamical system on [0, 1] generated by f has the POTP+ if and only if F = H. Remark. It follows from Theorem 3.1.4 that if a function f has no interior fixed points (for example, it behaves like f(x) = v/~), then the corresponding semi-dynamical system has the POTP+. Note that the dynamical system on S 1 generated by the homeomorphism corresponding to r = v/~ does not have the POTP on S 1 (see the remark after Lemma 3.1.3). Let us mention some papers devoted to one-dimensional shadowing. Coven, Can, and Yorke [Cov] studied shadowing properties for families of tent maps. Mizera [Mi] showed that POTP+ is a generic property in the space of semidynamical systems on [0,1] with the C~ The POTP+ for mappings of [0,1] was also studied by Gedeon and Kuchta [Ge] and Slackov IS1].
3.2 Linear and Linearly I n d u c e d S y s t e m s We begin with linear dynamical systems on C ~ and IR". Let A be a nonsingular matrix, complex in the case of C", and real in the case of IR'~. We consider the dynamical system r = Ax. As usual, the matrix A is called hyperbolic if its spectrum does not intersect the circle {)~ : I~1 = 1 }.
Theorem 3.2.1. For the system r the following statements are equivalent: (1) r has the POTP; (2) r has the LpSP; (3) the matrix A is hyperbolic. Remark. In the case of ]R'~, the equivalence of (1) and (3) was published by Morimoto [Morim3], later the proof of the implication (1) ~ (3) given in [Morim3] was refined by Kakubari [Ka]. For a linear mapping in a Banach space, an analogous statement was proved in [Om2]. We prove here Theorem 3.2.1 only in the case of C n. We begin with a lemma.
Lemma 3.2.1. Let (X,r) be a metric space. Assume that for two dynamical systems r an r on X there exists a homeomorphism H of X such that r
H =
3.2 Linear and Linearly Induced Systems
183
H o r and H, H -1 are Lipschitz. Then r has the POTP (or the LpSP) if and only if r has the same property. Proof. We prove the l e m m a only for the P O T P , in the case of the LpSP the proof is similar. Assume that r has the POTP. Fix arbitrary e > 0. Find el > 0 such that the inequality r(x,y) < q, x,y 9 X, implies that r ( H - l ( x ) , H - l ( y ) ) < e. Take A > 0 such that any A-pseudotrajectory {~k} for r is (el, r Now we find 5 > 0 such that r(x, y) < 5 implies r(H(x), H(y)) < ,5. Consider a 5-pseudotrajectory {xk} for r Set ~k = H(xk) for k 9 77. Since r(xk+l, r < 6 for k 9 77 and r o H = H o r we see that r(~k+l, r
----
r(H(Xk+l), r
= r(H(xk+l), H ( r
< A,
so that {~k} is a A-pseudotrajectory for r Hence, there exists ~ such that
r(~, r
< ~, k 9 77.
Let x = H - I ( ~ ) , then for any k we have
r(Xk, Ck(X)) = r(H-l(~k), C k ( H - l ( ~ ) ) )
r(H-l(~k), H - ' ( r
=
(we take into account here that r o H = H o r implies H -1 H -1 o Ck = Ck o H -1 for any k).
o r =
< e ~) o H -1 and [3
Now we prove Theorem 3.2.1. First we prove the implication (1) :=~ (3). let us assume that r has the P O T P . To obtain a contradiction, assume that the matrix A has an eigenvalue A such that [At = 1. Find a nonsingular matrix T such that J = T-1AT is a Jordan form of A. Then, for the dynamical system r = Jx and for the homeomorphism H(x) = Tx, the equality r o H = H o r holds. Since the homeomorphisms H , H -1 are Lipschitz in C '~, L e m m a 3.2.1 implies that ~ has the P O T P . We can choose the matrix T so that the matrix J has the form
(,0) 0
D
'
where B E GL(m, C) is of the form A 0 1A
.. ..
0 0 0 0
i
:
"'.
:
0
0
...
1A
:
Fix d > 0 and consider the vectors xk = ( x ~ , . . . , x k ) , k C 77, such that k satisfy the relations z~ = kAkd, z~,..., zm x~+1 = Ax~-t- x~_ 1 for i = 2 , . . . , m ,
k E 77,
184
3. Systems with Special Structure
and x~ = 0 for i = m + 1 , . . . , n , k E 7]. It follows from the equalities Xk+l - Jxk = (Ak+ld, 0 , . . . , 0 ) that IXk+l -- Jxkl = d, hence ~ = {xk} is a 2d-pseudotrajectory of r For any vector x = (Yl,.-., y~) we have Ck(x) = Jkx = (.kkyl,...), and this implies the inequality ICk(x) - xkl _> I k d - Yll. Since the right-hand side of the last inequality is unbounded for any Yl, we see that r does not have the POTP. The obtained contradiction proves the implication (1) =~ (3). Obviously, (2) ~ (1). To prove the remaining implication (3) ~ (2), we assume that the matrix A is hyperbolic. We show that r = A x has the LpSP with any finite do. It will be also shown that the shadowing trajectory is unique in the following sense: if for a sequence {xk : k E 7/} we have sup IAxk - xk+al < cr
(3.17)
kE7
then the inequality sup IAkx - xkl < c~
(3.18)
kEZ
holds for not more than one x. Denote by S the invariant subspace corresponding to the eigenvalues ~j of A such that IAjl < 1, and by U the invariant subspace corresponding to the eigenvalues ~j of A such that I~jl > 1. It follows from our assumption that C'~ = S @ U. We can find a natural m and A E (0, 1) such that IIA'~ Is II, IIA-m Iu II-< A.
(3.19)
By Lemma 1.1.3, to prove that r has the LpSP, it is enough to show that r = A m x has this property. To simplify notation, we assume that inequalities (3.19) hold with m = 1 (another possibility is to introduce a norm in C ~ equivalent to the standard one and such that inequalities (3.19) hold with m = 1, we leave the details to the reader). Now we find N > 0 such that the projectors P and Q onto S and U with the property P + Q = I satisfy the inequalities IIPIh llQIl 0 such that any d-pseudotrajectory of r is (e, r Take a d-pseudotrajectory ~ = {Xk} of r Since ~ is a d-pseudotrajectory of r it is (e, r by a point x E S ~, and it follows from the choice of e that x ~ P. The property of r mentioned above implies the inequalities
xk) _
0; z-1 = z +'m~ zk = Ck+l(z-~) for k < 0. Let s be a Lipschitz constant of r Since r
r(r
= p, we have
) _ 7r/2. Since ~ contains both points ( 1 , 0 , . . . , 0 ) and ( - 1 , 0 , . . . , 0), and d is arbitrary, we see that r does not have the POTP. The lemma is proved. [] Now we can establish the implication (1) =~ (3). First assume that the matrix A has two real eigenvalues A and A' with ]A] = I '1. Since the linear spherical transformation r generated by A 2 has the P O T P if and only if r has, we may assume that A = A' > 0, then by Jordan's theorem A is conjugate to a matrix of form (3.23), where B is one of the matrices A1 or A3 (see Lemma 3.2.3). If A has a pair of conjugate complex eigenvalues, then in a Jordan form (3.23) of A, the matrix B coincides with A2. In all these cases, Lemmas 3.2.2 and 3.2.3 imply that r does not have the POTP. Obviously, (2) =~ (1), hence it remains to show that (3) =~ (2). We may assume that A = diag(A0,..., A,,), where A0 > A1 > ... A,~ > 0. In this case, r has fixed points P~, i = 0 , . . . , n, corresponding to the eigenvectors of A, P~ = (a0,...,aN) with ai = 4-1, aj = 0 for j # i. Let us identify Tp,~:S n with the subspace {x E IR~+1 : xi = 0}. Direct calculation shows that for y = (Yl,-.., Yi-1, Yi+l,..., Y~) we have (A0
Dr
=
-~iYo,...,
Ai-1
~i+1
AN )
Ai yi-x,---~-i Yi+l,...,-~iYn
,
hence any fixed point of r is hyperbolic. It is easily seen that, for a fixed point P+, the stable and unstable manifolds are given by
Ws(P +) = {x e Sn : Xo . . . . .
z i - 1 = 0, z i > 0},
3.3 Lattice Systems
+) =
{x e
x , > 0, X,+l . . . . .
189
xn = 0},
and for a point P/- these manifolds are given by
W ' ( P [ ) = {x 9 S'~: Xo . . . . .
x,-1 = 0, x, < 0},
W~'(P:) = {x 9 S " : xi < 0,Xi+l . . . . .
x~ = 0}.
It follows that every point of S ~ belongs to the intersection of a stable manifold of a fixed point and an unstable manifold of a fixed point. Hence, the nonwandering set $2(r consists of fixed points of r and this means that r satisfies Axiom A. Let us show that r satisfies the geometric STC. Take a point x = (x0,. 9 x~) 9 S ", let x 9 W ' ( P ) fq W~(Q). If x0 r 0, then P = P+ or P = P o . Since both manifolds W s ( P +) and W S ( P o ) are n-dimensional, x is a point of transverse intersection of W ~ ( P ) and W~(Q). Similarly, if zn r 0, then Q = P+ or Q = P~-, and W ' ( P ) , W~(Q) are transverse at x. If x = ( 0 , . . . , 0 , x , , . . . , x , ~ , 0 , . . . , 0 ) with xl r 0 and xm r 0, then P = Pt+ or P = Pl-. In this case, T , W ' ( P ) is the intersection of the ( n - / + l ) - d i m e n s i o n a l subspace { 0 , . . . , 0, y l , . . . , yn} of IR~+1 with T , S ~, hence dimWS(P) = n - I. Similar reasons show that dimW~(Q) = m and
d i m ( T , W ' ( P ) N T~W~(Q)) = m - l, hence dimWS(P) + dimW~(Q) - d i m ( T ~ W ' ( P ) M T~W~(Q)) = n - l + m - m + l = n, and this means that W s ( P ) , W~(Q) are transverse at x. Now Theorem 2.2.7 implies that r has the LpSP. This proves our theorem. []
Remark. In [Sas], the equivalence of statements (1) and (3) of Theorem 3.2.2 was established. In this paper, Sasaki showed that the same conditions on the matrix A are equivalent to the P O T P for the corresponding real projective linear transformation. In [Katol, Kato2], Kato established similar results for Grassmann transformations and Poincar~ diffeomorphisms on spheres, he also considered the corresponding flows.
3.3 Lattice Systems We work in this book with a particular class of autonomous lattice dynamical systems defined as follows. Consider the Banach space B = {u = { u j } : uj E IRk,j E 77} with the norm
190
3. Systems with Special Structure Ilulr = sup luJl.
jEz
Fix a natural number s and denote
{uJ s= NJ-,...,~J,...,u~+s) e (iR~)~s+l. Consider a smooth mapping
F : (iRk)2~+l __, irk and define a corresponding operator 7- as follows:
[T(u)Ij = F({uj}~).
(3.24)
Under appropriate conditions on the mapping F, 7- maps B into itself, hence it defines a semi-dynamical system (7-~,B) called a lattice dynamical system with discrete time. A sequence {u(n) = { u j ( n ) : j 9 71}: n > 0} is a trajectory of this system if and only if the relations
uj(n + 1) = F(uj_,(n)...,uj+,(n)), n > 0, j 9 7],
(3.25)
hold. Lattice dynamical systems are models for a wide class of physical phenomena in space-time (see [Cou], for example). Another source of lattice dynamical systems are discretizations of partial differential equations. Consider, for example, a parabolic equation vt = v~,: + f(v, v~:), (3.26) where v,x 9 i r (we do not fix boundary conditions in this example). Let us discretize Eq. (3.26) with space step D and time step h. Denote by uj(n) the corresponding approxirnate values of v(jD, nh) for n > 0,j 9 7/. Taking standard approximations uj(n + 1) - uj(n)
vt(jD, nh) ~
h
'
and
v~:(jn, nh) ..~ u j + i ( n ) - u j ( n ) D
v~,:(jD, nh)
Uj+l(n ) - 2 u j ( n ) +Uj_l(n) D2
we obtain the relations
uj(n + 1) - uj(n) h
Uj+l(n ) -
2uj(n) + uj_a(n) + f(uj(n),UJ+l(n)-~- u j ( n ) ) D2
easily reduced to the following lattice dynamical system:
uj(n + 1) = ~-7(Uj+l(n) + Uj_l(n)) + hf uj(n), ui+i(n)~ - u~(n) +
3.3 Lattice Systems
191
(here s = 1). We take the explicit scheme above only to simplify presentation. These schemes are not of real practical interest, usually implicit or semi-implicit discretizations are applied (see Sect. 4.3). In [Cho8], Chow and Van-Vleck established a finite-time shadowing result for lattice dynamical systems and applied it to discretizations of some classes of partial differential equations, such as Burger's equation,
v, = a(x, t)v= + b(x, t)vv~, and the Korteweg-de Vries equation,
v, = a(x, t)v=~ + b(x, t)vv~. In this section, we introduce some special classes of pseudotrajectories for lattice dynamical systems and show that it is possible to reduce the shadowing problem for them to the same problem for auxiliary finite-dimensional dynamical systems [Af2]. Three types of solutions are usually studied for lattice dynamical systems. Steady-state solutions. These solutions do not depend on time n, we denote u j ( n ) = Vj. They satisfy the equations
vj = F ( v j - s , . . . , vj+s), j E 71.
(3.27)
Travelling wave solutions. Fix integer numbers l and m and consider solutions of the form uj(n) = v(lj + mn). They are called (l, m)-travelling waves and satisfy the equations v(lj+mn+m) = F(v(lj-ls+nm),...,v(lj+Is+mn)),
j E7I, n > O. (3.28)
Spatially-homogeneous solutions. They do not depend on the spatial coordinate j, we denote uj(n) = v(n), and they satisfy the equations + 1) =
> 0.
(3.29)
These types of solutions are governed by finite-dimensional dynamical systems (see below). The definition of a pseudotrajectory for a lattice dynamical system is similar to Definition 1.1. We say that = {zj(n): zj E IRk,j E 71, n >_ 0} is a d-pseudotrajectory for system (T =,/3) if
Izj(n + 1) - F(zj_8(n),..., zj+n(n)) I < d, j E 7/, n > O.
(3.30)
Now we define three types of pseudotrajectories corresponding to the three types of solutions introduced above.
192
3. Systems with Special Structure
d-static pseudotrajectory. A d-pseudotrajectory {zj(n)} is called d-static if Izj(n + 1) - zj(n)l < d for j E 77, n > 0
(3.31)
(i.e., it almost does not depend on time). d-travelling pseudotrajectory. A d-pseudotrajectory {zj(n)} is called a d(m, /)-travelling wave pseudotrajectory if
Izj_m(n + l) - zj(n)l < d for j E 77, n > - l .
(3.32)
d-homogeneous pseudotrajectory. A d-pseudotrajectory {zj(n)} is called dhomogeneous if
Izj+l(n) - zj(n)[ < d for j E 77, n > 0.
(3.33)
We begin with some standard results connected with the global inverse mapping theorem. Let f : ]Rp --~ ]Rp be a mapping of class C 1. We say that f satisfies the Hadamard conditions (and we write f E HC(p) in this case) if (HC1) d e t D f ( x ) ~ 0 for x E ]RP; (HC2) If(x)l ~ oo as IxI ~ ~ . Note that if p = 1 and there exists k > 0 such that If'(x)l >_ k for all x, then f E HC(1). A proof of the following statement can be found in [Z]. T h e o r e m 3.3.1. If f EHC(p), then f is a diffeomorphism of lR p onto ]Rp. Now consider a mapping f : lRq x IRp --~ IRp of class C 1, and let y , x be coordinates in ]Rq, ]Rp. We say that f satisfies the generalized Hadamard conditions (and we write f E GHC(q,p) in this case) if (GHC1) d e t ~ ( y , x ) # 0 for all y,x; (GHC2) for any fixed y, If(y,x)I--~ cc as Ix]--~ co. If p = 1 and there is k > 0 such that
Of(y, 5)
>__k,
then f E GHC(q, 1). It follows from Theorem 3.3.1 that if f E GHC(q,p), then for any fixed y, the mapping f ( y , .) is a diffeomorphism of IRp onto IRp, hence a mapping : ]Rq • lRp --* IRp is defined such that f(y,
z)) = z.
By the implicit function theorem, ~ is of class C 1. Hence, the following statement is true. T h e o r e m 3.3.2. If f E GHC(q,p), then
(a) for any y E IR q, z e IRp there exists a unique 4)(y, z) such that
3.3 Lattice Systems
193
f ( y , ~(y, z)) : z, and the mapping ~(y, z) is of class C1; (b) for any compact susbet K C IRq x IRP there exists a constant Co = co(K) such that fol" (y, zi) , (y, z2) ~_ K we have I~(~,Zl) - ~(~, z~)l < ~01zl - z21.
S h a d o w i n g of d-static p s e u d o t r a j e c t o r i e s Let us assume that for the mapping F ( u i _ , , . . . , uj+s) defining the system (T ~, B) we have F E GHC(2sk, k) (here u j _ s , . . . , uj+,-1 are coordinates in IR2sk, and uj+8 are coordinates in IRk). It follows from Theorem 3.3.2 that in this case there exists a mapping e E cl(]et, 2`k, In k)
such that for any
(Yl...,
Y2,+1) E ]R (2`+l)k (here y~ E IRk) the equality Ys+l
=F(yl,...,y2,+I)
implies that Y2,+l = G(yl, . . . , Y28).
Hence, it follows from Eq. (3.27) that Vj+, = G ( v j _ , , . . . , vj+,_l).
(3.34)
Let us introduce, for j C 77, ! (2s) X 1) = VJ- s , ' ' ' , x j : Vj+s-1.
Then Eq. (3.34) may be represented as a dynamical system (with time j) on the set of steady-state solutions, X~1:1 = X} 2) , "" " , X(2s--1)j+I =
xj(2"), Xj+l(2")-- a ( x ! 1), . .. , xj(2")').
We assume, in addition, that Ia(x!l),...,x}2"))[ ~ (:K) aS tX!I)[-"+ OO. Denote
r Since
g2,)) : (x(2),..., x(2,), a(g'),..., g2,))).
(3.35)
194
3. Systems with Special Structure
De=
l
0
I
...
0
: 0
: 0
"" ...
: I
'
where I is the unit k x k matrix, and
OG --
(~X(1)
,
it follows from T h e o r e m 3.3.1 that r is a diffeomorphism IR2sk -~ ]R2sk. 3.3.3. Assume that the diffeomorphism r has a hyperbolic set A. Then there exist numbers L,do > 0 such that if (a) ~ = {zj(n) : j E 77, n >_ 0} is a d-static d-pseudotrajectory of ( T n,13) with d < do;
Theorem
(b) dist((zj_s(n),...,zj+~_l(n)),A)
< do for j E 7], n >_ O,
(3.36)
then there exists a point x E ]R2~k (independent of n) with the property I ( z j - ~ ( n ) , . . . , z j + ~ - l ( n ) ) - CJ(x)l _< Ld f o r j C 77, n > O.
Proof. A p p l y T h e o r e m 1.2.3 to find numbers L', d' > 0 such t h a t r has the LpSP on the set U = Nd,(A) with constants L', s It is shown after the r e m a r k to T h e o r e m 1.2.3 t h a t r is expansive on a neighborhood of A, we assume t h a t r is expansive on U with an expansivity constant b. Let y be coordinate in IR 2sk and let w be coordinate in IRk. There exists N > 0 such that A C {lYl < N}. Let
g = { l y l < N + d ' } x { I w l < g + d ' } c ] R 2skxlR k . Apply T h e o r e m 3.3.2 to find a n u m b e r Co = Co(K) for the m a p p i n g f = F (with q~ = G) and the c o m p a c t set K . Set
L = 2CoL', do = min
L ~ 1' 2Co' 2s + 2s
"
(3.37)
Fix d < do and assume t h a t ( = {zj(n)} is a d-static d - p s e u d o t r a j e c t o r y for ( T '~, B). Obviuosly, we have
zj(n) = F ( z j _ s ( n ) , . . . , zj+~(n)) + (zj(n) - zj(n + 1))+ +(zj(n + 1) - F ( z j _ ~ ( n ) , . . . , zj+~(n)). It follows from (3.30) and (3.31) t h a t
F ( z j _ ~ ( n ) , . . . , zj+s(n)) = zj(n) + F1,
(3.38)
3.3 Lattice Systems
195
where IFll
O, m > Is.
197 (3.44)
If we introduce, for q E 7], X~1) = v(q), . . . , x_(lq-/) q = v(q + 1 ) , . . . , _(1+2Is) = v ( q + ~q
21S),.
_(ts+m) .. , ~q
:
v ( q + ls q- m -
1),
then Eq. (3.43) may be represented as a dynamical system (with time q) on the set of (l, m)-travelling waves, X(1) = a,q _(lsTm) , q+l ~" X~2), .. " , x(lsTm-1) q+l
(ts+m) = F(x~I), 9 , xq(1+21s) ).
Xq..}.1
(3.45)
We assume that F satisfies the generalized Hadamard conditions, so that (3.45) defines a diffeomorphism r : ]R(l"+m)k ~ ]R(Z~+m)k. Now we consider the following equation: 17 + mu = 1, 7, v E 7].
(3.46)
We assume that (3.46) has a solution (% v) with v > 0 (for example, if l, m are relatively prime, Eq. (3.46) has an infinite set of solutions with u >_ 0). Let for such a solution x = max(l'fl, u). Consider a d-(m,/)-travelling wave pseudotrajectory {z/(n)}, fix a solution (7, v) of Eq. (3.46) with v >__0, fix j E 77, n > 0, and for integer t _> 0 define vectors Z, = ( Z } I ) , . . . ,
Z} Is+m)) E ~(ls+m)k
by setting = z,(,.o(Z(i,
t)),
(3.47)
where a ( i , t ) = j + (i + t - 1 ) 7
, /3(i,t) = n + (i + t - - 1 ) u
(3.48)
for i, t _> 0. T h e o r e m 3.3.4. Assume that the diffeomorphism r defined by (3.45) has a hyperbolic set A and take L, do > 0 such that r has the LpSP with these constants on U = Ndo(A) (see Theorem 1.2.3). There exist numbers L1,L2 > 0 having the following property. If d(L1 + i2a) < do (3.49) and if for a d-(l, m)-travelling wave pseudotrajectory {zj(n)} and for a solution (7, v) of (3.46} with v > 0 there exist j, n such that for the vectors Zt we have
198
3. Systems with Special Structure
(3.50)
dist(Zt, A) < do, t ~ 0,
then there exists x C IR(t~+m)k such that [Zt - r
< d(L~ + L ~ ) L ,
t >_ O.
Proof. The hyperbolic set A is compact, hence there exists N > 0 such that the s(s + 1)d0-neighborhood of A is a subset of the set
{(X(1),... ,X 0 such that max l_(j_~2s+l
< M
for all ( Z l , . . . , z2~+~) with Izd _< N. Let L1 = M s ( 2 s + l) + l, L 2 = l + s . Consider a d-(l,m)-travelling wave pseudotrajectory {zj(n)} and j , n such that (3.49) and (3.50) hold. If we show that { Z t , t > 0} is a d(L1 + n2a)pseudotrajectory of r this will reduce our theorem to Theorem 1.2.3. Since
r
= (Z[2),..., Z[ Is+m), F(Z[1),..., z(l+2ls))),
we see that = Iz~(a+,~+l,t)(/3(Is + m + 1 , t ) ) -
]Zt+l - r
-F(z~(1,t)(/3(1, t ) ) , . . . , z~(l+2,s,t)(/3(1 + 21s, t)))[. It follows from (3.32) that
Izj_,m(n + pl) - zs(n)l < Iptd for all p E 77 such that n, n + p > 0. Hence, for i = 2 , . . . , 2s we have Iz~(l+ti.,)(/3(1 + li,t)) - Z~(l+t,,t)+,~m(/3(1 + li,t) - ivl)[ < iud. Since a(1 + li,t) + i v m = j + t7 + 17i + i v m = (~(1, t) + i and /3(1 + li,t) - ivl = n + tv + ium - i v m =/3(1, t), we see that Iz,(l+ti,t)(/3(1 + li, t)) - z,o,t)+i(/3(1, t)) I < ivd. Now it follows from the choice of M that
(3.51)
3.3 L a t t i c e S y s t e m s
199
[F(zc~(l,t)(]~(l, t)),..., Za(l+21s,t)(/3(1 + 21s, t)))- F ( z ~ ( ~ ) , z~+a(~),..., z~+2s(/3))] < < M d ( 1 + s + . . . + 2s) = M d s ( s + 1)
(3.52)
(in the estimates above, a = a(1, t),/3 =/~(1, t)). Similar arguments show that if a' = a(Is + m + 1, t), /3' =/3(ls + m + 1, t),
then Lz~,(D') - z~,+,~(w-~)(D' - l(~s - ~))1 < dl vs - ~1 < xd(s + 1),
(3.53)
and . ' + m ( ~ s - ~) = .
+ s, y - t ( ~ - ~) = ~ + 1.
It follows from (3.30),(3.51)-(3.53) that
[St+ 1 -- ~(Zt) I < IZ~Ts(~-~ +ids(s
1) -
F(z~(~),...,Zc~+2s(~))l-[-
+ 1) + tcd(s + 1) = d(L1 + L2x).
This completes the proof.
[]
The cases of m, l which do not satisfy (3.44) (but for which Eq. (3.46) has solutions (% u) with v _> 0) can be treated similarly. We discuss here in detail only the case m = l = 1. In this case, the "travelling coordinate" is q = j + s - m , and Eq. (3.43) has the form v(q + s + 1) = F(v(q), v(q + 1 ) , . . . , v(q + 2s)).
(3.54)
For brevity, we consider here the case k = 1. If s = 1, we assume that ~(z)-l[>__c0>0 for all z E ]R3, and if s > 1, we assume that OF Oz--~+l(Z) > co > 0.
Under these conditions, there exists a mapping G such that the equality vs+2 = F ( y l , . . . , y2s+l) is equivalent to Y2~+1 = G ( y l , . . . , y2~). Hence, Eq. (3.54) is equivalent to
(3.55)
200
3. Systems with Special Structure
v ( q + 2s) = G ( v ( q ) , v ( q + 1 ) , . . . , v ( q +
2s - 1)).
(3.56)
Let
X~a) = v ( q ) , . . . , x~2s) = v(q + 2s - 1), q E 77, then Eq. (3.56) m a y be represented as a dynamical system with time q,
X(') q+l = X~2),
Xq(2s) ~
,..,
e(x~i)
... ,x~2S)).
(3.57)
We assume, in addition, that
~
(z) > g > 0,
then it follows from T h e o r e m 3.3.1 that (3.57) defines a diffeomorphism r : ]R2~ --+ ]R 2s. For l = m = 1, Eq. (3.46) has the form 7+u=l. Obviously, its solutions with v > 0 form the set {(1 - u , u ) : u
> 0}.
Fix j E 7/, n, u E 77+, define fl(i, t) for i, t > 0 similarly to (3.48), and set c~(i, t) = j + (i + t - 1)(1 - u). Define Z~0 by (3.47), and consider z, =
.., z ? s)) 9
3.3.5. Assume that the diffeomorphism r defined by (3.57) has a hyperbolic set A and take L, do > 0 such that r has the LpSP with these constants on U = Ndo(A) (see Theorem 1.2.3). There exist numbers L1,L2 > 0 having the following property. If for a d-(1, 1)-travelling wave pseudotrajectory {zj(n) } and for some u > 0 (a) d(La + L2x) < do; (b) there exist j , n such that for the vectors Zt inequalities (3.50) hold, then there exists x E ]R2s such that
Theorem
]Zt - r
O.
Proof. T h e proof is based on the same ideas as the proofs of two previous theorems, so we give only a sketch for the case s = 1. Due to the compactness of the hyperbolic set A, we can find M > 0 such that in all our considerations below we have
3.3 Lattice Systems
201
O~-~zF j _<M. Let
M+I
M
L1 - m
+1, L2=--+2. Co Co By (3.50), it is sufficient to show that {Zt} is a d(L1 + L2u)-pseudotrajectory of r We have IZ~+, - r = ~~t+l ( ~ - a ( z } 1), z } ~ ) l . Denote • =
G(Z~ 1), Z~=)), then r = F(Z} 1), Z}=),r
Set a = a(1, t),/~ =/~(1, t). It follows from the equalities ,~(2,,) = ,~ + 1 - v, •(2, t) = Z + v, z} ~) = z~t~,,)(Z(2,
0),
and from (3.32) that IZ[ ' ) - Zo+l(Z)l < yd.
Hence,
( = F(z~,(~), z~+l(fl), ~) + F1, where [FI[
0 such that
dist(S(t)B,.4) < e for t > T; (a3) .A is Lyapunov stable, i.e., for any neighborhood W of .4 there is a neighborhood U of .A such that for u E U we have S(t)u E W, t > O. Note that (a3) is a consequence of (a2). Condition (3.64) implies [Hel] that S(t) has a global attractor. The existence of a global attractor implies the existence of a bounded open set 7-[0 such that for any bounded set B C 7-I there is a time T(B) such that S(t) C 7-lo for t > T(B) (a set with this property is called absorbing). Property (b): S(t) has Morse-Smale structure on .A, i.e., conditions (a4)(aT) below are satisfied. First we give some definitions. Note that a fixed point p = p(x) of S(t) is a solution of the following boundary-value problem:
p** + f(p) = 0, p(0) = p(1) = 0.
(3.66)
As usual, we say that a fixed point p is a hyperbolic fixed point of S(t) if the spectrum of the derivative DS(t)(p) does not intersect the unit circle [He1]. One can reformulate this condition as follows. A fixed point p = p(x) of S(t) is hyperbolic if and only if 0 is not an eigenvalue of the linear variational operator, v ~ v,~ + f'(p(x))v, with the Dirichlet boundary conditions. In this case, we call the solution p(x) hyperbolic. If p is hyperbolic, then the stable manifold of p with respect to S(t) defined by w
(p) =
e
: s ( t ) u --, p as t
and the unstable manifold of p defined by
W~(p) = {u 9 TI: S(t)u exists for t < 0 and S(t)u ~ p as t ~ - c ~ } are smooth immersed submanifolds of 7"l [Hel]. If p, q are hyperbolic fixed points of S(t), we say that W~(p) and W~(q) are transverse if, for any v 9 W~(p) D W~(q), the sum of T~W~(p) and T,W~(q) equals 7"/[He2]. Now we formulate the definition of the Morse-Smale structure on .4. (a4) .A contains a finite number of fixed points 7rl,..., r g of S(t), and these points are hyperbolic;
(as) n=
U l 0. We denote by 7-[* a ball in 7"[ centered at the origin and containing the 1neighborhood of 7"/r. Write problem (3.62)-(3.63) as an evolution equation,
(3.72)
= Au + n ( u ) ,
on 7"l. It is known [Fo, Chob] that it is possible to modify the nonlinearity R in Eq. (3.72) outside the ball 7-l* so that the modified nonlinearity vanishes on a neighborhood of infinity, and the system has an inertial manifold. It follows that the trajectories of the modified system beginning at points of 7-[0 coincide with the corresponding trajectories of the original system. Below we work with the modified system preserving notation (3.72) for it. Let inequality (3.71) be satisfied for u E 7"/*. Take an integer T > 0 such that 1
u = Cexp(-aT) < ~
(3.73)
(below we impose one more restriction on T; this restriction is "absolute", i.e., it depends only on the attractor .A).
3.4 Global Attractors for Evolution Systems
207
T h e o r e m 3.4.1. Assume that S(t) has properties (a) and (b). Then there ezist constants do, Lo > 0 and a neighborhood W of .A in TI such that if for a sequence {uk:k>0}CWwehave la(uk) - Uk+ll < d, dist(uo, A4) < 2d,
(3.74)
and d < do, then there is a point u E .A4 such that I k(u) - ukl _< Lod, k > O.
Remark. As was mentioned, it is enough to assume that S(t) has property (a), and every fixed point of S(t) is hyperbolic. To prove this theorem, first we reduce our shadowing problem to an analogous problem on A//, and then we establish the desired shadowing property on A4.
Reduction For Uo E 7-/let po = Puo and denote by p(t, po) the solution of the finitedimensional system on PT-/,
= Ap + PR(p + ~(p))
(3.75)
such that p(O, p0) = P0. Then (see [Fo]) for u0 e AJ we have
u(t) = S(t)uo = p(t, po) + ~(p(t,p0)).
(3.76)
We introduce the following notation. For Po = Puo, Uo C 7-/, we set
al(PO ) = p(1,po), a~(po) = p(T, po) (the number T was fixed by (3.73)), and for mo E .A~ we set
r
= o{~(po) + ~(o~(po)), r
= o:~(po) --~ ~(dr~(po)) ,
where Po = Pmo. It follows from the inclusions f , 9 E C 1 that ~r~,a~- (and r r are Ckmappings of PT-/(of .A4, respectively) to itself. In addition, by (3.76) for u0 E A/l we have r S(1)uo a(uo), =
--
i.e., r is the restriction of a on M . Obviously, ,A is the global attractor for a in 7-I and for r in M (the definitions are parallel to the one for a). Set C, = C(1, n o ) , CT = C(T, 7-/0) (see (3.65)). Fix a neighborhood ,s C 7-10of ,4 in ,~A. Since ~4 is Lyapunov stable, there exists a neighborhood A//1 C Ado of A such that S(t)u E ~,4o for u E ,~,41 and t > 0. It follows that for p0 E I/1 = P,~,41 we have p(t,po) E IIo = P • o for
208
3. Systems with Special Structure
t >_ 0 and that r is a diffeomorphism of class C 1 of A/t1 onto its image. Let K1 be a Lipschitz constant of a~r = p(T, .) o n / / 1 . Due to the same reason there exists do > 0 such that the inequality dist(u, ,4) < do implies the inclusions an(u) E 7-10 for n > 0. Let W1 be the do-neighborhood of .A. Assume, in addition, that the do-neighborhood of W1 is a subset of 7-10 and that P projects this neighborhood of W1 i n t o / / 1 . Take a sequence {uk} C W1 such that (3.74) holds with d < d ~ Denote Zk ~
ItTk.
L e m m a 3.4.1. Let C2 = 1 + C1 + ... + C T - l , C3 -
2C2 1--v
Then (1) lar(zk) - Zk+l[ _< C2d; (2) /fdist(z0, A/t) < 2d, then dist(zk, A/f) < C3d, k > O. Proof. First note that the choice of do and W1 implies that an(uk) E 7-lo for n > 0. Since C1 is a Lipschitz constant of a on 7-/0, statement (1) follows from Lemma 1.1.3. To prove statement (2), denote bk = dist(zk,.tel). By assumption, we have b0 < 2d. Now we estimate bk+l 0 such that, for v , v ~ E A41, the inequalities r(v, v') < Iv - v'l < r(v, v') K3
hold. It follows from (3.83) that r(vk+l, r _< KaCgd. Let us show that Theorem 3.4.1 is a corollary of the following statement. T h e o r e m 3.4.2. There exists a neigborhood M of A in M and numbers d', L > 0 such that i f { v k } C M and r(r vk+l) < d < d', then there is a point v E M with the property r(r vk) O. Indeed, assume that there exists v C M such that
r(r Then
~
Ld, k > O.
3.4 Global Attractors for Evolution Systems ICk(v) - vkl 0, Ai E (0, 1) such that
[Dr IDr
[ O,
(3.89)
_< K, ATIvl, v E E~', m > O.
(3.90)
Take Ao E (0, 1) and a natural number To such that K~AT~ 0. D e f i n i t i o n 4.1 A finite set (Y,h) = ({Yk e IR": 0 < k < N}, {hk > 0 : 0 < k < N})
is called a "periodic discrete ( d, h )-pseudotrajectory" (pd( d, h )-pseudotrajectory
below) for sustem (1.1 S) if the inequamies ]Yk+l
--
~(hk,yk)] < d, 0 < k < N,
(4.7)
and X(yk) r
O < k < N,
(4.8)
hold. Here and below, to make our formulas shorter, for any variable ak with index 0 < k < N, we introduce ag+l = ao, so that, for k = N, the inequality in (4.7) has the form ly0 - Z ( h N , N)I < d (and this is the reason to call (]i, h) a periodic pseudotrajectory). A pd(d,h)-pseudotrajectory is a natural model for the computer output in the process of computer realisation of a one-step method for numerical approximation of solutions for an autonomous system of ODE near a real closed trajectory. Below we describe a simplified version of the main result in [Coo2] (some estimates in our Theorem 4.2.1 are more "rough" than in [Coo2]; this enables us to make the proof more "readable"). Let us begin with some additional notation. We will simultaneously explain the meaning of the defined objects. Take a pd(d, h)-pseudotrajectory (Y, h). Assume that we are given a finite sequence {Yk : 0 < k < N} of n x n matrices such that
]lYk --D~(hk,Yk)ll < d f o r 0 < k < g .
(4.9)
The matrices Yk are approximations for the matrices D~(hk,yk). Since the variational flow 69(t) = D~(t, Yk) satisfies the variational initial-value problem d
-~0 = D X ( Z ( t , yk))O, 0(0) = I,
(4.10)
a natural way to find Yk is to apply a one-step method to approximate simultaneously the problem = X(x), x(0) = Yk,
4.2 Periodic Shadowing for Flows
225
and (4.10) for t 9 [0, hk]. Let us denote by L'k the subspace of IR ~ orthogonal to the vector X(yk). For 0 < k < N , let S~ be an n x (n - 1) m a t r i x chosen so t h a t its columns form an "almost o r t h o n o r m a l " basis in ,Uk, i.e., the inequalities
ISZX(y
)l, IIS;& - Zll
(4.11)
dl
hold with some dl > 0, and 9 denotes transpose. Next we choose (n - 1) x (n - 1) matrices Ak, 0 < k < N , so t h a t ItAk
S~+lYkSkll < da,
-
0 < k < N.
(4.12)
T h e q u a n t i t y dl in inequalities (4.11) and (4.12) is introduced to account for possible round-off errors in the necessary m a t r i x computations. Now we define some constants involved in the proof of T h e o r e m 4.2.1. Set hmin =
min hk, hmax = m a x hk.
0 0. Consider a mapping F : lRn --~GL(n, R) such that
lIDr
-
F(x)ll
d
(4.31)
for all x 9 lRn. We treat F(x) as an approximation of the value De(x) (for example, given by a numerical method). Let ~ = {xk : k > 0) be a d-pseudotrajectory of r We introduce the number
234
4. Numerical Applications of Shadowing ts(~,F) =
1
max l i n ~ - - l o g v~",lvl=l m
IF(xm_x)F(x,n_2)... F(xo)V I.
(4.32)
T h e o r e m 4.3.1. Let r : IRn --* ]R~ be a diffeomorphism of class C 2. Assume that A is a locally maximal hyperbolic set ore such that dimU(p) = 1 for p E A. There exists a neighborhood W of A and numbers L0, do > 0 with the following property. If ~ C W is a d-pseudotrajectory o r e such that d < do, dist(xo, A) < d, and inequality (4.31) holds for x E W , then there is a point p E A such that Its(p) - ts(5, F)I _< Lod.
(4.33)
Before we prove Theorem 4.3.1, we state an auxiliary result. We consider a sequence of linear mappings fi : ]Rn ~ lRn, i > 0, having the form fi(x) = Aix + gix,
(4.34)
where A~ EGL(n, IR), and gi are small n x n matrices. It is assumed that there exist constants M, Co, l > 0, and A0 E (0, 1) such that (al) IIA, II, IIA.7~II < M , i >_ o; (a2) there exist linear subspaces S~, Ui of ]R'~ such that (a2.1) S~ @ Ui = IR'~, i > 0; (a2.2) AiT~ = Ti+l for i > 0, T = S, U; (a2.3) if v E Si, then IAi+m_l... Aiv I < Co)C~H for m > 0; (a2.4) if v E Ui, then [A[2m . . . A 7 l v ] ~ CoA~tvl for m , i - m >_ 0; (a3) Iigii] < l, i > 0.
(4.35)
Remark. It follows from the remark after Definition 1.15 that under the above conditions there exists a positive number a (depending on M, Co, and ~) such that Z(Si, Ui) > a,i > O. The statement below is a special variant of the stable manifold theorem (Theorem 1.2.1) for small linear perturbations of hyperolic mappings, and it can be proved using Perron's method (see the proof of Theorem 12.5 in [Pil]). For fixed i > 0, we can introduce coordinates x = (y,z) with respect to the representation of lRn in (a2.1) so that y are coordinates in Si and z are coordinates in Ui.
4.3 Approximations of Spectral Characteristics
235
T h e o r e m 4.3.2. Assume that mappings (4.34) satisfy conditions (al)-(a3). Given C > Co,~ E (0,~o),b > 0, there exist numbers lo = l o ( M , C , ~ ) and
Ko = K o ( M , C , )~) such that if g~ satisfy (4.35) with l < lo, then there exist linear subspaces Si,Hi of ]R'~ such that (1) Si,Hi are given by z = ~ y and y = y~z, respectively, and
I1~,11,I1~11 -< Kol; (2) if Xo E Si, then IfiTm--1 0 . . . 0 fm(XO) I < c~rnlxol for m > O;
(3) if Xo E Hi, then )~-m
If,+m-x o . . . o
fm(xo)l >
-~-Ixol
for m >_ 0;
(4) fi(Si) = ~5i+1 and fi(Lli) = Ui+l. Now we prove Theorem 4.3.1.
Proof. Denote by Co, ~0 the hyperbolicity constants of A, let S(p), U(p), p E A, be the hyperbolic structure on A. Find numbers M, a such that
IIDr
IIDr
~ M
and
/(S(p), U(p)) > a for p E A. Apply Theorem 1.2.3 to find a neighborhood U of A such that r has the LpSP on U with constants L, d~. Since the set A is locally maximal, we may assume that the inclusion O(p) C U implies p E A. Let s > 1 and s be Lipschitz constants of r and D e on U. Find a neighborhood W of A and a number A > 0 such that N a ( W ) C U. Set L1 = L s and find dx > 0 such that
s
< do and Lldl < A.
Take a d-pseudotrajectory ~ = {xk : k > 0} C W such that d < dl and dist(x0, A) < d. Find a point y E A with [Xo - Yl < d. Set ~' = {yk : k E 77}, where Yk = Ck(y) for k < 0 and yk = xk for k > 1. Since x0, yo E W, the set ~' is an s of r It follows from the choice of W and dl that there is a point p such that ICk(p) - ykl < Ls The inclusion
k C 77.
236
4. N u m e r i c a l A p p l i c a t i o n s of S h a d o w i n g
O(p) C NLld(W) implies that p E A (and hence O(p) C A). Set Pi = r We can represent F(z~) in the form F(xl) = Dr I[Dr and IIF(xi) - Or
+ Hi. The inequalities
< Lls
- Dr
< d imply that ]lHill < n2d,
(4.36)
where L2 = L1/:1 + 1. Set Si = S(pi), U~ = U(p~). Obviously, the mappings
fi(x) = Aix + Giz, where Ai = Dr and Gi = Hi, satisfy the conditions of Theorem 4.3.2. Fix C > Co,s E (~o, 1), and apply this theorem to find the corresponding lo = / o ( M , C, A), Ko = Ko(M, C, ~). Take d2 ~ (0, dl) such that L:d~ < lo, then it follows from (4.36) that for d < d2 there exist the corresponding subspaces Si and/4i such that for the matrices ~i, 7]i we have the estimates II~ill, II~ill -< KoL~d, i >_O. It is geometrically obvious that there exists a constant La > 0 such that
L(S,, S,), L(l,li, Ui) < L3d.
(4.37)
Take n - 1 linearly independent unit vectors v ~ , . . . , v,~_1 E So and a unit vector v ~ E U0. By the definition of a hyperbolic set,
IDr
< Co~ylv~l for k = 1 , . . . , n - 1,
hence 1
lirr~__,~-- log IDCm(p)v~l < log Ao < 0 for k = 1 , . . . , n - 1. m Since
IDr
>
(Co)-l~omlvUh
we obtain the inequality l i m ~ - . ~ o ! l o g InCm(p)v~l >_ - l o g Ao > O, m
and it follows that 1
#(p) = li--~,,~_.o~--log ]De '~ (p)v~l. m
Similar reasons based on statements (2) and (3) of Theorem 4.3.2 show that if w = is a unit vector in L(o, then
4.3 Approximations of Spectral Characteristics #(~, F ) = limm_..oo1 log m
237
IF(xm_l)... F(Xo)w~J.
Fix m > 0, let v(m) be a unit vector in Urn, and let w ( m ) be a unit vector in Hm. It follows from (4.37) that if {v(m), w(m)) > 0,
(4.38)
(here (,) is the scalar product), then Iv(m) - w(m)l < Lad. Denote
am =
!
am --IF(xm)w(m)l.
IDr
Let
~m = DCm(p), ~, ,m+~ = DCm+l(p)~ ~. Since the space Um is one-dimensional, there exists a nonzero number b such that Vm = bv(m). Then Vm+l hence
Ivm+,l Ivml
=
Dr
= bDr
Ivm+,l - Ibl
-
[Dr
= am.
It follows that ]Vm+l] ~
amlVm] ~ a . . . .
ao[vU],
so that #(p) = limm--.ooI log(am-a.., a0). m
Similarly, #(~, F ) = lin~.-,oo 1 l o g ( a ~ _ l . . , a0). m !
Obviously, we can take v(m), w ( m ) in the definitions of am, am so that inequalities (4.38) hold. Set 5 = w ( m ) - v(m). Take d2 = min(dl, 1), then (4.36) implies for d _< d2 the inequalities IlHmll < L2 and IIF(xm)]] < M + L2. Since
F ( x m ) w ( m ) = (De(pro) + Hm)(V(m) + 5), we can estimate
]am - a" I = IIF(xm)w(m)l - IDr < IF(xm)~[-t-I(F(xm) - Dr
0 such that if ~ C W is a d-pseudotrajectory of r with d _< do, dist(x0, A) < d, and inequality (4.31) holds for x E W, then there is a point p E A such that I#(p) - #(~, F)I ~< e.
4.3.2 A p p r o x i m a t i o n of the Morse S p e c t r u m Morse spectrum. We begin with the definiton of a chain recurrent set [Con]. Let r be a homeomorphism of a metric space (X, r). Fix d > 0 and denote by P(d) the set of periodic d-pseudotrajectories of r i.e., of d-pseudotrajectories ~ = {xk : k E 77} such that there exists a number p with the property xk = xk+p for any k E 7] (this number p is called a period of ~). The chain recurrent set CR(r is defined by the equality
OR(C) = A
P(d).
d>0
It is easy to show that this set is closed and r Now let A be a compact invariant set of a diffeomorphism r : IR'~ ~ lR'~. We construct a special dynamical system connected with the pair (A, r as follows. Recall that the (n - 1)-dimensional real projective space P,~_aIR is defined by identification of one-dimensional subspaces of IR% For y E IR'~, y # 0, we denote by [y] E Pn-IIR the class of equivalence of the line {ky : k E IR}. Since for nonzero yl, Y2 E IR'~ with [Yl] = [//2] (i.e., for Yl = ky2 with k r 0) and for any x we have Dr = kOr we can define a mapping F ( x ) : P,~_IIR --~ P,~_IlR by F(x)[y] = [Dr Below we denote points of P,~-IIR by v.
4.3 Approximations of Spectral Characteristics
239
Denote T A = A x Pn_l]R. We fix a metric p on P n - I I R and introduce the corresponding metric r((x, v), (x', v')) = Ix - x' I + p ( v , v ' ) on TA. Now we define a m a p p i n g # ( x , v) = (r F(x)v). Obviously, # is a homeo m o r p h i s m of TA. Fix a point (x, v) E TA, take a vector w E ]R~ such t h a t Iwl = 1 and [w] = v, and define the n u m b e r
a(x, v) = IDC(x)wl. T h e n u m b e r a(x, v) is properly defined, since there are two vectors with the described properties, w and - w . Consider a sequence (xk, vk) E T A such t h a t (xk, vk) --~ (x,v). Obviously, we can choose vectors wk, w e IR~ so t h a t ]wkl = ]w[ = 1, [wk] = Vk, [w] = v, and Wk --* w. In this case, we have - w k --* - w , and it follows that the m a p p i n g
a :TAn
IR+
defined above is continuous. Let ~ = {(Xk,Vk) : k ~ 0} C T A be a d-pseudotrajectory o f # , as usual, this means t h a t the inequalities < d, k > 0,
are fulfilled. Fix a natural m and set 1 m--1
A(m, r = - - ~
log a(xk, vk).
m k=l
T h e Morse spectrum of the dynamical system # on the chain recurrent set C R ( # ) is defined as follows: Z ( ~ ) = {A = limk~oo A(mk, ~k)}, where mk ~ c o aS k ---+ o o , and ~k C C R ( # ) are dk-pseudotrajectories of with dk --~ 0 as k --* co. Note t h a t we give here the definition of the Morse s p e c t r u m in the simplest possible case of a diffeomorphism of IR~, for more general definitions see [Os3]. If ~ = {(xk,vk)} C T A is a periodic d-pseudotrajectory of 9 of period p, we define the n u m b e r 1 : log a(xk, vk). (4.40) Now we introduce the periodic Morse spectrum ~Pp(~) = {)~ = limk--,o~ A({k)}, where {k C T A are periodic dk-pseudotrajectories of 9 with dk --+ 0 as k --+ co. Colonius and Kliemann [Col] showed t h a t Z ( ~ ) = Z p ( ~ ) . Hence, to investigate
240
4. Numerical Applications of Shadowing
the Morse spectrum it is enough to study periodic dk-pseudotrajectories of with dk ~ 0. Symbolic image. Let us describe the concept of symbolic image of a dynamical system [Osl]. Let again r be a homeomorphism of a compact metric space (X, r). Consider a finite covering ~D = {~D(1),... ,7?(s)} (4.41) of X by closed sets. For i = 1 , . . . , s, we introduce the sets
c(i) = {j E { 1 , . . . , s } : r
M:D(j) # 0}.
The symbolic image of r corresponding to the covering 7) is a graph G with directed edges and with s vertices (the vertices are denoted by numbers 1 , . . . , s). The graph G contains an edge i ~ j if and only if j C c(i). We characterize the covering ~D by two numbers, ~(D) = max diam:D(i) l 0.
246
4. Numerical Applications of Shadowing
First we establish some properties of eh,N we need. In [O1], the following statement concerning r = eh,N was proved. T h e o r e m 4.4.1. (1) There exists a continuous function V(v) on ~t '~ such that V(r < ])(v), and the equality holds if and only if v is a fixed point ore. (2) If p, q are hyperbolic fixed points of r then the stable manifold of p is transverse to the unstable manifold of q. Let us begin with some definitions. For vectors v , w E IRN, let N
(v,w} = D ~
UmVm, IVl2 = (V,V).
m=l
We say that a diffeomorphism r : IRN --~ IRN is dissipative (in the sense of Levinson) if there exists a bounded set B C IRN such that for any v ~ C ]RW there exists no with the following property:
r
~ c B, n > n ~
It is well known (see [Ha], for example) that if r is dissipative, then r has a global attractor Jr, and .A C B (see the definition of a global attractor in Sect. 3.4). Let B~o,a1 be the set of functions f C CI(]R) such that for any u E IR we h ave uf(u) ~_ ao + alu 2. (4.51) Obviously, for f E B~o,~1, the inequality (f(v),v) 1 and for any v E IRg. T h e o r e m 4.4.2 Assume that f E Bao,~l with al < 7r2. Then there exist numbers h0, No, p > 0 depending on el and M (in (~.~9)) and such that
lim.-~oolr
~ p
for all v E ]RN, h E (0, h0], N > No. Easy calculation shows that the following statement holds. L e m m a 4.4.2.
The eigenvalues of the matrix A are -(~--q-) sin2 ( - ~ ) ,
and the corresponding eigenvectors are
m= 1,...,N,
4.4 Discretizations of PDEs
247
(sin m TrD , sin 2mTr D , . . . , sin N TrD ). Now let us prove Theorem 4.4.2. Proof.
Substitute ~)n+l _ vn§
-- vn 2
y n+l ~- v n + 2
into the left-hand side of
(Av n+l, u n+l) :
( A v n+l + f(vn), vn+l).
We obtain
Iv"+i12- I.-12 2h
+
Iv-+1- v-I~ 2h
_ ( A v n + l , vn+l) + (f_(v,~), ~)n+l).
(4.53)
Since If(ui) - f(u2)l _< M l u l - u21 by (4.49), it follows from the Cauchy inequality that (f(vn),~)n+l) ---- (f(vn+l),vn+i) + (f(v n) __ f(vn+l),vn§
~___(f(vn+l), yn+l) +
___~ (4.54)
M i v , + l _ v " ] . Iv"+'].
Obviously, for any a > 0 we have
iv-§ 1 _ v - i . iv-§
a
_< ~ iv~§
+ 1 I~"§ - v"l ~.
(4.55)
Fix b > 0 such that al + 2b < ~r2. Since the matrix A is symmetric, it follows from Lemma 4.4.2 that for all v E IRN we have ( A v , v) ,~-(~--2-)sin2(-~D-)M2 < _ - ( a l T 2 b ) H
2,
where the last inequality holds for D small enough (we assume that it holds for
g > Yo). Hence, for N _> No and for all v C ]RN we have
_< a0 - 2blvl 2.
(4.56)
Take a > 0 such that M a < 2b, then we deduce from (4.54)-(4.56) that the right-hand side of (4.53) does not exceed M ,+1 _ v=]2. a0 - b]vn+'{ 2 -{- ~alV
(4.57)
Now if we take ho = a / M , then (4.53) and (4.57) imply the inequality 2~[[~)n+112 _ [~)n]2] ~ ao -- b]v'~+ll2
(4.58)
248
4. Numerical Applications of Shadowing
for n >_ 0, N _> No, h E (0, h0]. It follows from (4.58) that ]v'~+1]2(1 + 2bh) 1, with two topologies. One of them is the C q strong Whitney topology introduced in Sect. 3.4, we denote the corresponding functional space by .T'~. Another considered topology is the standard topology of uniform Cq-convergence on compact subsets of IR. The base of neighborhoods of a function f in this topology consists of sets {g: Pqg(f,g) < e} for compact sets K C lR and positive numbers e (the numbers Pqg(f,g) are defined by (3.69)). We denote the corresponding functional space by 5c~. Fix a compact set K C ]RN and a diffeomorphism r = Ch,N. Let .T"q be one of the spaces 5c~ or 9v~. T h e o r e m 4.4.3. For q > 1, the set
7tq(K) = { f E ~q : fixed points of r in If are hyperbolic} is residual in .T"q. Proof. Fix q > 1 and L > 0 such that K C {v : - L < v,~ < L, m = 1 , . . . , N}. As usual, we say that a fixed point v of r is simple if det(Dr
- I) ~ 0.
(4.59)
Define 7-I = { f E ~'q : all fixed points of r in K are simple}. We claim that 7-I is a residual subset of .T"q. A point v is a fixed point of r if and only if
4.4 Discretizations of PDEs
249
9 (v) := A v + f(v) = O. For a mapping k~ : IRN --* IRN of class C 1 we say that a point x is critical if rankDk~(x) < N. We denote by S(k~) the set of critical points of ~. Since "1
9 (v) =
and det g # 0, for a fixed point v of r (4.59) is equivalent to v ~ S(~). First consider the set ]1~ = {v E IRg : v~ # v~ for i # j}, and define K0 = lR0N U K. The case of 1 ~ will be a particular case of IRN (see below), but we treat it separately to clarify the main idea. Let 7r = (7rl,..., 7rs) be a permutation of { 1 , . . . , N}. Take 1 > O, 1 1 , . . . , IN E lR such that -L
< ll -- 2l < ll d- 2l < 12 -- 21 < . . . ,
and denote/z = (l,
ll,...,
lN
-4- 21 < L,
(4.60)
IN). Set
Ro,,~,~, = {v E 1 ~
: l l - l < v~(i) < li + l,i = 1 , . . . , N } ,
and 7"/o,~,u = { f E 9t'q : fixed points of r in Ro,.,u are simple}. Simple fixed points are isolated, hence there is only a finite number of them in a compact set. This implies that the set 7-/o,~,u is open. Let us prove that 7-/o,~,~ is dense. By Sard's theorem (see [Hirs2]), mes 4~(S(~)) = 0 (here mes is Lebesgue measure in IRN). Therefore, given e > 0 there exists a E IRN , [al < c, such that a ~ ~(S(O)).
(4.61)
Consider fl E ~E-qsuch that f l ( u ) = f ( u ) - a~(,~)
for Im--l 0, 11,..., lm E IR such t h a t - L < 11 - 2l < 11 + 21 < 1 2 - 21 < . . . , l m
+ 21 < L,
denote # = (l, 11,..., Im),
RN
A , T r ~
={vEIR N:li-l 0 and take fl satisfying (4.67) and such that
f;(u,) = ~ Then we have
1 + h f ~ ( u , ) = 1--~e(1 + h f ' ( u l ) ) , i = 1 , . . . , s . Hence, for any fixed point v E K, Dr
1 i + e Dwkv)'~''
Therefore, the eigenvalues/~j of Dr (v) and the eigenvalues Aj of De(v) are related by 1 #J = 1 + cAJ"
4.4 Discretizations of PDEs
253
Obviously, we can find e so small that fl is in a given neighborhood of f in brq, and, for all #j, the inequalities
I#Jl-fi 1 hold. Hence, fixed points of Ca in K are hyperbolic. This completes the proof. [] Now we describe the properties of the global attractor .4 = .4h,N for a diffeomorphism r = Ch,N assuming that all fixed points of r are hyperbolic. T h e o r e m 4.4.4. If all fixed points of r are hyperbolic, then r has Morse-Smale
structure on .4 (see the definition in Sect. 3.4). Pro@ Since .,4 is compact and hyperbolic fixed points are isolated, .4 contains a finite number of fixed points, w l , . . . ,wm. This proves (a4) (we refer to the numbers of items in the definition of a Morse-Smale structure given in Sect. 3.4 working with IRN and r instead of 7-I and S(t)). As usual, we denote by WS(w ~) and W"(w i) the stable and unstable manifolds of w i. To prove (aS), we first show that the nonwandering set S2 of the diffeomorphism r coincides with the set { w l , . . . , wrY}. Obviously, fixed points are in ~?. Consider a point v such that r r v, denote a = V(v) and b = 12(r (the function 12 is given by Theorem 4.4.1). It follows from Theorem 4.4.1 that a > b. Set c = (a + b)/2. Since r and l) are continuous, there exists a neighborhood U of v such that v(r
> c and V ( r
< c for , ' e U.
(4.68)
It follows from (4.68) and from ])(r
< 1)(r
for k > 1
that Ck(U) 91U = 1~for k > 0, this proves that v q~ S2. Hence, f2 = { w ' , . . . , w i n } . Now we claim that for any point v E i _< m, such that Ck(v)
]R N
w i as k
(4.69)
there exists a fixed point w i, 1 _< -+
this will prove (a5). Denote by w(v) the w-limit set of the trajectory O(v), i.e., the set {limk--+~ Ctk(v): lk --+ ee as k --* e~}. It is well known that for any v we have w(v) C J2. Since -4 is the global attractor of r any positive trajectory tends to -4, hence any set w(v) is nonempty
254
4. Numerical Applications of Shadowing
and consists of fixed points of r To prove our claim, let us show t h a t every set
w(v) is a single fixed point. Assume that, for some v, the set w(v) contains two fixed points, w i and w j. Since the set of fixed points is finite, we can find their neigborhoods U 1 , . . . , U m such t h a t
u ku r
k) n u ~ = 0 for k # I.
It follows from our assumption that there exist two sequences, Ik, mk --~ c~ as k --~ oo, such that
& ( v ) -~ w', Cm~(v) -~ wL and we can choose these sequences so that Ik < mk. For large k we have Ctk(v) C U i and Cmk(v) E U j. Hence, for these k there exist numbers nk such t h a t lk < nk < mk and
r Set zk - r
u', r
9
~
u'.
and let z be a limit point of the sequence {Zk}. It follows t h a t
z 9 r
\
v'.
Our choice of the neighborhoods U k implies t h a t
Z ~ ul U . . . U Um, hence z ~ f2. On the other hand, nk --~ co, and it follows t h a t z 9 w(v) C f2. The obtained contradiction proves (a5). Now let us prove (a6). Since `4 is invariant, for any v 9 `4, the inclusions Ck(v) 9 ,4 hold for k _< 0. T h e same arguments as above (applied to r instead of r show t h a t for any v 9 .4 there is a fixed point w i such t h a t v 9 W~'(w~). Hence, ,4
c 0 W~(w~) 9 i=1
To prove the inverse inclusion, assume that for some fixed point w i there is a point v 9 W ~ ( w ') \ .4. T h e n a : = dist(v,.4) > 0.
(4.70)
Since .4 is L y a p u n o v stable (see (a2)), there exists a neigborhood U of .4 such that for any v' 9 U and k > 0 we have dist(r
< a.
It follows from Ck(v) ~ w i as k ~ - c o t h a t there exists k0 < 0 such t h a t v' = Ck0 (v) 9 U. T h e n for k > 0 we have
dist(r176
< a,
4.4 Discretizations of PDEs
255
and for k = - k 0 this inequality contradicts to (4.70). The obtained contradiction proves (a6). The second statement of Theorem 4.4.1 proves (a7). [] One can repeat the proof of Theorem 3.4.2 (taking IR N instead of M ) to show that if all fixed points of r are hyperbolic, then r has the LpSP+ on a neighborhood of .A. In a computer, the time step h takes rational values, so it is reasonable to consider a fixed countable set H of h-values. Theorem 4.4.3 and the reasons above prove the following statement. T h e o r e m 4.4.5.
For q > 1 there exists a residual subset F q of jr~ (or .T~) such that if f 9 F q, then r = Ch,N has the LpSP+ on a neighborhood of its global attractor ,4 for any (h, N) 9 H x IN. Remark. Let U be a neighborhood of .4 such that r has the LpSP+ on U with constants L,d0 (of course, these characteristics depend on h , N ) . Consider a bounded set B C IRN. One can find positive numbers dl = dl(B) _< do and no = no(B) such that if a sequence {w" : n > 0} is a d-pseudotrajectory of r with d < dl and w ~ E B, then w ~ 9 U for n > n0 (see [Pi3] for details). Set u ~ = w ~+~~ There exists a point v' such that ICk(v ') - ukl < Ld for k > 0. Since r is Lipschitz in IRN with a Lipschitz constant depending on h, N, there is a constant L1 = L I ( B ) > L such that ICk(v' ) - u k] < L l d f o r
-n0 Ko, N >__No, and write 9 instead of ~K,N. T h e o r e m 4.4.6. There exists m = m ( D + h) such that for any Vo E Wo M 7-IN we can find u with the property [S(nh)u - ~mKT+"(VO)I _ 0, where L = Co( LoCo + 1).
4.4 Discretizations of PDEs
257
Proof. Take v0 E W0 f-1 7/N and construct a sequence {v~ : n > 0} setting v,+l = ~ g T ( v , ) . By the choice of W0, the inclusions
s(t)v, e w, t > 0, and ~k(v.) ~ W, k > 0, hold. We can apply statement (2) of L e m m a 4.4.3 (with n = K T and u = v = vk) to show that
I~(vk) - v~+ll
=
IS(T)~k
-
CKT(vk)I < d,
where d = Co(D + h). Since hgo < 1 and D(No + 1) _< 1, inequality (4.71) implies t h a t d < do. It follows from L e m m a 3.4.4 t h a t there exists m0 = mo(D + h) such t h a t dist(vk, M ) _< 2d for some k < m0. Set w,~ = vk+,~ for n > 0. It follows t h a t the sequence w,~ satisfies all the conditions of T h e o r e m 3.4.1, hence there exists y such t h a t
I~"~(y) - w,~I < Lod for n > 0. Set m = m0 + 1, u' = rm~ l u ' - w,~0-kl = Ir'~~
and u = "rm-k(y). Since
- Wmo-k] < Lod = LoCo(D + h) < A,
it follows from statement (1) of L e m m a 4.4.3 and the choice of W2 and A that S(t)u' C W for t > 0. Hence, we can apply statement (2) of L e m m a 4.4.3 to show t h a t
IS(T+nh)u'--q~gT+'~(Vmo) ] < C o ( ] U ' - V m o ] + D + h ) = L ( D + h )
(4.72)
for 0 < nh < T. Note that
S ( n h ) u =- S ( T + nh)u' and ~mKT+"(Vo) = ~gT+"(vmo) , hence inequality (4.72) establishes the statement of our t h e o r e m for 0 < n < K T . T h e same reasons and the inequality
l u - vm]
