Shadowing in Dynamical Systems

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]

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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

<

hold. Thus, the existence of a shadowing point for a pseudotrajectory ~ means that ( is close to a real trajectory of r The mostly studied shadowing property of dynamical systems is the POTP (the pseudoorbit tracing property). A system r is said to have the POTP if given e > 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

<_ d <_ do

(0.1)

there exists a point x with the property

r(bk(x), Xk) <_ Ld.

(0.2)

In Subsect. 1.2.1, we describe main properties of hyperbolic sets. It is shown that there exists a Lyapunov metric in a neighborhood of a hyperbolic set. In Subsect. 1.2.2, we give a detailed proof of the Shadowing Lemma applying the approach of Anosov [Ano2].

IX

Let A be a hyperbolic set of a diffeomorphism ff : IR'~ --+ IR~. Assume that, for a sequence {xk} belonging to a small neighborhood of A, inequalities (0.1) hold. Following Anosov, we study a functional equation to find a sequence {vk EIR '~} such that r + vk) = xk+l + vk+l (0.3) (i.e., the sequence {xk + vk} is a trajectory of r and the point x = x0 + v0 satisfies inequalities (0.2). Subsection 1.2.3 is devoted to the so-called "theorem on a family of etrajectories" [Ano2] and some of its applications. This statement says that if A is a hyperbolic set for a diffemorphism r and if a neighborhood of A contains a family of approximate trajectories of a diffeomorphism r Cl-close to r then it is possible to shadow all this family by trajectories of r In Subsect. 1.2.4, Bowen's proof of the Shadowing Lemma is given. We also describe an abstract construction modelling the method of Bowen (the so-called Smale space) introduced by Ruelle [Ru]. In Sect. 1.3, we introduce the main technical tool applied in this book "abstract" shadowing results for sequences of mappings of Banach spaces [Pi4]. In the case of a hyperbolic set of a diffeomorphism r we can introduce "local" mappings =

+ v) - xk+l,

then Eqs. (0.3) have the form Ck(vk) = v k + l .

(0.4)

Generalizing this approach, we consider sequences {Ha} of Banach spaces and mappings r : Ha --~ Ha, and look for sequences {vk E Ha} satisfying (0.4). Obviously, inequalities (0.1) are equivalent to the estimates ICk(0)l _< d < do,

(0.5)

and inequalities (0.2) are equivalent to the estimates

Ivk] < Ld.

(0.6)

In Subsect. 1.3.1, we study sequences {r with (possibly) nonivertible linear parts. Theorem 1.3.1 (Subsect. 1.3.1) gives sufficient conditions under which there exist constants do and L such that inequalities (0.5) imply the existence of a sequence {vk} satisfying (0.6). Such a sequence is considered as a local shadowing trajectory for the sequence {@,}. In Subsect. 1.3.2, we obtain conditions for the uniqueness of a shadowing trajectory in this case. Subsection 1.3.3 contains a general scheme of application of Theorem 1.3.1. This scheme is applied to establish an analog of the Shadowing Lemma for one-sided sequences. Subsection 1.3.4 is devoted to two theorems on shadowing for sequences of mappings of Banach spaces proved by Chow, Lin, and Palmer [Chol] and by Steinlein and Walther [Stel].

Pliss [Pli2, Pli3] obtained necessary and sufficient conditions under which a family of systems of linear differential equations has uniformly bounded solutions. We "translate" these results into the shadowing language in Subsect. 1.3.5 and apply them to "affine" mappings Ck(v) = Akv + Wk+l, v E IR'~. It is shown that the developed method can be also applied to nonlinear mappings Ck (Theorem 1.3.7). In the statement of the shadowing problem above, the values dk =

r(Xk+l, ~)(Xk))

are assumed to be uniformly small. One can impose another condition on these values, dk --* 0 as k --+ cr and look for a point z such that the values hk = r(r

xk) -~ 0 as k ~ ~ .

The corresponding shadowing property (called the limit shadowing property [Ei2]) is studied in Sect. 1.4. It is shown in Subsect. 1.4.1 that in a neighborhood of its hyperbolic set a diffeomorphism has this property. In Subsect. 1.4.2, we investigate the rate of convergence of the values hk in terms of dk. It is shown that if a sequence {xk : k > 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 <_ m - 1, then there is a point x 9 X with r(r < e for 0 < k < m - 1. Then r has the P O T P on Y . (b) Assume that a set Y C X is negatively invariant, i.e., r 9 Y for x E Y and 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) <

e, Ikl _< m .

Let w be a limit point of the sequence wm (for simplicity we assume that w.~ converges to w). Passing to the limit as m --* oc in the last inequality, we see that

< e, k e 7/, hence ~ is (2e)-shadowed by w. (b) Let e > 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] <_ Ld, k e 77, are possible only if x = 0, since the trajectories of points x # 0 leave W. Hence, our assumption leads to the inequality v/-d < Ld which is contradictory for small d. Of course, the main reason for the absence of the LpSP in a neighborhood of the fixed point x = 0 in our example is the equality De(0) = 1. This equality means that x = 0 is a nonhyperbolic fixed point. One of the main goals of this book is to show that "hyperbolicity implies the LpSP" for a wide class of dynamical systems (we hope that the exact meaning of this statement will be clear to the reader after reading the book). Other shadowing properties are also considered. We define some of them below. In addition, let us mention the rotational shadowing property [Bar], the

1.1 Basic Definitions

5

asymptotically shadowing property [Che], the bi-shadowing property [A1N]. We refer the reader to the original publications for the details. Now let us pay some attention to the problem of uniqueness of the shadowing trajectory. We give the following natural definition. Let again r be a dynamical system on a metric space (X, r). D e f i n i t i o n 1.7 We say that r has the "SUP" (the "shadowing uniqueness property") on a set Y C X if there exists a constant e > 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)) <

k e x.

Since ~ = {r is a trajectory of r it is a d-pseudotrajectory for r with any d > 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[ <_ s + L i d , k E 77. Proof. The proof is straightforward. Take m E 77. By our conditions, (,~ = ym~. It follows from (1.5) that iym~+, - Cm~(~)i

Since r

--- d.

is Lipschitz on V~m.+l, we see that

< Ly~+~ - r

+ Ir

- r

or

< d + Z:d.

Repeating this process, we obtain the estimate i~m~,~(~) - ~§

= lem~,~(~m~) - ~m§

_< d(1 + s + . . . + s

<

= Lid.

This proves (1.6). Now we take arbitrary k E 77 and represent it in the form k = m~ + ml, 0 ~ m l _ ~ - l .

By the conditions of our lemma, i~

- y~i

= tz~ - ~ i

<

Let us estimate [Xrav+l

Similarly we estimate

-

-

Ym~+l[ ~ [r

-- r

~.

8

1. Shadowing Near an Invariant Set I'Tm~'+2

--

YrnlJ..~21 ~__ 1~3rnl,,..}-I 0 Crnu(Zrn) -- ~.)mu+l(Yrau-l-1)l-~-

+lCm~+l(Y,~+:) - Y,~,+~I < Z:(/:e + d) + d = s

+ (1 + Z:)d.

Continuing this process, we see that

Ix.~.+.-: - ym.+~-ll _< z:~-~e + (1 + Z: + . . .

+ / : ~ - 2 ) d < L:'-:e + Lid.

This completes the proof.

[]

Let us apply this lemma to show that a diffeomorphism r of IR~ has the LpSP on a neighborhood of a compact invariant set A if and only if 4~ = r with some u 9 IN has the LpSP on a neighborhood of A. First we assume that r has the LpSP with constants L, do on a hounded neighborhood W of A. L e t / : be a Lipschitz constant for r on W. Find a neighborhood W0 C W of A such that

r

W, : < i < . - 1 .

Take a sequence {z~ : m 9 71} C Wo such that Ir

- zm+l I < d < do.

Construct the corresponding sequence {xk : k 9 71} by formula (1.4) with Ck = r It follows from the choice of W0 that {xk} C W. Since r = xk+l for k ~ m u - 1 , m 9 71, and

there exists a point x such that ICk(x) - xkl <_ Ld, k 9 7/, then

I~m(x) - zml = ICm~(x)

-

xm~,l

<_ Ld,

m E

71.

Now we assume that the diffeomorphism ~ = r has the LpSP on a bounded neighborhood W of A with constants L, do. Let a g a i n / : be a Lipschitz constant for r on W. Find a neighborhood W0 of A with the same property as above. Take a neighborhood W1 of A and a positive number A such that W2, the A-neighborhood of W1, is a subset of W0. Now we set Hk = IR'~, Ck --- r Vk = W2, V~ = W for k E 7]. Take a sequence {yk : k G 71} C W: such that Ir where

- Yk+ll ~ d ~ d:,

(do,A)

d:=min\L:

LL:

"

(1.7)

1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism

9

It follows from the first statement of Lemma 1.1.3 that the sequence {~,~}, where ~,~ = ym~, belongs to W and satisfies the inequalities

[~(~m) - ~m+l [ _ Lid. Hence, there exists a point z0 such that for zm = ~m(z0) we have [zm - ~m[ <_ L L l d , m E 71.

By (1.7), the inequalities Izm -

< A

hold. Since ~,~ e W1, we see that Z,n E W2 C W0. Hence, we can apply the second statement of Lemma 1.1.3 to show that, for the sequence {xk} constructed according to (1.4), the inequalities [xk - yk[ <_ L2d, k 9 71,

hold, where L2 = f-)'-ILL1 + L1. Note that by construction the sequence {xk} is a trajectory of r This shows that r has the LpSP on W1.

1.2 S h a d o w i n g N e a r a H y p e r b o l i c Diffeomorphism

S e t for a

In this section, we describe two classical proofs of the so-called Shadowing Lemma, the main result about shadowing near a hyperbolic set of a diffeomorphism. These proofs were given by Anosov [Ano2] and Bowen [Bo2]. Different proofs of similar statements were given later by Conley [Con], Robinson [Robi3], Newhouse [Gu2], Ekeland [Ek], Lanford [Lanf], Bronshtein [Bro], Meyer and Sell [Me], Shub [Shu2], Palmer [Palm2], Blank [Bll], Kruger and Trubetzkoy [Kr], Katok and Hasselblatt [Katok2], Reinfelds [Re], and other authors. Below we mention some methods to establish shadowing which are of interest for the practice. The method of Anosov reduces the shadowing problem to a functional equation in a proper Banach space. We show in this book that the approach based on solving a functional equation is applicable to a very wide class of shadowing problems. That is why we describe its origin, the classical proof based on the ideas of Anosov, in Subsect. 1.2.2 with all the details (we hope that reading the proof of Theorem 1.2.3 will help the reader to understand its generalizations). In [Ano2], Anosov established the Shadowing Lemma as a particular case of a more general statement, the so-called "theorem on a family of e-trajectories'. This statement and some of its applications are described in Subsect. 1.2.3. The method of Bowen is "geometric", and it uses more detaled information about the behavior of the investigated system near its hyperbolic set. Subsection 1.2.4 is devoted to the method of Bowen and to an abstract construction modelling this method (the so-called Smale space) introduced by Ruelle [Ru].

10

1. Shadowing Near an Invariant Set

To avoid unessential technical difficulties, we consider in this section diffeomorphisms of IR~ (instead of the general case of a smooth manifold). The generalization to the case of a manifold (with application of exponential mappings) is described in detail in Chap. 2. 1.2.1 H y p e r b o l i c S e t s This subsection is preliminary, we devote it to the study of some basic properties of hyperbolic sets. Let r be a diffeomorphism of class C 1 of IR~. We denote by I.] the usual norm of ]R~. D e f i n i t i o n 1.9 We say that a set A is "hyperbolic" for a diffeomorphism r if

(a) A is compact and r (b) there exist constants C > 0, A0 9 (0, 1), and families of linear subspaces S(p), U(p) of IR~, p 9 A, such that (b.1) S(p) @ g(p) = IRn; (b.2) nr = T(r p 9 A, T = S, U; (b.s) ]Dem(p)v] < CA~]v] for v 9 S(p), m > 0; (1.8) IDr

< C~Ylvl for v 9 U(p), m > 0.

(1.9)

Remark. It is easy to show (see [Pil], for example) that the families of subspaces S(p), U(p) (usually called the hyperbolic structure on A) are continuous on A. Below we call the numbers C, A0 the hyperbolicity constants of A. In many proofs below, we apply the so-called Lyapunov (or adapted) norm [Anol] on a neighborhood of a hyperbolic set. With respect to this norm, estimates (1.8),(1.9) hold with C = 1 and A E (A0, 1) instead of A0. Let us prove its existence. L a m i n a 1.2.1. Let A be a hyperbolic set of r with hyperbolicity constants C, A0. Given e > 0, A E (A0, 1) there exists a neighborghood W = W(e, A) of A with the following properties. There exist positive constants Nt,6, a C ~ norm [.l~

for x C W, and continuous (not necessarily Dr extensions S', U' of S, U to W such that (1) S ' ( x ) ( ~ U ' ( x ) = ] R ~, x e W; (2) for x,y E W with [y - r < 6, the mapping II~Dr (II~Dr is an isomorphism between S'(x) and S'(y) (respectively, between U'(x) and U'(y)), and the inequalities

111~Dr

<- ~,t',,1~, IIZ~Dr

<- elvl=, v e s'(x);

IH~Dr

1 >_ ~[vl~, III~Dr

~

~lvl~, v ~ U'(x),

(1.10) (1.11)

1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism

11

hold, where H~ is the projection onto S'(x) parallel to U'(x), and H i = I - H~;

(s)

1

~ ; I v l ~ < Ivl < g'lvl~ for x 9 W, v 9 ~t n.

Remark. Take x 9 A , y = r norm on W.

(1.12)

in L e m m a 1.2.1 to see t h a t 1.1- is a L y a p u n o v

Proof. First we construct a continuous n o r m I.I- with the desired properties. Fix # 9 (~0,)~) and find a natural u such that C

< 1.

(1.13)

Take a point p C A and a vector v 9 IRn, represent v = v ~ + v ~ 9 S(p) | U(p), and set

Ivl~, _- (Iv'l~ + Iv~12p)112, where

b,

Iv'l, = ~ ~-JlDr

Iv"l, -- ~ ~-~ I D r

j=O

j=O

9

For v s we obtain the estimate

IDr162

= s I~-j ID# ( r

)Dr

=

j=O

= #

#-JlDCJ(p)vq + #-"-llDr

<

k j=l

t~-JlDr

< iz

#-u-lC/~;+ll/)s

\j=l

) ~ ]21vSlp

(we applied (1.13) in the last inequality). Similarly, for v ~ we have lJ

IDr

= y ~ tz-JlDr

(r

)Dr

=

j=O

=It-1 (~l.t-JlDr162 Since w : = D r

~ = Dr162162

Iwl < c ~ +' IDr hence we arrive to the inequality

~, it follows from (1.9) t h a t

12

1. Shadowing Near an Invariant Set

IDr > [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

<_~, II~Dr = O, >_ 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 <_ ~lvl~, v 9 V'(x).

(1.15) (1.16)

Now we describe the main geometric structures generated by a hyperbolic set, the so-called local stable and unstable manifolds W~(x) and W~(x).

1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism

13

Let A be a hyperbolic set for a C 1 diffeomorphism r of IRn, we assume that the Euclidean norm is Lyapunov in a neighborhood U of A, so that inequalities (1.10), (1.11) hold for x , y 9 U. A proof of the following statement (the generalized stable manifold theorem) can be obtained from Theorem 2.1 of Chap. 1 and Theorems 1.2,1.3 of Chap. 4 in the book [Plil]. To formulate the theorem, we need the following construction. Take a point p 9 IR~ and a number A > 0. Assume that ]R~ is represented as IRk X IR~-k with coordinates y in IRk and z in IR~-k so that p = (0, 0). We say that a set D C IR'~ is a C 1 A-disk centered at p if there is a Cl-mapping f from the set {y 9 IRk: lyl < A } into IRn-k such that f(O) = 0 and D = { ( y , f ( y ) ) : y 9 IRk ,lyl -< A}.

T h e o r e m 1.2.1. There exists a neighborhood U of A and numbers A, A1,/C > 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 <- Ax, then the disks W~(x), W~(y) have a unique point z of intersection, and Iz - xl, Iz - ul < ~ l x - ul.

14

1. Shadowing Near an Invariant Set

1.2.2 The Classical Shadowing L e m m a The following statement is usually called the Shadowing Lemma. T h e o r e m 1.2.2. If A is a hyperbolic set for a diffeomorphism r then there exists a neigborhood W of A such that r has the P O T P on W . In addition, we can find a neighborhood W1 of A such that r has the SUP on W1. We apply here the method of Anosov to prove that in a neighborhood of a hyperbolic set a diffeomorphism has a stronger shadowing property (the LpSP instead of the P O T P ) , and then reduce Theorem 1.2.2 to Theorem 1.2.3. It should be noted that both classical proofs of the Shadowing Lemma by Anosov and by Bowen really provide Lipschitz dependence of e on d in (1.1). T h e o r e m 1.2.3. If A is a hyperbolic set for a diffeomorphism r then there exists a neigborhood W of A such that r has the LpSP on W . In addition, we can take the number do such that, for d < do, the point x with property (1.2) is unique. Remark. A similar statement for the LpSP+ (without the uniqueness of a shadowing trajectory) is also true (see Theorem 1.3.3). Note that since a small neigborhood of a hyperbolic set is not always negatively invariant, we cannot refer to an analog of Lemma 1.1.1 for the LpSP and LpSP+. One of the main technical problems in the proofs of shadowing statements below is to establish that some operators in Banach spaces are invertible. We will often apply the following statement.

Lemma 1.2.3. (I) Let B be a Banach space. Consider a linear operator A : B ---* B such that I[A]] = A < 1. Then the operator I - A is invertible, and

I1(I

1 *' "~]-~11~ l-A"

(2) Let B be a Banach space represented as B = B ~ | B ~'.

(1.17)

Assume that for a linear operator A : B --~ B represented as [ A~, A ~ ] A~* A ~ according to (1.17), we have IlA**[I, I[(A~)-lt[ -~ A for some A ~ (0, 1).

(1.18)

1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism

15

If f o r u > 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 <_ u, I]A'Colt < v, < 1. (1.21) Define the operator C = Co ~ ( - 1 ) k ( A ' C o ) k. k=0

By the second inequality in (1.21), the series in the definition of C converges, and 1 IIcii < l I C o l l - < 1 - v, Let us show that ( I - A ) C = I. Indeed, since A"Co = I , we have ( I - A ) C = ( A " + A')Co Y ] ( - 1 ) k ( A ' C o ) k = k=0 oo

oo

= ~ ( - 1 ) k ( A ' C 0 ) k + y~](-1)k(A'C0) k+l = I. k=O

k=O

It is easy to see that if ( I - A ) C = I and C maps B onto B, then C = ( I - A) -1. Indeed, in this case for any x E B we can find y E B such that x = C y . Since (I-A)x = (I-A)Cy = y, we see that C ( I - A ) x = C y = x, hence C ( I - A ) = I. Thus, to complete the proof, we have to show that C maps B onto B. Represent C = CoC1 and set C2 = A~Co. By construction, Co maps B onto B. Since [IC2]1 _< vl, the same reasons as in the proof of (1) show that oo

C, = ~ ( - 1 ) k C ~

= (I + C2)-',

k=0

hence C1 also maps B onto B. Now we prove Theorem 1.2.3.

[]

1.2 Shadowing Near a Hyperbolic Set for a D i f f e o m o r p h i s m

17

Proof. Let A be a hyperbolic set with hyperbolicity constants C, ~0. Take numbers ~ E (~0, 1) and e > 0 such t h a t the inequality s

i < l

holds. Set R = R(~, e) (given by L e m m a 1.2.3). For the fixed )~, e, we can find, by L e m m a 1.2.1, a neighborhood W = W ( ~ , e ) , a n o r m ].Ix, and families of subspaces S ' , U ' such t h a t inequalities (1.10), (1.11) hold. Let us show that r has the LpSP on W. Take a sequence ~ = {xk} in W such t h a t Ir

- xk+ll ~ d.

By (1.12), the inequalities Ir

-- Xkl~k < N ' l r

-- xkl < N ' d

hold. A s s u m e t h a t we can find 6o, Lo such t h a t for any sequence ~ = {xk} satisfying the inequalities above with N ' d < 6o, there is a point x with the property ICk(x) - xal~:k < LoN'd, k C 7]. Then ICk(x) -- xkl < Lo(g')2d, k E 7], hence r has the LpSP on W with do = 6o/N', L = L o ( N ' ) 2. Thus, without loss of generality, we m a y assume t h a t the usual Euclidean n o r m I. I is L y a p u n o v in W. To simplify notation, we write S, U instead of S', U'. Fix a sequence ~ = {xk} such t h a t ]r

- Xk+l] ~ d,

take a sequence r/ = {Yk E IRn : k E 7/}, and set yk = xk + vk. T h e n ~ is a t r a j e c t o r y of r if and only if vk+l = r

+ vk) - z k + l , k ~ 7].

(1.22)

Obviously, we have r

+ vk) - Xk+l = [r

- Zk+l] + D r

+ hk(vk),

(1.23)

where hk(0) = 0 and Dhk(O) = O. Let B be the Banach space of sequences 9 = {vk E ]Rn : k E 7]} with the norm I1~11 = sup Ivkl, kEZ

Recall that for r > 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 <_ d + (C~ + b(d,))lvkl,

hence F is an operator from B(d~) to/3. By (1.22), r/ is a trajectory of r if and only if 0 is a fixed point of F. Now we represent /3 = B" @ B ~, (1.24) where s 9

= {~ 9 s : v~ 9 s ( ~ ) , k 9 ~z},

s ~

= {~ 9 s : v~ 9 U(x~), k 9 7z}

(recall that we denote by S(x), U(x) the extended families on W). It follows from (1.23) that F is differentiable at 0 with (DF(0)O)k+I = Dr Take vk = vk + vk, vk 9

and represent wk+l = Dr

~ 9 U(zk),

as

s 1 "91-W k'u.+ l , Wk+ 1 = Wk+

9 S(xk+,),

W ks+ l

W v. k+l 9 U(Xk+l).

Denote z = Xk+l. Then we have W~+l

8a 8~ W k + 1 + W k + 1,

where 88 s wk+, = II~Dr

8

8u s wk+, = IZ~Dr

Similarly, Wk+ 1 -~ Wk+l AI- W k + l ,

where U3

I.L

wk+ 1 = H I D r

8

U

Wk+l"l' = H i D r

Take d < 5 (where 5 is given by Lemma 1.2.1 for the fixed )~, e), so that the inequality Iz - r < 5 holds. Then it follows from Lemma 1.2.1 that

1.2 S h a d o w i n g Near a H y p e r b o l i c Set for a D i f f e o m o r p h i s m uu

lW~;ll _< ~1,~1, Iw~:~_l[ < ely;l, Iw~.ll < ~1,~1, IW~+ll _>

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 -

<

ICk(y)

-

<

e.

Then we have ICk(x) - Ck(y)] < Ldo. The sequence ~' = {r is a dpseudotrajectory with any d > 0. We have ~' C W. This d-pseudotrajectory is (Ld0)-shadowed by both points x and y. The uniqueness of a fixed point of G in B(Ldo) implies that x = y. This proves our claim. Now we reduce Theorem 1.2.2 to Theorem 1.2.3. Obviously, its first statement follows from the first statement of Theorem 1.2.3. Formally, the second statement (uniqueness) is not a corollary of Theorem 1.2.2, but it is easily proved using L e m m a 1.1.2. Indeed, by the remark above our system r has the SUP on W0 with a constant e. It follows from part (a) of L e m m a 1.1.2 that r is expansive on W0 with expansivity constant A ~ (0, e). Obviously, we can find a neighborhood W1 of A and a number 8 E (0, A/2) such that N26(W1) C W0. By part (b) of L e m m a 1.1.2, r has the SUP on W1 with constant 8. Let us describe an example of application of Theorem 1.2.2. Assume that p is a hyperbolic fixed point of a diffeomorphism r of ]R~ (i.e., the invariant set A = {p} satisfies Definition 1.9). It is easy to see that this condition is equivalent to the following one: all eigenvalues ~i of the matrix D e ( p ) satisfy the inequalities l~il ~ 1. We also assume that there exists a point q ~ p such that Ck(q) ~ P as Ikl ~ cr

(1.28)

A point q with property (1.28) is called a homoclinic point of the fixed point p. It follows from Theorem 1.2.1 that for fixed A > 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 --<1 1--~ holds. Set R = R(X, e) (see Lemma 1.2.3). For the fixed ~, ~ we find, by Lemma 1.2.1, a neighborhood W = W(~, ~), a norm I.l*, and families of subspaces S', U' such that inequalities (1.10), (1.11) hold. By the same reasons as in Theorem 1.2.3, we may assume that the usual Euclidean norm I.I is Lyapunov in W. We again write S, U instead of S', U'. Fix a number dl > 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 -

<

-

Now we take d3 E (0, d2) such that 8Rd3 < 1. If Pl,Wo (r C) < d3,

then the formula for he(w) similar to (1.33) and the inequality IIDr (which holds in W0) imply the inequality §

for p E W and Iwh Iw'l ~ d=,

< d3

24

1. Shadowing Near an Invariant Set

Denote Y = f ( X ) and let B be the space of continuous vector fields on Y. It is well known (see Chap. 0 in the book [Nil]) that B with the norm

Ilvll = sup Iv(p)l pEY

is a Banach space. Take f such that (1.32) holds (do will be chosen later). We will find the continuous mapping g : X ~ IRa in the form g(x) = f(x) + v(f(x)), where v E B. The equality g o T = r o g reduces to the following equation for the vector field v:

f(~'(x)) + v(f(r(x))) = r

+ v(f(x))).

(1.35)

We rewrite Eq.(1.35) at x, z = r - l ( x ) instead of r ( x ) , x,

f(x) + v(f(x)) = r

+ v(f(z))),

or, equvalently,

v(f(x)) = r

+ v(f(z))) - f(x).

(1.36)

It follows from condition (1.32) that for any x E X the inequality If(r(x)) - r

I< d

holds. This is equivalent to the condition If(x) - r

< d

or

If(x) - r

< d, x E X.

(1.37)

Now we define the following operator F: for v E B we set F(v) = ~, where

f~(f(x)) = r

+ v(f(z))) - f(x)

(recall that z = r - l ( x ) ) . It follows from (1.36) that, for the mapping g(x) = f ( x ) + v ( f ( x ) ) , statement (1) of our theorem holds if and only if v is a fixed point of the operator F. Let us write r + v(f(z))) - f(x) = = [r

- f(x)] + OO(f(z))v(f(z)) + h~(z)(v(f(z))),

(1.38)

where h~(z)(0 ) = O, Dh~(z)(O ) = O. Since the neighborhood W is bounded, there exists a constant C1 such that

IIDr The inequality

~ C1 on W.

1.2 Shadowing

N e a r a H y p e r b o l i c Set for a

Diffeomorphism

25

Pl,Wo(r162 < ~1 implies that IIDr

~

c~ = c1 -~- r on W.

Let ~(~) again denote the ban {v ~ B: II~ll -< r} for ~ > 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

<_ Ahwl, IH'~Dr

< elw[

hold. For w 9 U ( f ( z ) ) we obtain the inequalities

III~Dr

<_ elw I, III~Dr

>_ l l w [.

The same reasons as in the proof of Theorem 1.2.3 show that we can represent DF(O) with respect to the decomposition 13 = B ~ G B ~ in the form

A~ A~ with

iiA"I[, II(A"")-'II < ~, IIA'"Ii, IIA"'II-< e. Now L e m m a 1.2.3 implies that the operator I - DF(O) is invertible, and

II(I- DF(0))-'I[ < n. We again define

C(v) = (I - D F ( O ) ) - ' ( F ( v ) - DF(O)v) and consider the equation G(v) = v, equivalent to F(v) = v. In this case, for H(v) = F(v) - DF(O)v we have the representation

26

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

H(v)(f(x)) = [r

- f(x)] + h~(z)(v(f(z)) ).

By (1.37),

IIH(0)I[ _< d. By (1.34), for v, v' e L(d2) the inequality

IIg(v)-

1 Iv - v'll g(v')ll _< ~-~l

holds. Now we set L = 2R, do - rain(d3,51) 2R ' and repeat word-to-word the end of the proof of Theorem 1.2.3 (note that If(z) - g ( x ) [ = ]v(f(x))l for x E X). [] To obtain the Shadowing L e m m a (Theorem 1.2.2) from Theorem 1.2.5, one can proceed as follows. Take X = 7/ (with discrete topology) and set r(k) = k + 1. Let ( = {Xk} be a sequence in a neighborhood of a hyperbolic set A for a diffeomorphism r such that

Ir

- xk+, I < d.

Define f : X --~ IR" by the equalities f ( k ) -- Xk. Then

P(f o ~, r

o f ) = sup I~k+a - r

< d.

kEZ

Apply Theorem 1.2.5 to find a mapping g : X --~ IR" such that P ( g o T,r og) = 0 and If(k) - g(k)[ < t d . For the points Yk = g(k), the equality above means that Yk+l = r i.e., {Yk} is a trajectory of r The inequality above shows that Ixk - Ykl <- Ld. Following [Katokl], we describe below an important example of application of Theorem 1.2.5 (the reader can find other interesting applications in [Katokl]). Let us recall the definition of a nonwandering point for a dynamical system r on a topological space Y. D e f i n i t i o n 1.10 A point p is called "nonwandering" for (Y, r

if for any neighborhood V of p and for any No > 0 there is a natural number N such that N > 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 _<

N - 1,

have the form Pl = r

P2 = r

"'',

PO ----"r

i.e., P0 is a periodic point of r Since r

a

ly - xl < 2 ' Ip0 - vl <

we see that p0 E U. Note that the mapping g is not necessarily one-to-one, so possibly the minimal period of p0 is less than N. [] 1.2.4 T h e M e t h o d

of Bowen

We describe here the proof of the Shadowing L e m m a given by Bowen [Bo2]. As was mentioned, this approach uses not only the definition of a hyperbolic set, but applies more detailed information about the behavior of a diffeomorphism near its hyperbolic set. Later the ideas of this method were generalized [Ru], we describe this generalization at the end of this subsection. Bowen's proof of the Shadowing Lemma.

28

1. S h a d o w i n g Near an I n v a r i a n t Set

We will prove only the existence part of Theorem 1.2.3, the proof of the uniqueness part is the same. Let A be a hyperbolic set for a diffeomorphism r Fix a neighborhood U and numbers K:, A, A1, u given by Theorem 1.2.1. Find a natural number m such that 2u'~K: < 1.

(1.39)

We can take the neighborhood U bounded, hence r is Lipschitz on U. It follows from L e m m a 1.1.3 that there is a number L1 with the following property: if for a sequence ~ -- {xk} C U we have - Xk+,l <_ d,

Ir then

Ixk+j - eJ(xk)l _< L i d for k E 7/, 0 < j < m.

(1.40)

Since A is a compact invariant set, there exists e > 0 such that if y C A and Ix - y] < 2e, then era(x) e U for 0 < k < m. Let Uo be the e-neighborhood of A, it follows from the choice of e that U0 C U. Find do > 0 such that the inequality 2Lid0 < A1

(1.41)

is fulfilled, and take d < do. Now take a sequence ~ = {xk} C U0 such that

Ir

- xk+,l < d,

fix a natural number r, and consider the set { x o , . . . , x,m}. Let us construct points yo,.. 9 y~ as follows. Set y0 = x0. We deduce from (1.40) and (1.41) that Iem(yo) - xm] = Iem(Xo) - Xml <__L i d < 2Lldo < A1, hence by Theorem 1.2.1 there is a point

yl e w:,(r

cl

such that ]Yl --

~)m(yo)l ~-- ICLld <

2](~Lld,

lYl - xm] <_ 1CLld < 21CLad. We use induction to construct points Yk, 1 ~ k < r, such that

yk

n ws lYk -- era(Yk-1)] ~

2lCi~d,

]Yk -- xkm I <-- 2K.Lld.

(1.42) (1.43) (1.44)

1.2 S h a d o w i n g N e a r a H y p e r b o l i c Set for a D i f f e o m o r p h i s m

29

Assume t h a t we have constructed points y a , . . . , yk such t h a t (1.42)-(1.44) hold. Since yk E W~(xk~), the inequality -- r m ( X k-~)l < 2u"~Lad < Lad

Ir holds. Since

[r

-- X(k+l)m[ ~__Lid,

we see t h a t ICm(yk) - x(k+l)ml < 2Lid, thus we can find points

Yk+l E Wz~(~)m(yk)) ~ W~(X(kT1)m ) such t h a t analogs of (1.43), (1.44) are true. Take x = r and consider s,i such t h a t 0 < s < r, 0 < i < m. Since r

= r ....

- r

< Ir

+,(y~),

we can write

Ir +1r

- r

+...

= ~

]r

- r

+ Ir

- r

=

__ ~)(s-tT1)rnTi(Yt_l) I =

t=s+l

~-- ~

]r

-- r162

~

t=s+l

_<21CLld ~

u(t-')'~-i < 2]CLldu-i ~-~ uira _<21CLldi _lu "~

t=s+l

j=l

Here we take (1.42), (1.43) into account. Hence, Ir - r where L2

< L2d, 1

2~L1-1 _ u~"

It follows from (1.44) and (1.40) that

Ir

- x ~ + , l < lr

- r i ( x ~ ) f + Ir

- x~+,t <

<_2~Lld + Lld, O < i < m (to e s t i m a t e the first t e r m we take into account t h a t Ys E W~(xs~)). Finally we obtain, for k E { 0 , . . . ,rrn}, the estimates

30

1. S h a d o w i n g Near an Invariant Set

Ir

- xkl < Ld,

where L = (2K; + 1)L1 + L2. Consider an arbitrary natural n u m b e r r and set !

X k ~

Xk_rm

~

--rm < k < rm.

It follows from our reasoning above t h a t there exists a point x' such t h a t ICk(x ') - x'k] < Ld, 0 < k < 2 r m . Hence, for the point x T = CTm(x') the inequalities ICk(S) - xkl < Ld, - r m

< k < rm,

hold. Let x be a limit point of the sequence x * as r --~ cr (to simplify notation, we assume that x ~ ~ x). Then, passing to the limit for r ---* cx~, we see that ICk(x) - Xkl < Ld, k E 7/. This completes the proof. Smale spaces. Ruelle developed in [Ru] an abstract theory of the so-called Smale spaces. Let us describe this approach (with slight modifications). Let (X, r) be a c o m p a c t metric space. It is assumed t h a t (SS1) there exists a positive n u m b e r 7/and a continuous m a p p i n g

[,]: {(~,y) 9 x x x: ~(~, y) < , } -~ x such that [x, x] = x, and

[[x, y], z] = Ix, zl, [x, [y, ~11 = [x, z], when the two sides of these relations are defined. Define, for ~ > 0 and x E X , the sets V~8(x) = {u : u = [u,x] and r ( x , u ) < 6}, V s ' ( x ) = { v : v = Ix,v] and r ( x , v ) < ~}.

Now we state the second assumption. (SS2) There exists a h o m e o m o r p h i s m r of X and n u m b e r s g > 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 <

i <<_m. The existence of a number d with the formulated property is established similarly to the proof of Lamina 1.1.3 but since r is not necessarily Lipschitz, one has to refer to the uniform continuity of r (we leave the details to the reader). Let us show that this d has the property described in Definition 1.3. Like in Bowen's proof above, we again fix a natural number r and construct points Y0,.. 9 Y~ such that Y0 = x0,

32

1. Shadowing Near an Invariant Set

yk e Vs"(zk,~) n

yl~ ~

(r m (yk-,)),

and

r(yk, r

< ~0, r(yk, xkm) < ~0

(1.50)

for 1 _< k _< r. Assume that yl,...,yk-1 are constructed. Since yk-1 E VaS(X(k_l)m), we deduce from the second inequality in (1.50) for k - 1 that

r(r

r

< ~m~0.

By the choice of d,

r(Xkm,r

~__r(Xkrn,r

71- r((~rn(x(k-1)m),r

< ~1.

Hence, we can set

Yk = [~gm(yk-i), Xkm]. It follows from the choice of 61 that inequalities (1.50) are satisfied. To show that Yk E V6s(xkm), we apply the definition of the set VZ(zkm ) and (SS1). We can write

[yk,Xkm] : [[(~rn(yk-1),Xkm],Xkm] --~ [r

: yk.

Now the desired inclusion follows from the second inequality in (1.50). Similarly one shows that yk C V~(r We again take x = r The estimate of

r(r

r

0 < s < r, 0 < i < m,

is obtained similarly to the one in Bowen's proof, and we give here the final expression, r(r

r

< 1- u

(we take into account that 1 - u m > 1 - u). Finally we see (taking (1.48) into account) that

r(r

~ + , ) < r(r +~(r

r <

+ r(r

r

~0 + ~0 + ~<~.

It remains to repeat the passing to the limit for r -+ oo to complete the proof. []

Remark. Since [, ] and r are continuous (and not Lipschitz), we can prove only the P O T P (instead of the LpSP) for the Smale space. Let us show that the model of Ruelle has one more important property characteristic for neighborhoods of hyperbolic sets.

1.2 Shadowing Near a Hyperbolic Set for a Diffeomorphism T h e o r e m 1.2.8. If (X, r

33

is a Smale space, then r is expansive.

Proof. Find A E (0, ~) such that

r(~, [x, y]), r(y, [x,~]) < ~ for ~(x,y) < ~. We claim that A is an expansivity constant for r Take x, y E X with the property r(r

r

_~ A, n e 7].

It follows from (1.45) that for n > 0 we have

z := r

[r

r

By the choice of AI,

r(r

z) < ~.

r

r

Since by (SS1) we have

[z, r

= [[r

it follows that z e V~(r

= [r

r

= z,

Hence,

r(~, [y, x]) _ ~ r ( r

z) _ ~n~.

The last inequality holds for any n > 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 <- II~;'IIB (it is also said in this case that the norm II.IIB is compatible with the lattice structure), and the inequalities Ilslloo <

ilsllB

hold. L e m m a 1.3.1. Assume that for numbers No, g, A > 0

(1.52)

36

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 Set

(a) there exists a linear operator 6 : B ---*/3 such that

(al) I]6]] < go (here ]].]l is the operator norm generated by the norm ]].]1/3 of~3); (aZ) /f 2 = {zk} 9 13, then the sequence ~ = {uk} defined by e = 62 satisfies the relations Uk+l = Akuk + Zk+l; (1.53) (b) the inequalities ]WkTI(U) - -

WkTl(Vt)] ~ ElY -- Vtl for IU], IVt] ~ A

(1.54)

and ~No< 1 hold. Set

(1.55)

No A L - - do=--. 1 - aNo' L

/I II{r

d

do,

(1.56)

then there exist vk 9 Hk such that Ck(vk) = vk+a and II{vk}llB <_ Ld.

(1.57)

Proof. We consider the case k E 77+, the case k C 77 is considered similarly. Equalities Ck(vk) = vk+l, k > 0, (1.58) are equivalent to vk+l = A k v k + Wk+l(Vk), k > 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 <- Ld}. Take O,O'C D. Since Ld <_ Ldo = A, we have ]1~]1/3,1]~'11/3 <_ A. Condition (1.52) implies that II+ll~, I1+'1[~ < A. Since the norm of/3 is monotonous, we obtain from (1.54) the inequalities

1.3 Shadowing for Mappings of Banach Spaces

37

now it follows from condition (al) that

IIG~(o)

g~(o')llB < ,~Noll~

-

-

r

(1.61)

By (1.56),

II{~k+l(O))llz3

= II{r

<_ d,

hence for ~ E D we have IIg~(0)[IB < II~(0)IIB + II~z~(~) - ~ ( 0 ) I I B

< Nod + NoxLd = Ld. (1.62)

It follows from (1.55), (1.61), and (1.62) that ~t~ maps D into itself and contracts in D, hence there is a solution ~ of (1.60) that satisfies (1.57). This completes the proof. [] Now we prove the main theorem of this section. This theorem is widely applied below to establish various shadowing properties. T h e o r e m 1.3.1. Assume that (a) there exist numbers A E (0, 1), g > 1, and projectors Pk, Qk : Hk --* Hk (we denote below Sk = PkHk, Uk = QkHk) such that

(el) IIPklh IIQk[I < N, Pk + Qk = I;

(ae) I[Ak Isk II --- ~, AkSk C Sk+l;

(1.63)

(b) if Uk+x ~ {0}, then there exist linear mappings Bk : Uk+x --+ Hk such that

BkUk+x C Uk, IIBkll --< ~, AkBk Ivk+l= 1;

(1.64)

(c) there exist numbers tc, A > 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_
lu~] < N ~ k=0

I< N 1-

11~[I.

Similarly, we apply (1.64) to show that the series defining u,~ 3 converges, and the inequality

lu31_
A

- N1.

(1.66)

Let fi = G2. It follows from the equalities n-1

A,~u~ = AnP,~zn, A,~u~ = ~

A . . . . AkPkzk,

k=O

A,~u 3 = - ( I - P,~+l)z,~+l -

~

B,~+I...BkQk+lZk+l

k=n+l

that Anun

=

Z/,n+l --

Zn+l,

hence fi satisfies relations (1.53). Now our theorem follows from L e m m a 1.3.1. []

Remark. to take

The case k E 97 is considered similarly, in the definition of G one has n--1

u,~2 =

~_, A n - 1 . . . A k P k z k . k~--oo

This does not change the estimates. It follows from the remark after Theorem 1.2.3 that a neighborhood of a hyperbolic set has the following property. For any fixed L' > L there exists do(L') such that for any d-pseudotrajectory {xk}, k E 7/, with d < do(L') there is a unique point x with the property

1.3 Shadowing for Mappings of Banach Spaces

39

ICk(x) - xkl <_ L'd. Let us show that under the conditions of Theorem 1.3.1 the situation is qualitatively different, and the shadowing trajectory is not necessarily unique. E x a m p l e 1.12 Let Hk = l R , k E 7I. Set Pk = 0 (so that Sk = {0},Uk = J R ) for k < 0, and Pk = I (so that Sk = IR, Uk = {0}) for k > 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 <_ - 2 . Take d > 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~;
1 ~-2~>7>1,

A 2~ -+--<1 7 7

(1.69)

are fulfilled. Then the relations

~)k(Vk)

=

Vk+l, Ck(Uk)

----

Uk+l, Ivkl, lukl ~ A, k e ~,

(1,70)

imply that vk = uk, k E 7I. Proof. Assume that there exist two sequences {vk}, {uk} with properties (1.70). Set zk = vk - uk. It follows from (1.68) that zk+l = Akzk + W~+l,

(1.71)

IW~+ll < NolZkl.

(1.72)

where Represent zk = z~, + zL where z~ = Pkz~ E Sk, z~ = Q~zk E Uk.

Since Izkl < 2A, we see that

141, Iz~l _< 2 N A . By (a),(b'), Akz~ E Sk+l, Akz~ E U~+I, and it follows from the equalities

(1.73)

1.3 Shadowing for Mappings of Banach Spaces Z~+ 1 =

41

Pk+lAk(z~ + z'~) + Pk+lWlk+l

and from (1.72) that

Izs

< tPk+lA~z~l + IPk+lwLll < Alztl + ~Izkl.

(1.74)

Similarly, we obtain the estimate 1

Iz~+,l _> ;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~ --<1

IZ~+ll-

~lz~,l

- ~

(see (1.69)). This proves the second part of (1.77). Repeating the same reasons, we see that the inequalities

Iz~+k+ll > 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 _<... _< AlklZ~_kl _< A~NA

(1.79)

for any k _> 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[ [ <_ N (1.80) for x E W. Obviously, we can find e > 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)[] _<

_< II(Pk+, - I)DkPkll + II(Qk+l -- I)DkQkll.

(1.83)

Take v ~ Hk. Inequalities (1.80) imply that IPkvl G g l v ]. Since Pkv E Sk, it follows from Lemma 1.2.1 (or from Lemma 1.4.1 for the WS case) that the estimate I(Pk+, - I)DkPkvl = IQk+lDkPkv] < Nelvl holds. Hence, the first term in (1.83) does not exceed Ne. Similarly, the second term is also estimated by Ne. By (1.81), this leads us to the inequality

IIAk - Dk[I _< 2Ne <: ~o. Now we represent Ck(v) = Akv + wk+l(v),

where wk+l(v) = (Dk - Ak)v + Xk+l(V) 9

Note that IWk+l(O)l

=

We deduce from (1.82) and (1.83) that

Ir

< d.

(1.84)

1.3 Shadowing for Mappings of Banach Spaces

Iwk+l(V)

-

Wk+l(V')l

2 0lv

-

v'l for Ivl, Iv'l

do.

45

(1.85)

This completes the description of the common part of the proofs for the SL, and WS cases. Now let us formulate and prove the analog of Theorem 1.2.3 for the LpSP+. (Note that one can also apply Bowen's method described in Subsect. 1.2.4 to prove the statement below.) s

T h e o r e m 1.3.3. ff A is a hyperbolic set for a diffeomorphism r exists a neighborhood W of A such that r has the LpSP+ on W .

then there

Proof. It follows from Lemma 1.2.1 that AkSk = Pk+xDkSk = Sk+l, AkUk = Qk+IDkU~ = Uk+l, and that for A~ = Aklsk, Bk = (A~lvk) -1 conditions (1.63),(1.64) are fulfilled. Hence, we deduce from (1.85) and from the equality 1 2~0N1 = 2 that Theorem 1.3.1 is applicable to ~ = {xk : k > 0} if

- x +11 < d ___ do. Thus, for the point x = x0 + Vo we have

Ir

- xk[ <_ Ld, k >_ 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[ <_ g)~ n-k+1

(1.87) (1.88)

hold; (5) there exists A > 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 <_ do, where L1 = 1 + s + ... + s and consider a sequence yk E Tk such that (1.89) holds with d _< dl. Denote Set era(v) = e m ~ , ~ ( ~

+ v) - r

m ~ 7/.

By Lemma 1.1.3, the inequalities

ICm(O)l = I~m~,~(~.~)- ~m+~[ _< dnl <_ do hold. We can represent r

= A m v + w,~+~(v),

where Wm+I(V) : kl~mu,u(~m) -- ~m+l "-[-Wlm+l(Y) 9

Since IWlm+l(V) -- Wlm+I(V)I : I W m + I ( V ) - - Wm+l('O)[,

inequalities (1.54) hold for Ivl, Iv'l _< Ao. Applying Theorem 1.3.1, we see that there exists a sequence vm C Gm with IVml <---LLId such that for zm = (,~ + vm we have ~'m~,~(z,d = zm+~, m e 7/.

We obtain the estimates

Iz,~ - ~.~l <- LLld, m C 7/.

(1.92)

Represent arbitrary k E 7 / i n the form k = kou + kl, where 0 < kl < u - 1, and set xk = •k-1 o . . .

o ~k0~(Zko).

It follows from Lemma 1.3.1 that Ck(xk) = xk+l and

I x k - Ykl < L2d, where L2 = (/:~-IL + 1)L1. Thus, if we take d = c/L2, then the sequence {zk} satisfies the inequalities in (1.90).

48

I. Shadowing Near an Invariant Set

To prove the uniqueness statement, take x0 such that for x = N~0 inequalities (1.69) hold. Find d' > 0 such that for Ivl, Iv'l < d' the inequality in (1.54) is fulfilled. It follows that if d < d', then the sequence zm that satisfies the inequalities

IZm--~ml < ~0, meTZ, with ~o = L2d is unique. Obviously, this implies the uniqueness of a sequence xk for which (1.90) holds. An "abstract" shadowing result based on another definition of a hyperbolic set was proved by Steinlein and Walther [Stel]. This approach can be applied, for example, to study homoclinic trajectories for differential delay equations [Ste2]. We show that this result can be obtained as a corollary of a theorem similar to Theorem 1.3.1. Let us begin with the definition of a hyperbolic set in the sense of SteinleinWalther (our variant is a slight modification of the definition given in [Stel]). Let H be a Banach space and let r : V -* H be a mapping of class C a on an open subset V of H. It is assumed that D e is uniformly continuous on V and there exists a constant K0 _> 1 such that IIDr162 _< K0 on v . D e f i n i t i o n 1.14 We say that a set T C V is a "Steinlein-Walther hyperbolic set" if (1) T is positively invariant for r (/.e., r C T) and there exists r > 0 such that Nr(T) C V; (2) there exist complementary projectors P(x), Q(x) on T (i.e., P + Q = I) and numbers )~o E (0, 1), K1, C > 0 such that P, Q are uniformly continuous on T, the inequalities IIP(x)ll, IIQ(x)ll < K1 hold, and the spaces

S(x) = P(x)H, U(x) = Q(x)H have the following properties: (2.1) Dr

C S(r

(2.2) S(r

+ Dr

= H;

(2.S) IDek(x),l < C~0klvl for v e S(x), k _> 0; IQ(r162

)'~ > -d-lvl/or v e U(z), k >_ o.

This definition of a hyperbolic set is more general than the usual ones (for example, than the one applied in [Chol], see above), since it does not require the "unstable" spaces U to be Dr A detailed treatment of various definitions of hyperbolic sets for noninvertible mappings of Banach spaces can be found in [Lani]. Below we prove the following statement.

1.3 Shadowing for Mappings of Banach Spaces T h e o r e m 1.3.4. the LpSP on T.

49

If T is a Steinlein-Walther hyperbolic set for r then r has

Remark. The main result in [Stel] states that under some additional smoothness assumption r has the P O T P and the SUP on T, here we prove only the existence statement. Now we formulate and prove a result close to Theorem 1.3.1. Consider again a sequence of Banach spaces Hk, k E 7 / and a sequence of mappings Ck : Hk ---+Hk+l of form (1.51), where Ak = Bk+Ck, and Bk, Ck are linear mappings. T h e o r e m 1.3.5. Assume that (a) there exist numbers .~ E (0, 1), g >_ 1, and projectors Pk, Qk : Hk ---+Hk (we denote below Sk = PkHk, Uk = QkHk) such that

(el) [ICklh IIPkll, IIQkll -< N, Pk + Qk = I;

(ae) (1.93)

BkSk C Sk+, and IIBk Is~ II ~ ix; B k g k = gk+l and ]](Bk) -1 IUk+l

II

~;

Ck [&= 0, CkUk C Sk+l;

(1.94) (1.95)

(c) there exist numbers ~, A > 0 such that inequality (1.54) and the inequality ~N2< 1

(1.96)

hold, where N2 =

N 1 + N2.~ (-1~--~-~)2

Then there exist constants do, L > 0 with the following property: if

ICk(o)l ~ d < do for k E 7/, then there is a sequence vk E Hk such that Ck(vk) = vk+l and Ivkl < Ld. Proof. The proof is similar to the one of Theorem 1.3.1. We take the space B = / : c o , denote the norm of 0 E 13 by II011, and define a linear operator G on B as follows: we fix 2 C 13 and set

where we first define ~j by

yn = - ~ k=n

B~ 1...BklQk+lzk+l,

50

1. Shadowing Near an Invariant Set

and then set n-1 U n1

= P,~(z,~ +

2 Cn-lyn-1), Un

Z

=

B~-a ... Bk(Pkzk + Ck-lYk-1).

k=-oo

Let us show that • maps 13 into B and estimate its norm. Take ,~ 6 /3. Since Qk+lZk+l G Uk+l, we have

k=n

hence

11~11-
(

1+ l-A]

II~'ll"

By construction, yk-1 E Uk-1. It follows from condition (b) that Ck-ayk-1 E Sk. Since Pkzk E Sk, we see that

k=-m

--

1

--

Finally, we arrive to the inequality

IIGII_<~_

)~+I-),+N~+),+I_a]_
(here we take into account that _A2 _ N ~ + NA2 < 0). Let ~ = ~ . It follows from the relations B,~y~ = -Q,+lz,~+l + y~+l, B,~u I = BnP,~zn + B,~PnG',~_,y,~-i = BnP,~z,~ + B,~Cn_ly,~_l, C,~u~ = 0, and

B.~4 = u,,+l~- B,.(P.~. + c._,y._,), c.,4 = o, that 2

(Bn -~ Cn)(Yn ~- u I or-122) -~- -Qn+lzn+l -~ Yn+l J- 1~n+l J- CnYn = 2

= --zn+l + Pn+lZn+l + Un+l + yn+a + P,~+IC,@. hence Ur~+l -= (Bn + Cn)un + Zn+l, n 6 7]. Now we set

1.3 Shadowing for Mappings of Banach Spaces L-

51

N2 A a-------~2' 1d o = -~.

It follows from L e m m a 1.3.1 that if [r _< d _< do, then there exists a sequence vk E Hk such that Ck(vk) vk+x and Ivkl < Ld. [] =

Now let us reduce T h e o r e m 1.3.4 to T h e o r e m 1.3.5. First we note that condition (2.2) in Definition 1.14 implies the relation

Q(r162

= U(r

(1.97)

for x E T. Indeed, we have

U(r

= Q(r

= Q(r162

+ Dr

=

= Q(r162 since Q(r162 = {0}. To simplify our notation, we assume that C = 1 in condition (2.3) (this can be achieved passing from r to Cm with a proper natural m and applying L e m m a 1.1.3). Set K = max(K0, K1). Take a number A E (A0, 1) and find e > 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 <_

_<~01vl + ~Klvl _< alvr. This shows that Bk satisfies condition (a2) of Theorem 1.3.5. Let i = n4(xk)U(xk). Since BkU(xk) = Qk+lL and Q'ki = U(C(xk)) (see (1.97)), (s2.2) implies that BkUk = Uk+l. Take v E Uk. By (2.3), we have '

[QkDr

>_

vl.

It follows from (1.98) and (s2.1) that

IBkv[ = IQk+lDr >_ 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

<_ eI~;~

We see that if d _< A, then

ICk(0){

d,

and Iwk+l(v) - wk+l(v')l < x for Ivl, Iv'l

A

(here we take into account (sl), the estimate above, and inequality (1.99)). Thus, it follows from Theorem 1.3.5 that if d _< do, then for x = x0 + v0 we have ICk(x) - xk] = Ivkl. This completes the proof of Theorem 1.3.4. 1.3.5 F i n i t e - D i m e n s i o n a l C a s e In [Pli2, Pli3], Pliss studied conditions under which a family of linear systems of differential equations has uniformly bounded solutions. We show in this subsection that these results "translated" into the shadowing language give necessary and sufficient conditions of Lipschitz shadowing for a sequence of mappings of IR'~. Let A k : I R ~--+IR ~, k E 7 / , be a family of invertible linear mappings, consider the corresponding equations uk+l = Akuk + zk+l, k C 77,

(1.100)

where the sequence {zk} is given and the sequence {uk} is unknown. Let B = s be the Banach space of sequences o = {vk E IR= : k e 77} with the usual norm

I111 = sup Ivkl. kEZ

We begin with some definitions. Let < a, b > be a segment of 77 defined as follows. If a, b are finite with a < b, then < a,b > = { k E 77 : a <

k
if a is finite, then < a, cx) > = { k E 7 ] : a <

k < cx~}

and < - c o , a > = {k e 7] : - e c < k _< a};

54

1. Shadowing Near an Invariant Set

finally, < - 0 % oe > = 77. D e f i n i t i o n 1.15 The family {Ak} is "hyperbolic on a segment" < a, b > with constants N > 0, A 9 (0,1) if for any k 9 a,b > there exist complementary projectors P ( k ) and Q(k) of IR n such that

(1) IlP(k)ll, ]lO(k)ll

<

N;

(2) the spaces S ( k ) = P(k)IR ~ and U(k) = Q(k)IR ~ have the properties AkT(k)=T(k+l)forT=S,U

ifk, k+l

9

(1.101)

(3) for n > k such that k, n G< a, b >, the following inequalities hold:

I[A. . . . AkP(k)ll _< NA ~-k+l

(1.102)

IIA~-l... A=_~iQ(n)[[ _ NA ~-k.

(1.103)

and

Consider two linear subspaces La and L2 of IR~. As usual, we define the angle Z(L1,L2) as follows: let /141 and M2 be the orthogonal c o m p l e m e n t s of La Yl L2 in L1 and L2, respectively, then Z(L1, L2) = min /(ca, v2). vl E Mi

Remark. It is assumed in Definition 1.15 that the norms of the projectors P ( k ) , Q ( k ) are uniformly bounded for k C< a , b >. Let us show t h a t it is possible to substitute this condition by the following one: the value b - a is large enough and the norms of Ak are uniformly bounded. Indeed, fix N > 0, A E (0, 1), and find a natural n u m b e r T such that .~-T - - -- NA T ~ 1. N Assume t h a t b - a > T and that IIAkll _< L. Take c E < a,b > such t h a t c + T _< b. Fix vectors v s e S ( c ) , v ~ 9 U(c) with Ivq = Iv~l = 1 and denote ~(k)

= A~_I...

Ac(v ~ -

vs);

according to this notation, A(c) = v ~ - v s. Obviously, we have l a ( k ) [ _< D - c l a ( c ) [ .

(1.104)

On the other hand, it follows from (1.102) and (1.103) that U-k [A(k)l >_ [ A k - 1 . . . A c v ~ [ - I A k - 1 . . . A ~ v s 1> - -N - N A k-c > 1

(1.105)

if k 9 a , b > , k = c + T. C o m p a r i n g (1.104) and (1.105), we see t h a t if c 9 a, b - T >, then we have

1.3 Shadowing for Mappings of Banach Spaces

55

IA(c)l = Iv~ - v" I > L -T for any v s E S ( c ) , v ~ E U(c) with 0/1 = s l ( L , N, A) > 0 such t h a t

Ivq

--

Iv=l =

1. It follows that there exists

/ ( S ( c ) , V ( c ) ) > Sl for c e < a , b -

T >.

Since IIAkll G n and relations (1.101) hold, the inequality above implies that we can find s0 = s 0 ( L , Sl) such t h a t / ( S ( c ) , U ( c ) ) > So for c e < b - T,b > . Obviously, it follows from the last two inequalities t h a t the norms of P(c), Q(c) are b o u n d e d for c E < a, b > by a constant depending only on L, N, ~. D e f i n i t i o n 1.16 The family {Ak} is "piecewise hyperbolic" on 77 with constants

T, N, )~, s if there exist segments I0 = < -r

>, I1 =< Q,t2 >, . . . , Im = < tm, CXD>

of 77 (with finite t l , . . . , tin) such that (1) t i + l - - t i > T, i = 1 , . . . , m - 1; (2) the family {Ak} is hyperbolic with constants N, ~ on every segment Ij (we denote below by PJ(k), QJ(k), SJ(k), UJ(k) the corresponding projectors and spaces); (3) dimUJ(tj+x) >dimUJ+l(tj+a),j = 0 , . . . , m - 1; (4) the spaces UJ(tj+l) and SJ+l(tj+x) are transverse, and

/(UJ(tj+l), SJ(tj+l)) >_ s,

j = 0,... ,m-

1.

In our notation, the main result of [Pli2] can be stated as follows. 1.3.6. There exist functions T ( N , ~, s) and M ( N , ;~, s) such that if a family {Ak} is piecewise hyperbolic on 77 with constants T ( N , )%s), N, )%s, then Eqs. (1.100) have a solution fL such that

Theorem

I111 <-- M(N,A,s)II21I.

(1.106)

To prove this theorem, we first establish some auxiliary statements. 1.3.2. There exists a constant L = L( N, ~ ) such that if a family {Ak} is hyperbolic on a segment < a,b > with constants N , ~ , then one can find vectors un, n E< a, b >, with the properties

Lemma

un+l = A,~u~ + Z~+l, n E < a, b - 1 >, and

(1.107)

56

1. S h a d o w i n g

Near

an Invariant

Set

[u,[ .

(1.108)

Proof. Let P(k), Q(k), k E < a, b >, be the projectors given by Definition 1.15. Set ~n ~

3 U n1 7i- U2n Jl- U n ,

where

n-1

1 = P(n)z~, u,~ = ~ A n - 1 . . . AkP(k)zk, u,~ k=a b-1

u~3 = - ~

A ; 1 . . . A-~IQ(k + 1)zk+l.

k=n

The same reasons as in the proof of Theorem 1.3.1 show that relations (1.107) are satisfied, and inequalities (1.108) hold with L=N

l+A 1-A' []

The following statement is geometrically obvious. L e m m a 1.3.3. There exists a function p(a) > 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-<

Consider two linear subspaces S and U of ]R'~ such that IR~ = S @U.

Fix nonnegative numbers R, l,g. We say that a set D C IR~ is an (R, l,g)-ball with respect to (S, U) if D = {x + F x : x

E U, Ixl ~ n} +p,

(1.109)

where F is a linear mapping such that F U = S, IIFII ~ l, and p E S, Ipl ~ g. If sl = S + x , Y1 = Y + x , w h e r e x c IR~ , w e s a y that a s e t D1 C IRn is an (R, l, g)-ball with respect to ($1, U1) if D = D 1 - x i s a n (R, l,g)-ball with respect to (S, U). Now we consider linear subspaces S, U, S t, U ~ of IR~ such that S @ U = S ' ~ U' = IR~ and denote by P, Q, P', Q' the projectors such that

1.3 Shadowing for Mappings of Banach Spaces

57

S = PIR '~, U = QIR", S' = P'IR", U' = Q'IR =.

L e m m a 1.3.4. Assume that for the spaces S, U, S', U' above (1) [IPll, IIQII, IlP'll, iIQ'II < N; (2) U and S' are transverse, and L(U, S')

> 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 <-/liXl 9 We see that

IIr'lJ < h.

(1.113)

Set LI = L' + p (p is the vector in the definition of D similar to (1.109)). Since ]Pl -< d and $1 = S' + z', [x'[ < 2d, it follows from (1.112) and from Lemma 1.3.3 that there is a vector q E L1 M $1 such that Iql < 2p(a)d.

(1.114)

[q - x'[ < 2(p(a) + 1)d = gd.

(1.115)

Now we have It follows from the inclusion q E $1 that q - x' C S', hence the set D ' = {x + F ' x : x E U',lxi <_Rd} + q -

x'

is an (Rd, l;,gd)-ball with respect to (S', U') (here we take estimates (1.113) and (1.115) into account). This implies that the set D1 = D' + x' is an (Rd, ll,gd)-ball with respect to (Sx, U1). We claim that D1 C D. Note that L' + q = L' + p = L1. By construction, D~ C {x + F ' x : x E U'} + p -

x' + x' = L' + q = L' + p C L + p.

(1.116)

It follows from (1.114) that

IQql <- 2 N p ( a ) d (recall that Q projects ]R'~ to U). For any v E D~ we have v - q = x + F'x,

where x E U', Ixl < Rd. Inequality (1.113) implies that [Q(v - q)l -< N R ( 1 + l~)d.

Finally, it follows from the relation 2p(a) = R(1 +/1) that IQvl <- IQq[ + IQ( v - q)l <- N[R(1 + / 1 ) -~- 2p(oL)]d

= 2N(1 + l l ) R d = 7Rd.

--~

(1.117)

1.3 Shadowing for Mappings of Banach Spaces

59

Now we deduce from (1.116) and (1.117) that Da C D. Our lemma is proved. [] Now let us prove Theorem 1.3.6. Fix numbers N, A, a and find the corresponding R,'7,1o, ll,g given by Lemma 1.3.4. Find a positive number T such that the following inequalities hold: N ~ T g < 1,

(1.118)

N2~2Tll < lo,

(1.119)

~-T > 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 <'TRd} VO(a)), and hence the set

Do1 = Do + u ~ is a (3,Rd, 0,0)-ball with respect to (S~176 Take a vector ua E n 1 and define uk, k E 77, by (1.100). Set vk = uk -- u ~ for k _< a (recall that u ~ k E Io, are given by Lemma 1.3.2). It follows from (1.100) and from an analog of (1.107) for u ~ that vk-1 = A ; : l v k for k _< a, hence vk = A-~I... A21_lva for k < a. Since v~ E U~

item (3) of Definition 1.15 implies that Ivkl < g)~a-klv~] < 7 g R d for k < a.

It follows from the inequality lu~ < d that

60

1. Shadowing Near an Invariant Set [ukl < (1 + "7gR)d for k < a.

(1.121)

0 T h e n we can Introduce new coordinates in ]R n moving the origin to u~. write ]R n = S~ G b/~ Since [uO[, [ul~[ < d, items (2) and (4) of Definition 1.16 show that the conditions of L e m m a 1.3.4 are satisfied for (S, U) = (S~176 and ($1, U1) = (S'(a),/d~(a)), hence there exists an (Rd, l~, gd)-ball 01 with respect to (S ~(a), /dl(a)) such that 01 C 001. Denote w a1 = S l(a) N D 1 and define w~, k E I1, by

w~+1 = Akw~ + zk+l. Let v~ = w ~ - u~. Since 01 is an (Rd, ll, gd)-ball with respect to ($~ (a),/d~(a)), we have [v1] _< gd and v~1 E Sl(a). Taking into account that b - a > 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[ <-~ll[X'a -- Xa[, the relations above and (1.119) imply the inequality

lY; -- Ybl <-- N2/~2(b-a)lllX; - Xb] <-- loIXlb -- Xb]"

(1.126)

If for v~ we have ]x~ I = Rd, then it follows from (1.120) and (1.125) that

)~-b ]xb] k - - ~ - R d > 7Rd.

(1.127)

1.3 Shadowing for Mappings of Banach Spaces

61

Combining (1.123), (1.126), and (1.127), we deduce that the set

D ' = Ab_I...Aa (DI - ul~) -[-u~ contains a subset Do~ being an (',/Rd,/o, d)-ball with respect to (Sl(b),lgl(b)). Take ub E Do2, define uk by (1.100), set vk = u k - u I for k E I1, and represent

Vk = ~k + r/k, where ~k C Ul(k), Yk E Sl(k). It follows from (1.101) that -1 ~k = Ak"1... Ab-l~b for k < b, k E I1,

hence we have [~k[ _< NAk-b[~b[ --< "TNRd, k E I1.

(1.128)

The inclusion Do2 C D t implies that

Va hence the values

I,kl

Aa 1 .. Abl_lVb E D 1

1

satisfy the same estimates (1.124) as ]Yk[, so that [r/k[ < N(IIR + g)d, k E Ix.

Combine the last inequality with (1.128) to show that

Ivkl

_< N(I1R + ,,/R + g)d, k c 11.

It follows that I~kl < N(IIR + R + g + 1)d, k c Ix.

(1.129)

Since u~ C D1, inequalities (1.121) imply that

[uk] <_ N'd for k _< b, where

N' = N(I~R + TR + g + 1). By Lemma 1.3.4, the set D~ contains a subset D 2 being an (Rd, 11, gd)-ball with respect to (S2(b),LtZ(b)). Repeating the described procedure, we finally obtain a set D "~ being an (Rd, ll,gd)-ball with respect to (Sm(t,~),LC~(tm)). Take Utm = S'~(t,~)fqD m, define uk, k E 7], by (1.100), and set vk = u k - u ~ . Repeating the previous arguments, we show that

Iv l

<

N(IIR + g)d

and

lukl

[N(llR + g) + 1]d

for k > tin, and that

lukl < N'd for k < tin.

62

1. Shadowing Near an Invariant Set

Hence, the sequence {uk} has the desired p r o p e r t y with M = L N ' . [] Let us prove an i m p o r t a n t corollary of T h e o r e m 1.3.6 giving sufficient conditions of Lipschitz shadowing for a sequence of m a p p i n g s of IR n. Here we formulate only a "global" result assuming that the nonlinearities have small Lipschitz constants everywhere. T h e o r e m 1.3.7. A s s u m e that the family {Ak} is piecewise hyperbolic on 77 with constants T, N, A, a, where T = T ( N , A, a) is given by Theorem 1.3.6. Fix tr > 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.

<

This means that

One can consider also the following "two-sided" variant of the LmSP on a set Y for a dynamical system r on (X, r): given a sequence ( = {xk: k E 7/} C Y such that the analog of (1.133) holds for Ikl --* oc, to find a point x with the property We show in the next subsection that in a neighborghood of a hyperbolic set a diffeomorphism has this "two-sided" LmSP (see Theorem 1.4.3).

68

1. Shadowing Near an Invariant Set

1.4.2 s

It was shown in the previous subsection that in a neighborhood of its hyperbolic set a diffeomorphism r has the following property. If for a pseudotrajectory = {xk} of r the "errors"

dk

[Xk+1 -- ~)(Xk) I

~-

tend to 0, then the pseudotrajectory tends to some trajectory {r this subsection, we study the rate of convergence for the values Ir

~kl.

-

For 1 _< p < cx~, denote by s {vk E IR~ : k E 71} with the norm

the Banach space of sequences ~ =

E

Ilvll, =

of r In

/ lip

\k~z

Iv~l"

/

We consider also the Banach space Ep,+ of sequences ~ = {vk E IR~ : k E 7]+ } with the norm I1~11, =

Ivkl'/

9

Denote, as usual,

II01too = sup MI kE7'.']

and

G ( r ) = {~ ~ G : I1~11~ r}. Take a diffeomorphism r of ]R~, a sequence ~ = {xk C IR~}, and a point x E ]R~, and denote

gk(~) = ~k+~ - r

g(~) = {g~(~)},

hk(x,~) = ek(z) - xk, h(x,~) = {hk(x,~)}. We use this notation in both cases k E 7] and k E 7]+. The main result of this subsection is the following theorem (we consider the case k E 7/). T h e o r e m 1.4.2. Let A be a hyperbolic set for a diffeomorphism r o f l R n. There exists a neighborhood W of A and numbers L, do > 0 with the following property. If for ~ = {xk} C W and some 1 < p < oo the inequality

Ilgff)ll~ ~ d ~ do holds, then there is a unique point x such that

1.4 Limit Shadowing

IIh(~,~)ll~ _< Ld.

69 (1.138)

Remark. For the case k E 71+, the statement of the theorem and its proof are similar, the only exception is that the shadowing point x is not necessarily unique. Proof. We begin by repeating the scheme described in Subsect. 1.3.3. We take B = s obviously, the norm II.llv is monotone and satisfies condition (1.52). It follows from L e m m a 1.2.1 that AkSk

=

AkUk

Sk+l,

= Uk+l,

and that for A~ = A k l s ~ , B k = (Aklu,) ~ -1 conditions (1.63) and (1.64) are fulfilled. Now we fix 1 _< p < oe and define a linear operator G on Ep in the same way as in Theorem 1.3.1, G5 = ~1 + ~2 + ~3, where n--1

1

tt n :

Pnzn,

2

tt n :

oo

~_~ An-1 ...AkPkzk,

u~3 = -

k=-oo

~_, B . . . . BkQk+lzk+l. k=n

Let us show that ~ m a p s / : v into s (1.80),

and estimate its norm. Take 5 C s

By

II~111,, _< NIl~llp. Since Pkzk E Sk, we deduce from (1.63) that n--1

lull _< N ~

~-klzk I

k=-oo

Let us estimate

Now we apply the Minkowski inequality (see [Har], Theorem 165) in the following form: if b,~,am,~ > 0 and p > 1, then

/

\ 1/p ap

We see that N

A~-klzk

= N

ATM z~_~

_<

70

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

11p

~n-ml"

<-N ~-] Am

< NI_--~ll~llp.

m=l

Similar estimates show that

II~II~ ~

N 1_

~ II~II,.

It follows that IIGIIp ~ N1, where N1 = NI(N, 1) is defined in Theorem 1.3.1. Now the existence statement of our theorem follows from L e m m a 1.3.1 with N~ d~ L- 1-~NI' d0=-~-. To prove the uniqueness of a point x for which (1.138) holds, note that conditions (a) of Theorem 1.3.1 and (b') of Theorem 1.3.2 are satisfied. Fix numbers no, A given by Theorem 1.3.2. Take a point x that satisfies (1.138). Let ~ be the corresponding solution of Eq. (1.60). It follows from the equalities

that II~llp ~ Ld, hence II011~ < Ld. Take do such that the inequalities for wk+1 in (1.68) are satisfied for Ivl, Iv'l _< Ldo. Now it follows from Theorem 1.3.2 that a sequence ~ solving (1.60) (and hence a point x with property (1.138)) is unique. Our theorem is proved. []

Remark. In [Ea], Easton studied the following property of a dynamical system r on a metric space (X, r): given e > 0 there exists 5 > 0 such that if a sequence of points {xk : k C 77} satisfies the inequality

r(x~+l, r

< ~,

kE7]

then there exists a point x such that

r(xk, r

< ~.

kEZ

This was called the strong shadowing property. In the case p -- 1, our theorem shows that a diffeomorphism has the strong shadowing property in a neighborhood of a hyperbolic set. In the case p = oe, it is possible to consider two different shadowing properties. We obtain the first one taking the usual space s with the norm I1.11r instead of s in Theorem 1.4.2. It is easy to see that the corresponding statement coincides with the classical Shadowing Lemma. The second one appears if we consider the space of bounded sequences with zero limits endowed with the standard supnorm topology.

1.4 Limit Shadowing

71

T h e o r e m 1.4.3. There exists a neighbourhood W of A and numbers L, do > 0 such that if for a sequence ~ = {xk: k ~ 7]} C W we have IIg(~)lloo < d < do and limN__,~ gk(~) = O, then there exists a unique x such that

IIh(x,~)lloo < Ld and

lira Ihk(x,~)l = 0. N-~oo

The proof of this statement is similar to the proof of Theorem 1.4.2, and we omit it. As an example of application of Theorems 1.4.1 and 1.4.2, we consider approximations of solutions for ordinary differential equations by one-step discretizations. E x a m p l e 1.20 Consider a system of differential equations

= F ( t , x ) , x C IR ~.

(1.139)

Assume that F and ~ag- are continuous and bounded and that F is l-periodic in t. Denote by x(t, to, Xo) the solution of (1.139) with initial value x(to) = Xo, and let r be the Poincar6 mapping of (1.139), that is, r

= x(1,0, x0).

Our assumptions on F imply that r is a diffeomorphism of class C 1. Assume that r has a hyperbolic invariant set A. If F is smooth enough, any standard one-step discretization with time-step h (e.g., a Runge-Kutta method) gives a mapping

eh(to, xo) = (to + h, x(to + h, to, Xo) +

~),

where q > 1 is the order of the method, and I~1 < C0h q+l. Take a point x0 and an increasing sequence of natural numbers {nk : k > 0} such that limk_~ nk = cx~ . Set hk = 1/nk and define a sequence {xk : k > 0} by ~r xk) = (1,Xk+l). Then ]r - Xk+ll < C n ; q. If we assume that xk C W (W is given by Theorem 1.4.1), then it follows from Theorem 1.4.1 that there exists x such that l i m k - ~ Iek(x) - xkl = 0. To get the/:p-shadowing described in Theorem 1.4.2, it suffices to have nk > no + k '~/pq with some a > 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~ <_ Alvin, pl//~DC(x)~l~ < ~lvl~, v e S'(~); 1

p]H~Dr

> ~lvl~, plH~Df(x)vly < ~lvl~, v E U'(x),

(1.143) 1.144)

hold, where H~ is the projection onto S'(x) parallel to U'(x), and H2 = I - H~;

(s) 1 N--71vl~ <_ Ivl _< g'lvl~ for x C W, v E ~'~.

(1.145)

Proof. T h e proof is similar to the one of L e m m a 1.2.1. For # E (Ao, ~) we find a natural n u m b e r v such that C

<1.

Take a point p E A and a vector v E ]R ~, represent v = v s + v ~ E S(p) | U(p), and set Ivl~ = (Iv~l ~, + Iv'l~) 1/~, where

j=O

IvSlp -= ~ pJ#-JlDCJ(p)vSh ]v~ip = ~ p-~#-JiDr j=O

For v ~ we obtain the e s t i m a t e

pIDr

v~ I,(p)

= ~

PJ+ltz-Y IDr j ( r

)DC(p)v~I

=

j=0

=/z(~-~PJ#-J' < -DeJ(p)vSl j =q-pv+llt-v-l'DeV+l(p)vS') <-[~(~-~Iz-Jl < - DeJ(P)VSl j = +#-v-l l c)~+llvSl) (we applied (1.140) with k = ,

+ 1 to e s t i m a t e the second t e r m )

1.4 Limit Shadowing

75

< ,lr (here we take into account the choice of v). Similarly, for v ~ we obtain the estimate

plDr

>

~Iv%

The rest of the proof repeats the proof of Lemma 1.2.1. Now we prove Theorem 1.4.5. We apply the scheme described in Subsect. 1.3.3 (for the WS case). Fix r,p _> 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, <_

\ p \)l l v

(~__o/~_ 1

--

l l

\k=0

m=l

\k=m

I

76

1. Shadowing Near an Invariant Set

= N

Am

rk-mlzk_mlv

< N

rrt~--1

Now let us write oo

P


(p-lAkl)Qk+lpk+lzk+l

.

Since Qk+lZk+l E Uk+l and (1.144) holds, we obtain the inequality


(~-~)~k-n4-1pk+l [zk+i [I p \k=rt /

Applying again the Minkowski inequality, we see that

II~Zll,.p < N~II~II,,~. Finally we get the estimate ,,~2,,.,p<_N (1 + 2 1 _ ~ ) [ [ 2 , , , . , < N l [ [ 2 [ [ , . v ,

(1.146)

where N1 = NI(N, $) is defined in Theorem 1.3.1. Now the existence statement of our theorem follows from Lemma 1.3.1 with N1 dl - - , do = - - . 1 - ~N1 L

L-

Let us prove the uniqueness statement. Take a neighborhood W and the corresponding numbers L, do given by the first part of the proof. If [p, cxD)fl S(A) = 0, statement (c) of Theorem 1.4.4 and Lemma 1.4.1 imply that U(x) = ]R= for x E W. Denote ~b(x, v) = r

+ v) - r

- DC(x)v

for x E W, v E lR n. It follows from Lemma 1.4.1 that in our case we can find numbers d~, ~ > 0 and a neighborhood W0 of A such that the Ld~-neighborhood of W0 is a subset of W, and the following inequalities hold: 1

Ir

pllDc~(x)[[ > -~ for x E W;

(1.147)

v) - ~b(x, v')l < plv - v' I for x 9 W, Ivl, Iv'l < Zdo;

(1.148)

1 3,=~-~>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

<

+ v~) = r

+ PC(x~)vk + r

v~)

+ wk) ----r

+ PC(xk)w~ + r

W~),

and X~+l + Wk+l = r we see that

Zk+x = Dr

+r

vk) -- r

wk).

Taking (1.147)-(1.149) into account, we obtain the inequality 1

Since p[Zl[ _> 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 <_ d <_ do there exists a unique point x such that []h(x,~)[[e,v <_ Ld. (1.152) Pro@ First note that taking min Po-Jr j , o<j
c =

we obtain the estimate (1.153)

rk > cpok.

As in the proof of previous theorems, we are looking for a sequence ~ = {vk : k _> 0} C s that satisfies the equalities r

+ vk) = xk+l + vk+l, k _> O.

This means that we want to solve the equation F(0) = S0, where (F~)k = r

(1.154)

+ vk) -- xk+x, and S is the left shift, S(VO, V l , . . . ) = (Vl,?J2,...).

Let us show that the operator S - DF(O) has a bounded right inverse in First we establish the following elementary inequality: if {aj : j >_ 0} is a nonnegative sequence, and p, rI > 1, then

s

[

. ~ XlP

E aj <_ COl,p ) I E q'aPj] j>o

\j>o

,

(1.155)

/

where c(~,p)

=

\~-~-~ ~ 1 Indeed, by Hhlder's inequality we have p-1

aj ----~_, rlJ/Pajrl-j/p < j_>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 <_

oo

ID~_~l,kwj]<_ E MJ+l-klwil = M E MJlwk+i I <_ j=k j=k j=O

(

_

J

P

=

j=O (1.158)

Since

y - k < y < p~9 < -rj, C

we see that (1.158) does not exceed

Mc-1/'C(~,p) l I@1I,,p. Hence, the sums in (1.157) converge. Further, we have ?)k+l ~--~ -

~ j=k+l

Dj~-l,kTlWj ~--Vk+l,k ~

(DjTl,k+lDk+l,k)-lwj --

j=k+l oo

= -Dr

Y~ Dj-~l,kW j + wk = Dr j=k

+ wk,

i.e., O satisfies (1.156). We estimate the norm of 9 applying (1.158),

80

1. Shadowing Near an Invariant Set oo

rklVkl" <_ MVC(rl,p)" k=O

~

rk E ~ J - k ] w j l p = j=k

k=O

= M ' C ( ~ , p ) ' ~ I L ~P

I ~Jl~Jl'.

d=o \k=o d

]

We can find cl > 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

~

-~ ~_
k=ko P0

po ~ .

Po -- P

Hence, the equation S~ - D F ( O ) 9 = ~ has a solution 0 depending linearly on t~ and such that t1~11,,, < MC(77,p)(c2 + c3)'/'11~11,,,, i.e., the operator S - DF(O) has a bounded right inverse. To prove that S DF(O) is invertible, it is enough to show that it is one-to-one. Assume that there exists 0 such that SO - DF(O)O = 0 and v m r 0 for some m. Then for k > 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 <--

r~--I

_< N ~ Y~-k sup Ip'~wkl < N k=0

1~

I1,~11~,~.

k_>0

Thus, we see that A IIv211~,oo ~ N 1 _ ~ Ilwlkoo. The remaining terms are estimated similarly. In the analog of Theorem 1.4.6, we assume for simplicity that for the weight sequence S the inequalities

rk+l ~ prk, k > O, are satisfied, and that condition (1.151) holds. To estimate the norm of the sequence defined by (1.157), we write (taking into account that rk < p j - k r j for j > k and putting p = 1 in the previous estimates)

oo

rk[vk[ <_ M ~

rk~j_k suprj[wj[ < 9=

rj

j>o

P j=k

P

--

P

We leave the remaining details to the reader. Let us note that this last type of shadowing was first introduced by Fe~kan in [Fe] (to be exact, two-sided sequences {vk: k E 7]} and weights of the form rk = w -Ikl with w > 1 were considered; this approach does not seem to be of real interest for applications, since the "shadowing errors" [r - xkl may grow to infinity). Fe~kan studied mappings r : JR" ~ IRn of special form, r

: x + f(x),

(1.159)

under the condition

I f ( z ) - f(x')l <_ K l x - x'l with K < 1.

(1.160)

82

1. Shadowing Near an Invariant Set

T h e following s t a t e m e n t is the main result of [Fe] (note t h a t the Lipschitz dependence of the value sup w-lktlck(X ) -- Xkl kEZ on sup kEiE

W -Ik'l(~(xk)

--

Xk+ 11

was not explicitly stated in [Fe]; in addition, we write condition (1.161) in the form of an inequality instead of the original equality. It is easily seen t h a t the proof in [Fe] establishes the result formulated below). Theorem

1.4.7.

Assume that I+K w > 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 <_

i=k'=

<- ~ P k - i

i=k

suprilvil =

i=k

P

i>o

p- 1

I1~11,,~.

This shows that IIBII <

~.

(1.165)

-p-1 It follows from the equality (A~)k

= --

v~ + V'z_.,v~ = v k

link-t-1

i=k

that .AB = I. Similarly one shows that B.A = I, hence B = (.A) -1. Set n(0)

= ~(G~

+

~).

Equation (1.164) is equivalent to = ~(~).

(1.166)

We easily deduce from (1.160) and (1.165) that for ~, 9' C/Z~,oo we have

IIr~(o)

-

n(0')ll,,oo < -

By condition (1.162),

If

p lie p-1

-

~'11,.oo.

84

1. S h a d o w i n g Near an I n v a r i a n t Set

p-pK>

1, h e n c e K

This shows that T~ is a contraction on s K

P p-1

< 1.

with contraction constant

p p-l'

hence for any @ C s there is a unique solution ~ of Eq. (1.166) (and of Eq. (1.164)). Since G(0) = 0, we see that I[~][~,oo < g[l~[koo, and it follows from the inequalities p

1 II~lkoo ->

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(~)) <

(the sequence g(~) was defined in Subsect. 1.4.2). Remark. In [Hirs4], a sequence with the property above is called a A-pseudoorbit.

Let K be a compact invariant set for a diffeomorphism r Fix a point x E K and define the expansion constant of r at x by the formula

EC(r

= min [Dr Ivl=l

= [[Dr

-1.

Now we define the number

EC(r K ) = min EC(r x). xEK

The main shadowing result in [Hirs4] can be stated as follows. 1.4.9. Assume that K is a compact invariant set for a diffeomorphism r of JRn. Let #o = EC(r g ) . / f ~ = {xk} C g is an asymptotic pseudotrajectory of exponent A for r and Theorem

0 < A < #0,

(1.167)

then there exists a unique point x such that

T~(h(x, ~)) _< (the sequence h(x,~) was defined in Subsect. 1.4.2). Proof. Fix a number # E (A, #0). It is easy to see that there exists a number p0 > 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 A
(1.171)

Obviously, we have Igkff)l = Ir

Xk-]-lt ~

--

1)k

for large k. Now it follows from (1.169) and (1.171) that

k+l _< uk + dk+l = dk [ ( d ) k + d] < #d k

Ir

for large k. The last inequality implies the inclusions (1.172)

Bk+l(d) C N~,dk(r Since Xk E K, we deduce from (1.168) that if d k < po, then

N.dk(b(xk)) C r

(1.173)

It follows from (1.172) and (1.173) that there exists ra such that (1.170) holds for k > m. Inclusions (1.170) imply that the set

Qm = n r k>O

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 <- p-a. It follows that 74(h(x,~)) < p~-l. It remains to apply (1.175) to show t h a t

_<

88

1. Shadowing Near an Invariant Set

1.5 Shadowing for Flows In the case of dynamical systems with continuous time (flows), the shadowing problem becomes technically more complicated. A lot of research was devoted to shadowing near a hyperbolic set of a flow. In this section, we consider a hyperbolic set A containing no rest points for an autonomous system of differential equations. Theorem 1.3.1 is applied to show that in a neighborhood of A pseudotrajectories are shadowed by real trajectories, and the shadowing is Lipschitz with respect to the "errors". First shadowing results for flows were obtained by Bowen [Boll. He developed a method close to the one described in Subsect. 1.2.4 to approximate pseudotrajectories by real trajectories for the restriction of a flow to a basic set. After that many authors studied the shadowing problem for abstract flows and for differential equations (autonomous and nonautonomous). Let us mention Franke and Selgrade [Frl], Bronshtein [Bro], Meyer and Sell [Me], Kato [Kato3], and Nadzieja [Na]. Note that in [Bro] and [Na] analogs of Theorem 1.2.5 for flows were proved. The mentioned papers were mostly devoted to the case of a neighborhood of a hyperbolic set. Komuro studied in [Ko2] the shadowing problem for a geometric model (introduced by Guckenheimer [Gull) of the famous Lorenz system [Lo], k = a(y - x) = rx - y-

xz

(1.177)

= x y - ~3z.

This geometric model is characterized by its return map f. It is shown in [Ko2] that the geometric model does not have the property of weakly parametrized shadowing (see Definition 1.25 below) except a special case f(0) = 0 and f(1) = 1. These conditions correspond to the value r = 24.06... in the Lorenz system under (a, b) = (10, 8/3). In this section, we apply the techniques developed in Subsect. 1.3 to establish shadowing in a neighborhood of a hyperbolic set containing no rest points. Consider an autonomous system of differential equations ic = X ( x ) ,

x 9 I R a.

(1.178)

We assume that X 9 el(IRa). Let ~ ( t , z) be the trajectory of system (1.178) such that ~(0, x) = x. Since we restrict ourselves to the case of a neighborhood of a compact invariant set for system (1.178), without loss of generality we may assume that any trajectory is defined for t 9 IR. Thus, we study the flow ~ ( t , x) of our system. We denote by D ~ . ( t , x ) the corresponding variational flow, i.e.,

0.~(t, x) D~(t,

x) =

Ox

Let us define pseudotrajectories for flows.

1.5 Shadowing for Flows

89

D e f i n i t i o n 1.24 For d, T > 0 we say that a mapping

~ : IR ~ IR~ is a '(d, T)-pseudotrajectory" of system (1.178) if, for any T C IR, I~(t,~'(r))

-

~(t + T)I < d, Itl _< T.

(1.179)

Remark. This definition is close to standard definitions of pseudotrajectories for flows (see [Kato3]). Note that we do not assume k~ to be continuous. Another possible definition of a pseudotrajectory for an autonomous system of differential equations is discussed in [Ano3]. We describe here a concept generalizing the one of [Ano3] (and close to it). Let us say that a mapping ~* : IR ~ IR~ is a (d)-pseudosolution of system (1.178) if there exists an increasing sequence {tk E IR: k E 7]} such that ~* E cl(tk,tk+l), ~*(tk) =

lim ~*(t),

t--+tk+O

and the inequalities < d, t e (tk,tk+l), and I *(tk)

< d

(1.180)

hold for any k G 7] , where ~_*(tk) =

lim ~*(t)

t--*tk-O

(of course, it is assumed that all the mentioned limits exist). It is easy to see that if the right-hand side X ( x ) of system (1.178) is Lipschitz (let s be its Lipschitz constant), and the values tk+l - tk are separated from 0 (say, they satisfy the inequalities

0 < To <

t k + l -- t k

for k E 7]), then the trajectory of a (d)-pseudosolution ~* is an (Lld, T)pseudotrajectory for the system, where L1 = (1 + T)

exp(is

,

and m is the least natural number with the property

(m - 1)T0 > T. To prove this, we apply the following standard estimate (an easy consequence of Gronwall's lemma): if ((t) E Cl(a, b) satisfies the inequalites

90

1. S h a d o w i n g Near an Invariant

Set

l~(t) -- X ( ~ ( t ) ) I < dl, and I~(t') - xol _< d2 for some t' E (a, b),

then I~(t) - 3.(t - t',xo)l < (d2 + dlI t - t ' l ) e x p ( s

- t'I)

for t E (a, b). Let us prove (1.179) for 0 < t < T. Take T E [tk, tk+l), it follows from the first inequality in (1.180) and from the estimate above that I~(t, ~*(r)) -- ~*(t + r)I < d l t [ exp(s

_< d(1 + T) exp(s

while t + r E [tk, tk+l). Thus, if T + r E (tk, tk+l), then our statement holds. In the case T + r E [tk+l, tk+2), the left-hand side of the previous estimate implies the inequality [~(tk+l -- % #*(7")) -- ~P_*(tk+l)[ ~ d T e x p ( s and hence the inequality IZ(t, ~ * ( r ) ) - ~*(t +T)l -< <_ (d + d T e x p ( s

+ d(t + r -- t k + l ) ) e x p ( s

+ r -- tk+l)) <

(we applied the second inequality in (1.180)) _< d(1 + T) exp(s

+ d(1 + T) exp(2s

while t + r E [tk+l, tk+2). In this case, our statement is proved. Continuing this process, we obtain, for t + r E (tk+,,~-l,tk+m), the estimate Is

~ * ( r ) ) - #*(t + r ) l _<

_< d(1 + T ) ( e x p ( s

+...

+ exp(ms

this gives the desired estimate, since by the choice of m we have r + T < tk+l + ( m -- 1)T0 < tk+m.

The case t E [ - T , 0] is considered similarly. Thus, we have proved that if the "jumps" tk are not very frequent, then the trajectory of a (d)-pseudosolution is an (s T)-pseudotrajectory. Of course, since our Definition 1.24 requires nothing except (1.179), the inverse statement cannot be true. Nevertheless, one can easily find, for a (d,T)-pseudotrajectory #, a (d)pseudosolution #* such that I#(t) - #*(t)l < d for t E IR. Indeed, take tk = kT, k E 7], and set 9 *(t) = Z ( t -

t k , ~ ( t k ) ) for t E [tk, tk+l).

(1.181)

1.5 Shadowing for Flows

91

Then obviously the first inequality in (1.180) holds with d = 0. Inequality (1.179) implies (1.181) and the second inequality in (1.180). Thus, if a solution ~ ( t , p ) has the property I - ~ ( t , p ) - ~*(t)l < Ld, then we have the estimate I~(t,p) - 4~(t)l _< (L + 1)d. This means that in the problem of Lipschitz shadowing (and this is our main problem) the two definitions are equivalent. Two types of shadowing for flows are usually considered. First let us define two sets of homeomorphisms ( reparametrizations) a : ]It --~ JR. Define Rep as the set of increasing homeomorphisms a mapping ]R onto ]R and such that a(0) = 0. Fix e > 0 and define Rep(e) as follows: Rep(e)=

aERep:

-

$

<efort~s

.

D e f i n i t i o n 1.25 System (1.178) has the "property of weakly parametrized shadowing" on a set Y C IR'~ if given e > 0 there exist d , T > 0 such that for any (d, T)-pseudotrajectory ql: I R ~ Y there is a point p and a reparametrization a CRep with the property

x(-(t),p))l _< D e f i n i t i o n 1.26 System (1.178) has the "property of strongly parametrized shadowing" on a set Y C IRn if given c > 0 there exist d , T > 0 such that for any ( d, T)-pseudotrajectory qI : IR ~ Y there is a point p and a reparametrization a ERep(e) with the property I~(t) - ~,(~(t),p))l ~ e.

Relations between these (and some other) shadowing properties for flows were studied by Thomas [T1], Komuro [Kol], and others. Now let us define a hyperbolic set for system (1.178). D e f i n i t i o n 1.27 A set A is called "hyperbolic" for system (1.178) if (h. 1) A is compact and ~-invariant;

92

1. S h a d o w i n g Near an I n v a r i a n t Set

(h.2) there exist numbers C > 0, A0 E (0, 1), and continuous families of linear subspaces S(p), U(p) of ]Rn, p E A, such that (h.2.1) the families S, U are D~-invariant, i.e., D ~ ( t , p ) T ( p ) = T ( ~ ( t , p ) ) f o r t e IR, p 9 A, T = S,U; (h.2.2) for p 9 A we have S(p) @ U(p) = IR~ if X ( p ) = O, S(p) G U(p)~ < X(p) > = ]R'~ if X(p) ~ 0

(here < X(p) > is the span of X(p)); (h.2.3) ID~(t,p)vl <_ C2olvl for v 9 S(p), t > O; IO~(t,p)v I < CAo'lV I for v 9 U(p), t < O. We call C, )~o the hyperbolicity constants of A, and the families S, U are called the hyperbolic structure on A. Now we state the main result of this subsection. 1.5.1. Assume that A is a hyperbolic set for system (1.178) and that X ( p ) ~ 0,p E A. Then there exists a neighborhood W of A and numbers do, L > 0 such that for any (d, 1)-pseudotrajectory ~ C W with d < do there is a point p and a homeomorphism c~ ERep(Ld) such that

Theorem

I~(t) - Z(~(t),p)l _ Ld, t e JR.

(1.182)

Remark. The theorem states that the homeomorphism a is "close" to the identity in the sense of the inequality

Let us show that, in general, a similar statement with an estimate

Ice(t)

-

tl

< Dl(d), where limDl(d) = 0, --

(1.183)

d---*O

is not true. Assume that system (1.178) considered in IR2 has a hyperbolic closed trajectory S corresponding to a 27r-periodic solution ~(t) = (sin t, cos t). Consider for d > 0 a mapping ~ given by

qt(t) = ~(kd/2 + t), t e [2rk, 2 r ( k + 1)), k e 77. Since

I (t)l

= 1, + is a (d, 1)-pseudotrajectory for system (1.178).

1.5 Shadowing for Flows

93

We claim that if d is small enough, then it is impossible to find a point p and an increasing homeomorphism a : ]R ~ IR such that estimate (1.183) holds together with

I~.(a(t),p) - ~(t)[ < D2(d), where limD2(d) = 0. d---*0

(1.184)

Note that the set { ~ ( 2 r k ) : k E 71} is d/2-dense in S (i.e., for any point x C S there is a point x' = ~ ( 2 r k ) such that r(x, x') <_ d/2, where r(, ) is the distance on S). Since the trajectory S is hyperbolic, it is isolated (i.e., there is a neighborhood U of S such that S is the only complete trajectory of system (1.178) in U). It is easy to understand that if d is small enough, then a point p for which (1.184) is satisfied belongs to S. Take d > 0 having the formulated property and such that 1

d < r, Dl(d) < ~;, D2(d) < 3 Assume that for this d there exists a point p E S and a homeomorphism a for which the inequalities in (1.183) and (1.184) are fulfilled. Since 3(2~rk, p) = p for any k E 7/, it follows from the inequality r(~(a(27rk), p), ~(27rk, p)) < -4 that

p), p) <

7r

But it follows from (1.184) that

r(Z(a(2~rk),p),~(2~rk)) < 2D2(d) < 4' hence for any point x of the set {~(2~rk)} we have r(x,p) < r / 2 , and this set cannot be ~'/2-dense in S. The obtained contradiction proves our statement. Now let us prove Theorem 1.5.1. Let A be a hyperbolic set of system (1.178) with hyperbolicity constants C, A0. Find T > 0 such that

r <_

Denote r = ~ ( T , x). We fix a bounded neighborhood W I of the set A and take a Lipschitz constant s for X in W I. We will take a neighborhood W in Theorem 1.5.1 such that ~(t,x) E W' for x E W, It I < T + 1. (1.185) The same reasons as in Lemma 1.1.3 show that there is a constant K = K(T, s such that if a (d', 1)-pseudotrajectory k0 of system (1.178) belongs to W, then is a (Kd ~, T)-pseudotrajectory. We will show that there exist constants do, L

94

1. Shadowing Near an Invariant Set

such that for a (d,T)-pseudotrajectory ~ C W with d < do we can find a point p and a homeomorphism (~ with the properties described in our theorem. Obviously, this proves the theorem for (d, 1)-pseudotrajectories with d'o = do/K and L I = KL. The first step of the proof is to extend the hyperbolic structure to a neighborhood of A. L e m m a 1.5.1. Given e > 0, A C (A0,1) there exists a neighborghood W = W(e, A), a positive constant ~5, and continuous (not necessarily Dr

extensions S', U~ of S, U to W such that

(1) s'(x) + g ' ( x ) + < x(x) > = n~-, x 9 w (we denote below by H~, H~, and II ~ the complementary projectors onto S'(x), U'(x), and < X(x) > generated by this representation of IR~); (2) for x,y 9 W with l Y - r < 5, the mapping II;Dr

(II~Dr is an isomorphism between S'(x) and S'(y) (respectively, between V'(x) and U'(y)), and the inequalities

[H;DC(x)vl < ~lvl, [H~DC(x)vl < elvl, v 9 S'(x); [H~DC(x)v[ >_ ~lvl, [II;DC(x).I <_elvl, ~ 9 v'(~),

(1,186)

(1.187)

hold; (3) ifh(x) = ~(T',x) with ]T'-T] <_ 1, then, forx, y 9 W with ]y-~(x)[ < and for v 9 S'(x) (~ U'(x), the inequality

[H~

< elvl

(1.188)

holds. Proof. We first extend S, U to continuous (not necessarily Dr families S', U' on a closed neighborghood W0 of A so that statement (1) holds. It follows immediately from Definition 1.27 that for points x C A, y = r the mapping II~Dr (II~Dr is an isomorphism between S(x) and S(y) (respectively, between U(x) and U(y)), and the relations

[IH;DC(x)ls<=)ll <_Ao, H~Dr _> 1/~0, H~Dr

IIH~Dr

= O,

=0

hold. For points x E A, y = ~(x), we have

/I~

= 0.

Since H~, H i , H ~ r and De(x) are uniformly continuous on A, and ~ is a shift along trajectories of system (1.178) for a bounded time, given arbitrary e > 0 we obviously can find a neighborhood W(e, A) C Wo and a number <5with the desired properties. []

1.5 Shadowing for Flows

95

Below we write S, U instead of S', U'. Take A E (A0, 1) and find the corresponding neighborhood W = W(1, A). We will decrease W in the proof. Since W C W0 and W0 is compact, there exists N > 0 such that

IX(x)t, III/~11, II//~ll ~ N for x E W,

(1.189)

and

IiD~.(t,x)-D~(t',x)[l<_NIt-t'lforxEW,

Iti, i t ' ] < _ T + l ,

(1.190)

For p E W set

Z(p) = S(p) + U(p). We will denote in this proof by ,g(p) the (n - 1)-dimensional hyperplane through the point p parallel to Z(p). Note that this hyperplane is transverse to the vector X(p). Let us formulate two auxiliary statements (Lemmas 1.5.2 and 1.5.3). In these statements, dist(x, Z(z)) is the distance between a point x and a hyperplane ~(z), and ~(b, x) is the ball in S(x) of radius b > 0 centered at X.

L e m m a 1.5.2. There exist constants dl > 0, e0 < 1,/(1 _> 1 such that if z,q C W and Iz - ql <- dl, then there is a unique scalar function f ( x ) in the dl-neighborhood of q such that [f(x)l <: e0 and

.~(f(z), x) E Z ( z ) . This function f is of class C 1 and has the properties If(x)] < Kldist(x, ~7(z)) and I z - ~ ( f ( x ) , x)] _< K l l X - z I.

Remark. The geometric sense of the function f is obvious. For a point x in a small neighborhood of z, f ( x ) is the time at which the trajectory through x reaches its nearest point of intersection with Z(z). Proof. If we assume, without loss of generality, that z is the origin, and that Z(z) is given by the relation y : Ls, s E ]Rn-l, then we can write the equation for f in the form

F(x, f, s) = ~ ( f , x) - Ls = O. Since F(0, 0, 0) = 0, and the matrix

A-

OF O,f,s,(O,O,O)( ) = [X(O),-L]

(1.191)

96

1. Shadowing Near an Invariant Set

is nonsingular, it follows from the implicit function theorem that Eq. (1.191) has a unique small solution f ( x ) , s(x) for small x. Since our neighborhood W is a subset of a compact set W0 on which statement (1) of Lemma 1.5.1 holds, for any z 9 W the angle between ~W(z) and X ( z ) and the norm IX(z)[ are separated from 0 by a constant independent of z. It follows from standard estimates in the implicit function theorem that f has the properties described in our lemma with constants dl, e-o,K1 independent of z.

L e m m a 1.5.3. There exist positive constants d2 < dl and b with the following property. If p, z 9 W and I r zl <- d2, then there is a unique function T(X) of class C 1 on ~(b,p) such that r

=

x) 9

and Iv(x) - Tr <

(dl and eo are from Lemma 1.5.2). Remark. We do not prove this lemma here, its proof is similar to the proof of the preceding lemma. We will study in detail properties of the functions f, r in Subsect. 2.2.1, Chap. 2, investigating equations analogous to Eq. (1.191). Take N1 = NI(N, A) given by Theorem 1.3.1, fix the corresponding t~ so that inequality (1.65) holds, and find e > 0 such that e < 12K1-~"

(1.192)

Below we work with a neighborhood W = W(e, ,k) (increasing it, if necessary, to satisfy (1.185)). The main technical part in the proof of Theorem 1.5.1 is the lemma below. In this lemma, we consider points p,z 9 W with Iz - r < d2, so that T(X),r are defined on X(b,p). L e m m a 1.5.4. There exists d3 <_ da such that for anyp, z e W with d < d3 and for any v 9 Z(p) with Iv] = 1 we have [ ( D e ( p ) - Dr

Iz-r

<

< -~.

Proof. Take v e Z(p) with Ivl = 1. By Lemma 1.5.3, we can define r(x) on Z(b, p). Denote TO = r(p) and define ((x) = ~(ro, x). Denote q = r w = ((p), and let vl = Dr v2 = D((p)v, va = Dr Take d~=min

d2, K1, 8

N

'

1.5 Shadowing for Flows

97

where eo and K1 are given by L e m m a 1.5.2, and d2 is given by L e m m a 1.5.3. If [z - ql < d < d~, then obviously dist(q, E ( z ) ) < d, hence if(q)[ < K~d by L e m m a 1.5.2, and If(q)l < eo by the choice of d~. T h e point w belongs to E ( z ) . It follows from the equalities w = ~,(To,p) = ~(TO -- T , ~ , ( T , p ) )

= Z(ro - T,q),

from the estimate [To-T[ < Co, and from the uniqueness of f that To--T = f ( q ) . Since eo < 1, we deduce from estimate (1.190) that IIDr

- D~(p)ll = [ [ D ~ ( T , p ) - DZ(To,p)[[ < N [ T

- TO[.

This leads to the inequality IV1 - - V2[ ___
< ~.

(1.193)

We want to estimate Iv2 - v31. Let "/(s) = p + vs, s > 0, be a ray in Z ( p ) . Define = ,~(To,'~(S)) ,

"~1(8) = ~ ( ~ ( S ) )

~(s) = r

= z(T(~(s)), ~(s))

for s >_ 0 with 7(s) e Z'(b,p). Now we take d~' = min

d2, -~-,

,

where ~ is given by L e m m a 1.5.1. If [z - q[ < d < dJ, then obiously there exists So > 0 such that, for s E [0, So), the function T*(S) = f ( 7 ~ ( S ) ) is defined. It follows from the equalities 3(f(71(s)),~/l(S))

= ~(f(*/l(S)) -1- To,"{(s)) = ~(T(~/(S)),')'(S))

that T(~(s)) = To + T*(S), hence d (0) = ~ ( V o , ~ / ( s ) ) I s = o ,

v2 =

v3 =

(0) = ~d= (_T o + T*(s),~(s)) Is=o= x ( w )

We can write, for small s > 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~

<- ~,

(1.194)

(0) + vs.

(1.195)

98

1. Shadowing Near an Invariant Set

hence dist(71(s), ~P(z)) < es + o(s), and consequently (see Lemma 1.5.2), I~*(~)1 = I f ( ~ ( ~ ) ) l < ~K,~ + o(~). This leads to the estimate d'r*(O) _~ eK1. It follows from (1.189) that

X(w

)d

T'(O) <_eNK1 < ~.

Comparing (1.194) and (1.195), we see that /r

IV2- V31 < g. This inequality and inequality (1.193) prove that for ]z - r d3 = min(d~, d~), we have

< d < d3, where

Iv~ - ~zl < -. 4 This completes the proof.

[]

Now we take a (d, T)-pseudotrajectory # for system (1.178) such that #(t) 9 W for t 9 IR and d < d3. Denote xk = # ( k T ) for k 9 7/. We choose coordinates in 2~(xk) so that xk is the origin of 2Y(xk), this identifies Z(xk) with X(Xk). Let Hk = Z(xk). Denote Pk = H ~ ]z(~k), Qk = / / ~ k ]z(~k), and Sk = PkHk, Uk = QkHk. Obviously, & = s(xk), uk = v(xk). Fix a pair xk, Xk+l and denote p = xk, z = xk+l. Let Ck(x) = r -- p) for x 9 ~(b,p) (the mapping r and the number b are given by Lemma 1.5.3). We can write Ck(v) = Dkv + Xk+l(v), where Ok = DCk(0), Xk+l(0) = Ck(0), DXk+l(O) = 0. Since the mapping Ck is a shift along trajectories of system (1.178) for time not exceeding T + 1, there exists d4 < da such that /Z IXk+l(v) - Xk+,(v')l < -~lv -- v' I for v,

"0t

e Z(p), Ivl, Ir -< d4.

(1.196)

Now we set 8 Ak = A~ + A~, where A~ = II~Dr

Represent Ck(v) in form (1.51), where

8

~ A Uk = IPzDr

~.

1.5 Shadowing for Flows

99

wk+l(v) = (Dk - Ak)v + Xk+l(V). It follows from L e m m a 1.5.1 that A~ maps S(p) = S'k to S(z) = ~k+l, and that ]lAk]sk]] = ]lA~[s~]l <_ ~.

(1.197)

Similar reasons show that A~lu * = AkIu, maps isomorphically Uk to Uk+l, and for Bk = (A~[u,) -1 we have

IIBkll < ~.

(1.198)

Now let us estimate ](Ok -- Ak)v] for v 9 Hk, Iv] = 1. Since Dk = DCk(0) = D e ( p ) and v = (II~ + II;)v, we can write

(Dk - Ak)v = ( D e ( p ) - Dr +(II; + II~' + II~162

+ II;)v -

nZDr

- II~Dr

It follows that ](Ok - Ak)v[ <_ I(DC(p) - DC(p))v]§

§176162

§ III:Dr

§ III~'Dr

(1.199)

By L e m m a 1.5.4, the first t e r m on the right in(1.199) does not exceed ~/4. Since v 9 Sk § Uk, (1.188) implies that the second t e r m does not exceed e _< n/12. Since lI~v 9 Uk, II~v 9 Sk and III;vh III~v[ << N, it follows from the second inequalities in (1.186) and (1.187) that the third and the fourth terms in (1.199) are each estimated by Ne < ~/12. Hence, if d _< da, then t~

I(Dk - Ak)vl < 3 ' Now it follows from this inequality and from (1.196) that Ixk+l(v) - Xk+l(v')l <_ ~[v - v' I for v,v' 9 Hk, [vl, Iv'[ <_ d4.

(1.200)

Finally, we deduce from (1.197)-(1.200) and from Theorem 1.3.1 that there exist constants d ~, L ~ > 0 such that if

Ir

< d"< d',

then one can find points Vk 9 Hk with the properties

~)k(Vk) = ?2k+1 and [Vk[ <<_L'd".

(1.201)

Since any Ck is a shift along a trajectory of system (1.178), we see that all the points Pk = xk + vk belong to the same trajectory ~(t,po). It follows from L e m m a 1.5.2 and from the definition of Ck that if ~ is a (d, T)-pseudotrajectory with d < d2, then

100

1. Shadowing Near an Invariant Set

Iek(0)l = Iz- ~,(f(r

r

~ KIIr

z[ ~_ Kid,

hence we deduce from (1.201) that if d _< do = min ( d,t, ~-1, d' 1 + Lodl~xp(CT)) , then

[Pk - xkl <_ Lod,

(1.202)

where Lo = L'K1 (recall t h a t s is a Lipschitz constant for X in the neighborhood W'). Define n u m b e r s tk by Pk+l = ~(tk,pk). Let us e s t i m a t e

I,~(T, pk) -- z[ < [-~(T,p) - zl + 13(T, pk) - ~ ( T , p ) [ (recall t h a t p = xk, z = Xk+l). T h e first s u m m a n d does not exceed d, the second one does not exceed L o d e x p ( s Since by the choice of do we have the inequality d(1 + L o e x p ( s

<_ dl,

it follows from L e m m a 1.5.2 and from the equality pk+l = ~ ( t k , p k ) t h a t Itk - TI <_ Lad, where 51 = / ( 1 ( 1 + i o e x p ( s Set To = 0, rk = t l + . . . + tk for k > 0, and T k ---- tk + 9 9 + t-1 for k < 0. Now we define a : IR --~ JR. For k 9 7] we set

a ( T k ) = rk, for t 9 [Tk, T ( k + 1)] we set ~ ( t ) = ~-~ + (t - T_ .k. t) k +yl .

Obviously, (~ is an increasing h o m e o m o r p h i s m m a p p i n g ]R onto JR. Take t 9 [Tk, T ( k + 1)] for k > 0, set t' = t - Tk. Since It'l < T, we obtain the e s t i m a t e (1.203) I,*)- -t'1 = t' _ 1)" _< Lid. Now let us show t h a t a 9 that t ~ s. Assume t h a t t < s and that

where L2 = L1/T. Take t, s 9 IR such

t 9 [Tk, T ( k + 1)), s 9 [Tm, T ( m + 1)). Since

tk+l = 1 T we obtain the inequalities

+

tk+l - -- T ~_ 1 + - T

L•d

= 1 + L2d,

a ( T ( k + 1)) - a(t) <_ (1 + L2d)(T(k + 1) - t),

1.5 Shadowing for Flows

101

a ( T ( k + 2)) - c~(T(k + 1)) < (1 + L:d)T,

a(s) - a ( T m ) < (1 + L2d)(s - Tin).

Adding these inequalities and dividing the sum by s - t, we prove the estimate a(s) - a(t) < 1 + L2d. s-t. -

Similar reasons establish an analogous estimate from below. This proves that a CRep(L:d). Set p = p0. Obviously, ~ . ( a ( T k ) , p ) = E ( r k , p ) : Pk. For t C [ T k , T ( k + 1)] we have

+l---(t',

-

x )l + I (Tk + t') - .V(t',

(Tk))l

(note that Z ( a ( t ) , p ) = S ( a ( t ) - rk,pk), xk = ~P(Tk), q](t) = ~P(Tk + t')). The first term on the right does not exceed N L l d (we apply (19 the inequality IXI _< N). The second term does not exceed IPk - xkl e x p ( s

and

<: L o e x p ( s

and the third term is not more than d (since # is a (d, T)-pseudotrajectory). This leads to the inequality r(~.(a(t),p),g~(t)l < L3d,

where L3 = N L 1 + Lo e x p ( s + 1. Take L = max(L:, L3) to complete the proof of Theorem 1.5.19

[]

Remark. Analyzing the proof of Theorem 1.5.1, it is easy to understand that a point p and a homeomorphism a such that (1.183) holds are not unique. This nonuniqueness is due to the absence of hyperbolicity "along the flow". Nevertheless, since the "linear parts" of the mappings r introduced in the proof obviously satisfy the conditions of Theorem 1.3.2, the following "uniqueness" statement for the considered problem holds9 We can take do so small that, for d < do, a sequence vk C Z ( x k ) that satisfies (1.201) is unique9 By Lemma 1.5.2, this means that there exist numbers d* and L* such that for d < d* a point p = P0 E ~U(x0) with the properties (1) pk e Z ( x k ) , IPk - zkl <_ L'd; (2) pk+l = z(rk,Pk); (3) irk - T I <_ L*d is unique. Another way to formulate the uniqueness of a shadowing trajectory in a neighborhood of a hyperbolic set for a flow is described in [Coo3].

2. Topologically Stable, Structurally Stable, and Generic Systems

Everywhere in this chapter, M is a C ~ smooth n-dimensional closed (i. e., compact and boundaryless) manifold, and r is a Riemannian metric on M.

2.1 Shadowing and Topological Stability This section is devoted to relations between shadowing and topological stability. Following Walters [Wa2], we show that a topologically stable homeomorphism has the POTP. It is also shown that if an expansive homeomorphism has the POTP, then it is topologically stable. Various authors used different definitions of topological stability. In this book, we work with the following one. We consider the space Z ( M ) of discrete dynamical systems on M with the metric P0 defined by the formula

po(r162162162162162 xEM

Definition 2.1 We say that a dynamical system r E Z ( M ) is "topologically stable" if given e > 0 there exists a neighborhood W of r in Z ( M ) such that for any system r E W there is a continuous mapping h of M onto M having the following properties: (a) r(x,h(x)) < ~ for x E M; (b) r 1 6 2 This definition of topological stability is close to the corresponding ones in [Wal, Y1]. In [Ni2], the defined property is called the C o lower semi-stability in the strong sense. Walters [Wa2] and Morimoto [Moriml] showed that topological stability implies the the POTP. Walters proved Theorem 2.1.1 for n = d i m M _> 2, the proof given by Morimoto is valid for n _> 1.

Theorem 2.1.1. If a system r E Z ( M ) is topologically stable, then r has the POTP.

104

2. Topologically Stable, Structurally Stable, and Generic Systems Walters also proved the following statement.

T h e o r e m 2.1.2. Let r E Z ( M ) be expansive with expansivity constant (~. If r has the POTP, then given ~ > 0 with 3e < ~ there exists d > 0 with the following property. If r E Z ( M ) and P0(r < d, then there is a unique continuous mapping h: M --* M such that h o r = r o h and r(x, h(x)) < e for all x E M. If e is small enough, then h maps M onto M. Remark. This last result shows that a topologically Anosov homeomorphism is topologically stable (and even has a stronger property, the corresponding mapping h close to identity is unique). We follow [Wa2] in the proofs of Theorems 2.1.1 and 2.1.2. The proof of Theorem 2.1.1 is given here only in the case n = d i m M ~ 2 (the remaining case dimM = 1 is simple but requires special treatment, the reader can find the corresponding proof in Chap. 2 of [Pi2]). Let us establish some preliminary results. First we prove a "C~ '' statement obtained by Nitecki and Shub in [Ni3]. L e m m a 2.1.1. Assume that dimM > 2. Consider a finite collection {(p~,q~)EMxM:

i=l,...,k}

such that (a) p~ ~ pj,q~ ~ qj for l ~ i < j ~ k; (b) r(p~, q~) < d for i = 1 , . . . , k, with small positive d. Then there exists a diffeomorphism f of M with the following properties: (a) p0(f, id) < 2d (here id is the identity mapping of M); (b) f(p,) = q, for i = 1 , . . . , k . Proof. Take the circle S 1 with coordinate ~ E [0, 1). Consider the following system of differential equations on M • S 1 with coordinates (p, ~): p=O,~=l. Its vector field is X = (0, 1), and its flow is given by ~(t,p,O) = (p,O + t(mod 1)). Obviously, ~(1,p, 9) takes M x 0 to itself and induces the identity mapping there. Given the points p~, q~ E M, we consider the points (p~, ~),1(q~, ~)3 in M • S 1. We take for each i a curve 71(t) in M, 0 < t < 89 of constant speed, joining p~ to q~ and having length less than d. We can change parameter t on ~/i(t) so that

~,(0) = #,(1) _- 0, I#,(t)l < 2d. s

(2.1)

2.1 Shadowing and Topological Stability

105

Consider the curves gi given by 1 1 g,(t) = ('r,(t),-~ + t), 0 < t <_ -~. The curves gi are one-dimensional, and dim(M x S 1) >__ 3. It follows from condition (a) and from the transversality theorem (see [Hirsl]) that we can slightly perturb the curves ~,~(t) so that (2.1) holds, and

1 l <_i < j < k . _ g,(t) # gj(t) for 0 < t < ~,

Hence, we can find tubular neighborhoods N~ of gi in M • S 1 such that N~nNj=O,

l < i < j < k.

Take the tubular neighborhood N~ of g~, extend g~ beyond its endpoints, and then extend N~ to be a tubular neighborhood of the extended curve (let this extended neighborhood be N*). This can be done so that

~-~ n ~-~j = 0, l < _ i < j < _ k .

(2.2)

Let Y~ be the vector field Yi = (~i(t), 1) on g, (we can take Y~ = (0, 1) on the extended part of g~). Extend Y~ to Ni* making it constant along fibers. Now take a "bump" function ~ which is 1 on g~(t), 0 < t < 89 and 0 off N~*. We obtain the vector field z, = x + o(y, - x) such that Z~ = (~h(t), 1) = gi(t) on g,(t), 0 < t < 88 hence g~(t) is a trajectory of the system generated by Zi, and IZi - X I < 2d.

(2.3)

It follows from (2.2) that we can obtain a vector field Z which coincides with X on M \ ( N ~ U . . . U N ~ ) , and with Z~ on N~,i = 1 , . . . , k . Let ~(t,p,O) be the flow of the corresponding system of differential equations. Since X = (0, 1), and inequalities (2.3) hold for i = 1 , . . . , k, we deduce from standard estimates for differential equations that for any (p, 0) E M • S 1 we have ~(~v(t,p, 0), z ( t , p, 9)) < 2d

for It[ _< 1. Let f be the time-one mapping of the flow ~P. It follows from our construction that f(p~) = q~, i = 1 , . . . , k, and po(f, id) < 2d. [] The following statement enables us to "improve" a finite segment of a dpseudotrajectory.

Let ~ = {xk} be a d-pseudotrajectory for a dynamical system r Let m >_ 0 be an integer, and let 71 > 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
<m-1.

Since

r(r

~ r(r162 +r(r

< 2d,

we can find ym such that Ym ~ Y~ for 0 < i < m - - 1, r(ym,xm) < y, and

r(r

< 2d.

[]

Now we show that it is possible to approximate a finite segment of a dpseudotrajectory for r by a trajectory of a close dynamical system. L e m m a 2.1.3. Let d i m M > 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
# r

, 0 _< i < j < m).

1;

(2.5)

2.1 Shadowing and Topological Stability

107

It follows from Lemma 2.1.1 that there exists a diffeomorphism f of M having the following properties: Po(f, id) < dl and

f(r

:

Yk+l for

k = 0 , . . . , m - 1.

(2.6)

Consider r = f o r and take y = Y0. Obviously, (2.5) and (2.6) imply (2.4). For any x 9 M we have r(r

r

r(f(r

:

and r(r162

=

r

< dl < .4

r(r162

<`4

(we take into account that

r(f-l(x),x) ~ Po(f,

id) < dl). []

This completes the proof.

Now we can prove Theorem 2.1.1 (let us remind that we consider the case dimM _> 2). Let r be topologically stable. Fix arbitrary e > 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

<_ r(h(r

+r(r

ek(x)) + r(r

h(r

h(r

_<

ek(y))+

< ~ + ~ +~ <

for ]k I < g . Therefore, the inequality r(x, y) < 7/implies r(h(x), h(y)) < A, and the continuity of h is proved. The mapping h is the only one with h o r = r o h and r(x, h(x)) < e since if l is another one, then

T(r

# ( h ( x ) ) ) = r(l(r

h(r

<

2.2 Shadowing in Structurally Stable Systems

< r(t(r

Ck(x)) + r(r

h(r

109

< 2~ <

for any k E 7/, and it follows that l(x) = h(x). It is well known that if M is a compact manifold and r(x, h(x)) < e for any x E M with e small enough, then h maps M onto M. This completes the proof. [] We consider here only the case of discrete dynamical systems. The same problem for flows was studied by Thomas [T2].

2.2 S h a d o w i n g in Structurally Stable S y s t e m s Structurally stable systems (both flows and diffeomorphisms) were the main objects of interest in the global qualitative theory of dynamical systems in the last 30 years. Now we know that structural stability is equivalent to Axiom A (for diffeomorphisms) or Axiom A' (for flows) combined with the strong transversality condition, see [Robb, Robi2, Ma] in the case of diffeomorphisms, and [Robil, Hay, We] in the case of flows. Various approaches were applied to show that a structurally stable diffeomorphism has the P O T P [Robi3, Mori2, Saw]. 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 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 nonivertible "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 LpSP. This result was first published in [Pi2] with a proof based on another approach. Here we show that a method similar to the one applied in Subsect. 2.2.1 gives a shorter and a more clear proof. Sakai noted that the P O T P is "uniform" in a Cl-neighborhood of a structurally stable diffeomorphism [Sakl] and that the C 1 interior of the set of diffeomorphisms with the P O T P consists of structurally stable diffeomorphisms [Sak2]. We prove (Theorem 2.2.8) that the LpSP is also uniform [Be] and that a diffeomorphism in the C 1 interior of diffeomorphisms with uniform LpSP is structurally stable. 2.2.1 The Case of a Flow

In this subsection, we consider autonomous systems of differential equations generated by vector fields of class C 1 on M. Below we identify a vector field X

110

2. Topologically Stable, Structurally Stable, and Generic Systems

on M and the system

= X(x)

(2.7)

generated by X. To define structural stability for the flows of considered systems, we introduce the Cl-topology on the set of vector fields on M as follows. By the Whitney theorem [Hirsl], for any closed smooth n-dimensional manifold M there exists an embedding f : M ~ ]R2"+1. Let us fix such an embedding f . Thus, we can consider M as a smooth submanifold of IR2n+1. For a point p E M we identify the tangent space TpM with a linear subspace of IR2'~+1 and denote by ]. I the norm in TpM generated by the Euclidean norm of ]R2'~+1. Now we can consider a vector field X on M as a mapping that takes a point p C M to X(p) E T~M C IR2'~+1. For two vector fields X and Y on M, we define p0(X, Y) = max IX(p) - Y(P)I. pEM

In this setting, DX(p) is considered as the derivative of the mapping above, and for p G M we can define

I[DX(p) -

DY(p)] I =

max

veTpM,H=I

IDX(p)v - D Y ( p ) v I.

Now we define the CLdistance pl between two vector fields X and Y on M by the formula

pl(X, Y) = p0(X, Y) + max

pEM

We denote by 2r topology induced by pl.

IIDX(p) - DY(p)]I.

the space of vector fields of class C a of M with the

Remark. Of course, one can consider metrics generated by coverings of M with local charts (see Chap. 0 in [Pill). Let ~ ( t , x) be the flow generated by system (2.7), i.e., ~ ( t , x) is the trajectory of (2.7) such that ~(0, x) = x. We denote by D~(t, x) the corresponding variational flow,

D~(t, x) -

O (t, Ox

Let H(t, x) be the flow of a system

= Y(x).

(2.8)

2.2 Shadowing in Structurally Stable Systems

111

D e f i n i t i o n 2.2 A homeomorphism h of M is called a "topological equivalence" of the flows ~. and H if h maps oriented trajectories of (2.7) onto oriented

trajectories of (2.8). Now we define structural stability for flows. D e f i n i t i o n 2.3 We say that the flow ~ of system (2.7) is "structurally stable" if given e > 0 there is 5 > 0 such that for any Y 9 2dl(M) with p l ( X , Y ) < 5 there is a topological equivalence h of the flows ~ and H with the property r(x, h(x)) < ~ for x 9 M.

Remark. Another possible definition of the structural stability for the flow requires the existence of a number ~ > 0 such that for any Y E rYI(M) with pl(X, Y) < 5 there is a topological equivalence h of the flows 3 and H (and h is not required to be close to the identity). It follows from [Hay, We] that these two definitions are equivalent. Before we state necessary and sufficient conditions for structural stability, we give some related definitions and discuss the appearing structures. We will apply the usual notation, for a set V C M we denote

z ( t , v) = U z ( t , ~). xEV

D e f i n i t i o n 2.4 A point p C M is called "nonwandering" for the flow ~ if for any neighborhood V of p and for any T there exists t > T such that

.~(t, v) n v # ~. It is easy to show that the set of nonwandering points of Z (we denote this set n ( ~ ) ) is compact and ~-invariant. Smale introduced in [Sm2] the following property of 2 . A x i o m A'. The nonwandering set n ( ~ ) of the flow ~ is hyperbolic and can be represented as the union of two disjoint compact sets, [2' and n ~ such that

(i) n ~ consists of a ~nite number of rest points of 2; (2) X(x) # 0 for 9 e n'; (3) closed trajectories of 3 are dense in n r. Remark. In the case of the flow 2 , we apply Definition 1.27 of a hyperbolic set with the following obvious modification. Everywhere in this definition, we take TpM instead of IR~, and, for v E TvM, Ivl is the norm generated by the

112

2. Topologically Stable, Structurally Stable, and Generic Systems

R i e m a n n i a n metric r. For a point p E ~ ( ~ ) we denote by S~ corresponding subspaces of the hyperbolic structure on ~(.~).

and U~

the

Take a point p E M and define its stable and unstable manifolds as follows:

W~(p) = {x E M : r ( ~ ( t , x ) , ~ ( t , p ) ) -~ 0 as t -~ + ~ } , W~(p) = {x E M : r ( ~ ( t , x ) , ~ ( t , p ) ) --* 0 as t ~ - c o } . For a t r a j e c t o r y -~ of ~ we define the stable manifold ld;s(7) = {x E M :

d i s t ( Z ( t , x), 7) ~ 0 as t ~ +c~}

and the unstable manifold 14;~(~/) = {x E M :

dist(~'(t, x), 7) -~ 0 as t ~ - c o } .

Let us also introduce the following notation. For a set V C M we denote

0 + ( V ) = U ~ ( t , V), O - ( V ) = U X.(t, V), t>O

t
o(v) = o§

u o-(v).

According to this notation, O(x) is the t r a j e c t o r y of a point x E M . T h e following s t a t e m e n t is a corollary of the stable manifold t h e o r e m for flows [Sml] for the hyperbolic set ~2(~). Theorem

2.2.1. (la) For a point p E J2(Z), the stable manifold WS(p) and the unstable manifold W~(p) are the images of Euclidean spaces IRk under immersions IRk M of class C 1. The equalities

TpW~(p) = S~

TpW~(p) = U~

hold; (lb) for any point x e W~(p), the tangent space T~W~(p) has the property D 3 ( t , x ) T ~ W ~ ( p ) = TyW~(r), t E IR, a = s , u , where y = ~ ( t , x), r = ~ ( t , p ) (below we use the notation S~

= T~WS(p), U~

= T~W~(p));

(lc) the stable manifolds WS(p) have the following property: if pm,p E A, p,~ ---* p as m ~ co, and for xm E WS(pm) we have xm ~ x E WS(p), then

so(xm) --, so(x) (and the unstable manifolds have the same property); (2) for a trajectory 7 C f2(~), the sets ),Y~(7), )4;~(7) are ~-invariant images under immersions N ~ M of class C 1, where N is IRk if the trajectory ~/

2.2 Shadowing in Structurally Stable Systems

113

is not closed, and N is a fiber bundle over the circle S 1 with fiber IRk if ~ is a closed trajectory. If p E 7, then S(p) := T, YW(7 ) = S~

+ < X(p) >,

(2.9)

U(p) := TpYY=(7) = U~

+ < X(p) >;

(2.10)

(3) for p e Y2(•) we have }'V~(O(P)) =

U

W~(r),a=s,

TM

reO(p)

To prove the main result about shadowing in structurally stable flows, we construct a special geometric structure connected with a structurally stable flow ("weakly upper semicontinuous" subspaces S(p), U(p) for p C M, see Lemma 2.2.10). To do this, we need a group of auxiliary results. A part of these results (Lemmas 2.2.1 and 2.2.2) are well-known, their original proofs are detailed or reproduced in books (see, for example, [Pil] for proofs of Lemmas 2.2.1 and 2.2.2). In contrast, the proof of Lemma 2.2.9 on the existence of "compatible extensions of stable and unstable bundles" (very important for us) is only sketched in the fundamental paper [Robil], thus we give here a detailed proof of this statement. L e m m a 2.2.1 [Sm2]. There exists a unique decomposition t2( ~)

= J31 u . . . u ~2m,

where ~2~ are compact disjoint invariant sets, and each S2i contains a dense trajectory. The sets S2i are called basic sets. For a basic set Y2~ we define the stable and unstable sets Ws(~21) = {x E M :

d i s t ( 3 ( t , x ) , 12~) ~ 0 as t --+ +oo}

W~(~,) = {x E M :

dist(.~(t, x), J2i) ~ 0 as t ---* - ~ } .

and The following statement easily follows from Definition 2.4 and Lemma 2.2.1. L e m m a 2.2.2 [Sm2].

M= U w'(n,)= U w"(n,). l
l
Below we formulate a very important property of basic sets. L e m m a 2.2.3 [Hirsl]. For a basic set 12~ we have

114

2. Topologically Stable, Structurally Stable, and Generic Systems

w'(~/) = U w'(,,/), w ~ ( ~ / ) =

7cYL

U w~('~) .

7c~2,

Remark. In [Hirsl], semi-invariant disk families are applied to prove L e m m a 2.2.3. Let us show that one can prove this lemma using an approach based on shadowing. We give here only a sketch of the proof, leaving the details to the reader. By L e m m a 2.2.1, we can find a neighborghood U of S2i such that u n ( ~ ( z ) \ n/) = O. Fix a point x 9 PW(D/). We can take U so small that an analog of Theorem 1.4.3 for 3 holds in U (one can prove it using the same ideas as in the proof of Theorem 1.5.1). Take a small d > 0 and find to such that ?(Xl, ~i)

<

d and ~ ( t , Xl) 9 U for t > 0,

where x I = ~,(t0, X). Find a point y E f2/ such that r(xl, y) < d, and consider the mapping ~P : IR ~ M defined by k~(t)

f ~ ( t , Xl) , t > O,

[

~ ( t , y), t < o.

Define a sequence g = {gk : k E 71} by gk = r(~(1,kD(k - 1)), k~(k)). Obviously, we have Ilglloo < d and

Igkl--+

0 as Ikl ~ ~ .

An analog of Theorem 1.4.3 for flows implies that there exists a point p close to Xl (so that we may assume p to be in U) and a reparametrization a such that

r(~(~(t),p),~(t)) --* 0 as Itl ~ co. Obviously, it follows that X1 E ]/~s(O(p)), and hence x E Ws(O(p)). Thus, it remains to show that p E ~2i. Of course, if ~2/ is a rest point, L e m m a 2.2.3 is trivial, so we consider a basic set /2i such that closed trajectories are dense in it. The trajectory O(p) belongs to a small neighborghood of a hyperbolic set /2i, hence it has properties similar to the ones described in Theorem 1.2.1. Take an arbitrary neighborhood V of p. The same reasons as in the proof of L e m m a 2.2.1 (see also the proof of L e m m a B.6 in [Pi2]) show that there exist closed trajectories Pl, P2 C S'2/ with the following properties: W~(pl) and l,VS(p2) have a point ql of transverse intersection; V~8(pl) and IJY=(p2) have a point q2 of transverse intersection such that -

-

q2EV.

2.2 Shadowing in Structurally Stable Systems

115

It follows that the set O(qt)U O(q2) is a transverse homoclinic contour [Pil], hence the point q2 is nonwandering [Pil]. Since V is arbitrary and the set $2(~) is closed, we have p E (2(~). By the choice of U, this proves that p E ~2+. By Lemma 2.2.2, any trajectory of the flow ~. tends to a basic set as It] --+ oc. By Lemma 2.2.3, for any trajectory 7 of .~ there exist nonwandering trajectories "Yl and 72 such that 7 belongs to the intersection of Ws(Vt) an 142~(V2). It follows now from our definition of the stable and unstable manifolds for a trajectory of ~, that ~]s(~)

=

~]S(~l) and YY~(7) = 14]"(V~).

Since the stable and unstable manifolds for nonwandering trajectories are the images under immersions of class C t of smooth manifolds, for any point x E M we can define the following subspaces of T~M:

S(x) = T,142"(O(x)) and

ld(x) = T.W'(O(x)). Now we formulate the second main definition in the theory of structural stability. D e f i n i t i o n 2.5 We say that the flow ~. satisfies the "geometric STC" (the "geometric strong transversality condition") if, for any trajectories Vt,'~2 C Y2(3), the stable and unstable manifolds )'Ys(-/t) and 142~(V2) are transverse.

The flow F. of system (2.7) is structurally stable if and only if ~. satisfies Axiom A' and the geometric STC.

T h e o r e m 2.2.2 [Robil, Hay, We].

Remark. Robinson proved in [Robi] that the conditions of Theorem 2.2.2 are sufficient for the structural stability of ~ , Hayashi and Wen established their necessity. We give a definition of a (d, T)-pseudotrajectory for ~ similar to Definition 1.24. A mapping ~:IR~M is called a (d, T)-pseudotrajectory for ~ if

r(~(t,~P(r)),~P(t + r)) < d, It] _< T,

(2.11)

for any T E IR. Let us state the main result of this subsection. T h e o r e m 2.2.3. Assume that the flow 3 of system (2.7) is structurally stable.

Then there exist numbers do, L > 0 such that for any (d, 1)-pseudotrajectory ql

116

2. Topologically Stable, Structurally Stable, and Generic Systems

of ~. with d < do there is a point p E M and a homeomorphism (~ CRep(Ld) such that r(~((~(t),p),q/(t)) <_ Ld for t 9 IR (2.12) (the set Rep(e) is defined in Sect. 1.5). Remark. If D' = 0 (i.e., the nonwandering set consists only of rest points), one can take ~(t) = t. Now we describe some additional properties of structurally stable flows. Below the flow ~, is assumed to be structurally stable. It is shown in [Sm2] that we can choose the indices of the basic sets so that W ' ( D i ) N W~(Y2j) it 0 with i r j implies the inequality i > j. L e m m a 2.2.4 [Pu]. There exists a smooth function G : M --+ [0, m] C ]R (a Lyapunov function) such that (1) g2i C G - I ( i ) , I < i < m; (2) if x q~ ~7(3), then the function g(t) = G(:~(t,x)) is increasing. The following statement is a consequence of Theorem 1.1 in [Hirsl]. We prefer to give here a simple direct proof based on the existence of a Lyapunov function described in Lemma 2.2.4. L e m m a 2.2.5. Fix 0 < d < 1 and define the set

= G - ' ( i - d) n W s ( ~ , ) . For any neighborhood Q of :D in H:=G - 1 ( i - d ) , the set 0 +(Q)UI/V~(~2,) contains a neighborhood of ~ . Proof. Fix a neighborhood Q of the set T~ in H and assume that the statement of our lemma does not hold. Then there exists a sequence of points {xm} such that xm ---* S?i as m ~ co, (2.13)

(2.14) and

x,~ ~ O+(Q). It follows from (2.14) that, for any m,

---(t, x,,~) ~ ~?j as t ---, - ~ with j ~r i. By (2.13), for large m we have the inequality

v(zm) < i + 1,

(2.15)

2.2 Shadowing in Structurally Stable Systems

117

hence it follows from L e m m a 2.2.4 that j < i. On the other hand, we have i - d < C(xm)

for large m, hence for any such m there exists a number t,~ > 0 such that ym = Z ( - t m , xm) E H. The set H is compact, let y be a limit point of the set {y,~}. If y e W'(9,),

(2.16)

then ym E Q for large m, hence xm 6 0 + ( Q ) , and this contradicts to (2.15). Thus, to prove our lemma it remains to establish inclusion (2.16). If it does not hold, then we deduce from the inequality 0 < d < 1 and from L e m m a 2.2.4 that y E ~42"(f2k) with k > 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 ) ) <_ i - d for x C H1, t E [O,T]. Since any trajectory on W8(/2~) \ ~ intersects H1 at exactly one point, we see that this number d has the desired property. [] Consider a family E = { E ( x ) : x E M } , where any E(x) is a linear subspace of T~M. D e f i n i t i o n 2.6 A family E as above is called "lower semicontinuous" at p E M

if.for any linear subspace L C E(p) there exists a neighborghood U of p and a continuous subbundle F of TMIu such that F(p) = L, and F(x) C E(x) for xEU. A family E is called "lower semicontinuous" if it is lower semicontinuous at every p E M. Remark. Let E be a lower semicontinuous family as above and let L be a continuous subbundle of T M on an open set U C M. It is easy to see that if E(x) + L ( x ) : T ~ M for some x E U, then

E(y) + L(y) = TyM for all y close to x. It is known [Nil] that on M there exists a C ~ Riemannian metric r' such that for the hyperbolic set J2 we can take C = 1 in item (h.2.3) of Definition 1.27 (for norms generated by r'). Of course, this metric is an analog of the metric generated by a Lyapunov norm (see Lemma 1.2.1), and we call it a Lyapunov metric. The same reasons as in the proof of Theorem 1.2.3 show that if we prove

2.2 Shadowing in Structurally Stable Systems

119

Theorem 2.2.3 for a Lyapunov metric r', then it is true for any metric r (with another constants do, L). Thus, below we assume that r is a Lyapunov metric. Let ~2i be a basic set. Fix a number d E (0, 1), consider the set

V = a - l ( i - d) n W~(~,), and let Q be a neighborhood of 7) such that Q n W"(s2~) = O. Assume that 1) is a continuous subbundle of TMIQ such that

S(x) @ V(x) = T~M for x E 7)

(2.17)

(the spaces S(x) are defined in item (2) of Theorem 2.2.1 and before Definition 2.5). Let us extend the subbundle Y to the set O(Q) u )/y~(/2,) as follows: for y = ~ ( t , x ) , where t E ]R,x E Q, we set -

V(y) = D~.(t, x)l;(x); -

for y E ~Y~'(~2i), we set v ( y ) = G~

(the spaces U~ are elements of the hyperbolic structure for z E /'2(~), for x ~ ~2(~.) they are defined in item (lb) of Theorem 2.2.1).

There exists a neighborhood Q of 7) such that the subbundle )) constructed as above is continuous on the set O(Q) u VV~(~2~).

L e m m a 2.2.7 [Robb].

Remark. Statements analogous to Lemmas 2.2.7 and 2.2.8 were established in [Robb] for a diffeomorphism, our proofs for a flow are based on similar arguments. In the proofs of Lemmas 2.2.7-2.2.9, for a basic set ~2~ we fix a small neighborhood V~ and introduce "coordinates" in TMIv ~in the following way. We take a neighborhood V of ~2i and fix continuous subbundles U r and 8 ~ of T M I v such that u'(x) = u~ for x n V and

s ' ( x ) = s ( x ) for x e

n v

(the continuity of U ~ on )/Y~(~2~) follows from item (lc) of Theorem 2.2.1, the continuity of S on ~/Ys(~21) follows from the same statement of T h e o r e m 2.2.1 and from formulas (2.9) and (2.10)). Note that, for x 6 )4;s(/2,), S'(x) does not necessarily include < X(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 2R]v~'l. Take a positive 6 such t h a t

6<

AR(1 - p)

4

6

' PA +-ff-2-~-~ < A"

It follows from the estimates above t h a t we can take the neighborghood Vi such that, for x E Vi, the inequalities

IIF-II ___2#, IIF=~II, IIF~=tl < 6, IF==v=l >_ 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 <_2~[vgl + 61v~l.

(2.23)

For x E 14~(J2,) we have S'(x) = S(x). Since the family S is D ~ - i n v a r i a n t , for v = v" E S(x) its image F(x)v E S(xx), hence Fs, = 0. This proves the equality v~' = F,,,,v'~ and the inequality

This inequality, inequality (2.21), and e s t i m a t e (2.23) imply t h a t

c~1 < 2#lvgl + 61v~l 6 RIvSI <_ p~o + ~ .

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2. Topologically Stable, Structurally Stable, and Generic Systems

Continuing this process, we show that for any m we have

am ~__pmao+

(l+p+...+pm-l)~pmoto+

R(l__p~.

It follows from the choice of 3 that we can find m0 such that, for any

x E D' = ~.(moT, 01) and any v E ~, the inclination a of v does not exceed AI/2. Since the family 1; is continuous on O(Q), there exists a neighborghood Q' of 7)' such that the inclination a < A for any v E )2(x), x E Q'. (2.24) Take a point x = x0 E Q', a vector v = v0 E Y(x0), and consider the point Xl = ~ ( T , x0) and the vector vl = F(x)vo. Let us estimate the inclination ax of the vector Vl. The value Iv~t is estimated by (2.23). Since now we cannot refer to the equality Fs~ = 0, we give a new estimate for ]v~']. We obtain the inequality

Iv~'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)) <_ i +dl for t E (--0% 0], and similar inequalities are true for G(~(t, yp)) on time segments with lengths tending to infinity, it follows from (2.25) that we can write yp = ~(m(p)T, xp), where xp E Q, and re(p) --+ oo as p -+ or.

2.2 Shadowing in Structurally Stable Systems

123

For x 9 ~V=((2,) N V~ we have U'(x) = U~ the same reasons as above show that in this case F=, = 0. Fix arbitrary e > 0 and find a neighborghood Y(e) of the set W~(J2,) n V~ such that for x 9 Y(e) we have e ( R - 6)(1 - p) IIF.,II < ~1 =

Since O - ( y ) C V(e), there exist numbers m,(p) such that ~ ( m T , xp) 9 V(e) for ml(p) <_ m < re(p) and u(p) = re(p) - ml(p) ---* c~ as p --* cx~ (in addition, we can take ml(p) > m0). Denote z v = ~ ( m l ( p ) T , xp). Take a vector Vo 9 ~2(zv) and let vk = Fk(zp)vo. For 0 < k < u(p) we have the estimates

and

]V~.4.11 >__Rival- 61v~l, hence for the inclinations we have el

al < pA + R---Z-~,

ak < pk A +

s

( R - 5)(1 - p ) '

and it follows that ak < e for large k. Since u(p) ~ oo as p --+ 0% we see that, for large p, the space V(yp) is e-close to l)(y) = U~ This proves the desired continuity of l). If the basic set Di is a rest point, the proof is easier, since for the corresponding operator F on J2i we have the estimates

IIFs~ll, II(F=~)-lll _< AoT9 We leave the details to the reader 9

L e m m a 2.2.8 [Robb].

The families {S}, {/4} are lower seraicontinuous.

Proof. We prove our lemma for the family {/4}, for {S} the proof is analogous. Let us apply induction in i to show that for a basic set ~2i there exists a neighborhood Zi of ~i and a continuous subbundle Gi of TM[o(zd such that

a,(x) c u~

o(z~)

(2.26)

for x 9 w ~ ( ~ ) .

(2.27)

for 9 9

and

~ ( x ) = u~

It follows from formula (2.10) and Definition 2.6 that in this case the family {/4} is lower semicontinuous in a neigborhood of )/Y~(Y21).

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2. Topologically Stable, Structurally Stable, and Generic Systems

To establish the induction base, we consider minimal (with respect to the order introduced before Lemma 2.2.4) basic sets. A set f2i is minimal if and only if WS(J2,) gl W"(J2j) = r for i r j, it is easy to see that in this case WS(12~) = 12i, and W~(12~) is a neighborhood of Y2i. If f2~ is a minimal basic set, then the subbundle Gi(x) = U~ for x 9 }4]u(•i) has the desired properties. Thus, we assume that, for j < i, the corresponding neigborhoods Z/ and subbundles Gj are constructed. Fix d 9 (0, 1) and consider the set

V = G-l(i - d)

n

W'(~?d.

Take a point x 9 :D, there exists an index j < i such that x 9 W~(g?j). It follows from the geometric STC that

U(x) + S(x) = T~:M. Hence, there is a subspace ~' of U~

such that

~' @ S(x) = T,M. By the induction hypothesis, there exists a neighborhood Z(x) of x and a continuous subbundle G} C ~j of TMIz(, ) such that

~}(y) C V~

for y 9 Z(x)

(2.28)

and =

Decreasing Z(x), if necessary, we may assume that Z(x) C V~ and that

~ ( y ) @ S'(y) = T~M for y 9 Z(x)

(2.29)

(recall that S' and U' are chosen to introduce "coordinates" in TMIv ~). It follows from (2.18) and from (2.29) that g~ can be written as the graph of a continuous bundle mapping :

--, s'la

).

Let B be a closed neighborhood of 79 in G-l(i - d) such that

B c U z(x). Take a smooth partition of unity {/~(x) : x 9 T~} subordinate to the covering {Z(x) : x 9 Since the set B is compact, all but a finite number of the/~(x) are identically zero. Define gB : UqB --* S'IB by

xeT)

2.2 Shadowing in Structurally Stable Systems

125

and let ~i C TM[B be the graph of gB. By construction, ~i is a continuous subbundle of TMIB. A fiber ~i(Y)is a convex combination of ~}(y), hence it follows from (2.28) that GI(Y) C V~ for y E B. (2.30) Now we define ~i on O(B) as follows:

G,(x) = D 3 ( t , y)Gi(y) for t E IR, y E B (this definition is reasonable, since any trajectory in O(B) intersects B at a unique point); ~,(x) = U~ for x E W=(~2~). Since the family U ~ is D~-invariant (see item (lb) of Theorem 2.2.1), we deduce from (2.30) that ~i(x) C U~ for x E O(B). It follows from Lemma 2.2.7 that if the chosen neighborhood B is small enough, then the constructed subbundle G~ is continuous on O(B) U 142'(s and by Lemma 2.2.5 this last set contains a neighborhood of f2~ (and hence of I,V~(D~)). Denote this neighborhoood by Z~ to complete the induction step. [] Now we are able to give a proof of the following statement from [Robil]. L e m m a 2.2.9 [Robil]. There exist disjoint neighborhoods V~ of the basic sets F2i, continuous subbundles Si, Ui o f T M ]V,uo(vd , and a number ~1 E (0, 1) such that (1) Si, Ui are DF.-invariant;

(2) w, I n = w o l a , , w = s,u; (s) sj(x) c sk(x), vk(~) c v~(x) for x e O+(V~) n O-(Vk); (4) Si(x) @ Ui(x) = T~M if s

C ~2~

Si(x) @ Ui(x) @ < X ( x ) > = T~M if Y21 C f2';

(5) if x E Vii,v" E Si(x), and v ~ E Ui(x), then

IDZ(t,x)v~l < )~I,~I, 0 < t < 1;

ID~(-t,x)v~l <~ ),~1r

o < t < 1.

Pro@ We show how to construct the subbundles Ui, for Si the construction is similar. We proceed by induction assuming that Uj and Vj have been defined for 1 _< j < i, and that they satisfy analogs of statements (1)-(5) (of course, we have in mind only the statements about Uj). In addition, we assume that one more induction hypothesis is satisfied, Uj(x) + $(x) = T~M for x E O(Vj), 1 _< j < i.

(2.31)

126

2. Topologically Stable, Structurally Stable, and Generic Systems We also assume that the constructed neighborhoods Vj are closed and that

where G is a Lyapunov function given by Lemma 2.2.4. We take closed neighboghoods ~' of ~?j such that ~' CIntVj for 1 _< j < i, and construct Ui on ~ so that properties (1)-(5) hold with the neighborghoods

VL..., V/i, By Lemma 2.2.3, any point x E M belongs to a stable manifold Ws(3'), where 3' C / 2 ( Z ) . By statement (3) of Theorem 2.2.1, there is a point p E ~ ( S ) such that x E WS(p) (obviously, W~(x) = WS(p)in this case). Choose a neighborhood Vii~ of ~i such that 1 i+ 88 qo c c - l [ i - ~, and fix continuous subbundles U' and S' on TMIv such that

U'(x) = U~

for z C ]'V~(~,) M V

and S'(x) = S(x) for x e Ws(n~) n Y. We take the neighborhood V~~ so small that (2.18) holds for x E V~~ Apply Lemma 2.2.6 to find d > 0 such that the sets

G-'( i - d')

M ~'Vs([2i)

are subsets of V/~ for d' E (0, d]. Denote z) = G - l ( i - ct) n w s (

It follows from Lemma 2.2.2, from our choice of indices for the basic sets, and from the induction hypothesis that V C W"(~2,) C U

]'v~'(/2J) c

i<j
U

O+(IntVj) 9

i<j
For q :> 0 and for a set V we denote

vq = U s ( t , v ) . O<_t<_q

Since the set D is compact, there exists q > 0 such that DC

U

~ 'q .

l<j
Take a point x E 7) and define j = j(x) by the conditions x E Vjq

(2.32)

2.2 Shadowing in Structurally Stable Systems

127

and x ~ V~q for j < k < i.

(2.33)

Since the neighborghoods ~' are closed, we can find a neighborghood Z(x) of x such that

Z(x) C VJ and Z(x) n Vff = O, j < k < i. We claim that

Z(x) rl O+(V~) = 0 for j < k < i.

(2.34)

Indeed, assume that there is a point y C Z(x) C Vjq such that

y E O+(V/~) \ Vk'q for some j < k < i. Then y E ~(s, Vk') with some s > q, hence y'=~(-s,y)

1 EV~, a n d G ( y ' ) > k - ~ > j + 4 .

1

The Lyapunov function G is increasing along trajectories, this gives 1

G(F.(t,y)) > j + ~ for - s < t < 0 . Therefore, ~.(t,y) ~ Vj for - q ~ t

<0,

y ~ VJ. The obtained contradiction proves (2.34). It follows from (2.31) that there exists a continuous subbundle B' of TMIz(= ) such that B'(x) C U~(~) and

T , M = B'(x) | 8'(x) (note that both conditions above are imposed only on the space B'(x)). Let

rr : T M I z ( , ) --+ Ujlz(, ) be the orthogonal projection. Set B (z) = ~rB'. If the neighborhood Z(x) is small enough, this gives B (~) with the properties

B(~:)(y) C US(y) and TyM = B(~)(y) @ $'(y) for y E Z(x).

(2.35)

It follows from (2.18) and (2.35) that B (~) can be written as the graph of a bundle mapping b(=) : U'[z(=) ~ $'[z(=). Let Q be a closed neighborghood of 1l) in G-'(i - d) such that

Q C U Z(x). xe7)

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2. Topologically Stable, Structurally Stable, and Generic Systems

Take a smooth partition of unity {/~(x) : x E 7:)} subordinate to the covering {Z(x) : x E T~}. Since the set Q is compact, all but a finite number of the ~(x) are identically zero. Define bz~ : U'IQ --~ S'IQ by b" =

~e:P and let B v C TM[Q be the graph of bv. By construction, B v is a continuous subbundle of TM[Q. Now we want to show that

BY(y) C Uk(y) for y E Q M O+(V~) with 1 _< k < i.

(2.36)

A fiber BY(y) is a convex combination of B(~:)(y), where y E Z(x), thus it is sufficient to show that

B(')(V) c u (y) for y e z ( z ) n o + ( v ; ) , x Let j = j(x) satisfy (2.32) and (2.33). By (2.34), we have k < j. Then it follows from (2.35) and from the induction hypothesis that

c u (v) c This proves (2.36). Now we define Ui on O(Q) as follows. For y = ~ , ( t , x ) , x E Q, we set Ui(y) = D~(t, x)BV(x). On 14;~(~2,), we let U, = U ~ By Lemma 2.2.5, the set O+(Q)UW~(J?i) contains a neighborghood V~ C V~~ of the basic set ~2i. By construction, Ui is D~-invariant, and it coincides with U ~ on 14;~(~2i), this proves statements (1) and (2) of our lemma. Since G is increasing along trajectories, we have O+(Vi) M Vj = 0 for j < i, now (2.36) implies statement (3). By Lemma 2.2.7, Ui is continuous on O(V~) if Q is small enough. Since for x E ~2i analogs of the inequalities in statement (5) of our lemma hold with )~o, we can take any A1 E ()~0, 1) and take a smaller neighborghood V~ (if necessary) to establish (5). To complete the induction step, it remains to check (2.31). Since for x E 1/Y~($21) M Vi~ we have

U~(x) + S(x) = U~

+ S(x) = T~:M

due to the strong transversality condition, since the family Ui is continuous, and since the family S is lower semicontinuous (see Lemma 2.2.8), it follows from the remark after Definition 2.6 that we can reduce V~ to satisfy (2.31). It remains to establish the induction base. Take a basic set I2~ such that Ws(s MW~(f2j) = 0 for i ~ j. Obviously, we can apply the same construction as above taking B v equal U' on Q. []

2.2 Shadowing in Structurally Stable Systems

129

If V is an arbitrary neighborhood of S2(~), then there exists [Bi] a number To > 0 (usually called the Birkhoff constant) such that if for x E M X(t, x) it V for t E (9, then mesO _~ To (here mes is the Lebesgue measure). Fix a Birkhoff constant To for V = V1U... Vm (where V~ are given by Lemma 2.2.9). Now we construct a family of linear subspaces S(p), U(p) C TpM, p E M, with special properties. Take p E M. There exists to E [0, To] such that q = -~(to, p) E Vl for some l, and ~ ( t , p) • Vk with l # k for t E [0, to]. (2.37) Set

W(p) = D ~ ( - t o , q)l/Vt(q), W = S, U.

(2.38)

We say in this case that the point p takes the subspaces (t.s. below) from Vt. Take points p, q = ~ ( t , p ) , a neighborhood W of q belonging to a chart of M, and introduce local coordinates in T M I w as follows. Let ~1 : W--.-~ Wo C ]Rn

be a coordinate mapping. We consider a coordinate mapping

~ : T M [w -~ Wo x IR'~ of the form Z(x, v) = ( z , ( x ) ,

B(x)v)

for (x, v) E T M I w . L e m m a 2.2.10. Let q = ~ ( t , p ) . Then

(1) D E ( t , p ) S ( p ) C S(q), t >_ 0; D Z ( t , p ) U ( p ) C U(q), t <_ 0; (2) if rm --~ q for m --~ 0% then for large m there exist linear isomorphisms IIm : IR'~ --~ IRn with the following properties: - III

o for m -,

and IIm(B(q)D~.(t,p)S(p)) C B(rm)S(rm) for t >_ To; Hm(B(q)DY.(t,p)U(p)) C B(r,~)U(r,~) for t < O. Proof. The first statement follows immediately from the definitions of S(p), U(p), and from statements (1), (3) of Lemma 2.2.9.

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2. Topologically Stable, Structurally Stable, and Generic Systems

Before we prove (2), note that all the limits below in the proof exist due to the continuity of the subbundles Si, Ui in V~. First we consider the case t > To. There exists to E [0, To] such that p' = ~(t0, p) C Vt with some l, and (2.37) holds, so that p t.s. from Vl. Set a = t - to, then , = .~(-a, r,~

rm) ---* p, as m +

c~,

hence r " E Vt for large m. The continuity of St(x) in Vt implies the relation

s,(/)

s,(p').

Consequently,

D~.(t, p)S(p) = D ~ ( x , p')&(p') = = mlira - ~ D.~(a, r'r~)&(rm) = lirnoo DZ(~,/m)S(r~). By the first statement, '

=

'

D-(tc,rm)S(rm) C hence for large m there exist isomorphisms//~ such that II/I= - Ill ~ 0 and llm(B(q)S(q)) C B(rm)S(rm). Let us prove the second statement in (9). Assume that the point p t.s. from V/, that is, there exists a point

p ' = ~(l,p) E Vi, /_>0, such that

U(p) = D~(-l,p')Ui(p'). Since ' rm

s(-t

+ l,

--+ p',

we have

limr U(r~) = U(p'). It follows from statement (1) that

D ~ ( t - l , r ' ) U ( r ' ) C U(rm), hence

O.~(t,p)U(p) = O ~ ( t -l,p')U(p') C lirnoo U(rm), and this implies the existence o f / / m . Obviously, there exists a constant Co such that IlOZ(t, x)ll _< exp(Coltt)

[]

2.2 Shadowing in Structurally Stable Systems

131

for all t E JR, x E M (since D 3 ( t , x) satisfies a variational system analogous to (1.207), one can take Co -- max IIDX(p)II). pEM

Set C = exp(CoTo),V~ T~ Take a point p E M. Let us show that for v E S(p) we have

ID~(t,p)v I < CA~Iv I for t > O.

(2.39)

Since D ~ ( O , p ) = I, for t = 0 our e s t i m a t e is obvious. Consider t > O, let O be the following subset of [0, t]:

o = {~ 9 [0,t] : 3(~,p) ~ v}. It follows from the definition of the Birkhoff constant that mesO < To. For arbitrary e > 0 there exist open segments (a~, b~), i = 1 , . . . , N, such that (bl - el) + . . . + (bN -- aN) < To + e

(2.40)

and N

0 C U [ai, bi]. i=l

Take v 9 S(p) and set

v7

=

DF.(ai,p)v, v +

=

D~(bi,p)v, v ' = D~.(t,p)v.

Since ~ ( r , p) 9 V for

9 (0, al) u (b,, a~) u . . . u (bN,t), statement (5) of Lemma 2.2.2 implies the inequalities

IvTI _< a;'lvl, Iv;I _< ~;2-bllv+l, ..., Iv'l _< ~-b~lv+NI. On the other hand, it follows from the definition of Co that

Iv+l <

exp(Co(b~ -

ai))lvTI.

Hence, we deduce from (2.40) that Iv'l < exp(Co{(bl - el) + . . . + (bN -- aN)})A~a'+~-bl+'"+t-bN)lvl _< _< exp(Co(To + e))At~-(T~

Iv I = A(e) exp(CoTo)A~-ToA~Iv I

with A(e)~lase~0. Obviously, this proves (2.39). Similarly one shows that for v 9 U(p) we have

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2. Topologically Stable, Structurally Stable, and Generic Systems

ID~(t,p)vl <_ CA-~tlvl for t _< 0.

(2.41)

Now we consider a (d, 1)-pseudotrajectory ~P of system (2.7). Let us fix a Lipschitz constant /: for the vector field X on M. Fix a natural number T. The same reasons as in the proof of Lemma 1.1.3 show that ~P is a (dT, T)pseudotrajectory of (2.7) with

dT = d(1 + exp(s

+ . . . + exp((T - 1)/~)).

It is easy to understand that if we prove an analog of Theorem 2.2.3 for (d, T)-pseudotrajectories, then the same result holds for (d, 1)-pseudotrajectories (with other constants do, L). Take # E (0, 1) and find a natural T such that T > To + 1, CA T-1 < #.

(2.42)

For p E M we set

Z(p) -- S(p) + V(p). Note that it follows from our constructions that

Z(p) = S(p) 9 U(p). Estimates (2.39), (2.41), the second inequality in (2.42), and the uniform continuity of the families Si, Ui on Vi imply the following statement.

The subspaces S(p), U(p) have the properties (1) there exists N > 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[ <_ T + l.

(2.43)

Now we fix local coordinates in T M as follows. For p E M let expp : TpM --* M be the standard exponential mapping generated by our metric r. For c > 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 <_ l hold, then there is a linear isomorphism II(p, z) : TzM --+ TzM such that I I / / - Ill <- fl, lI(p,z)(Dexp;'(q)DZ(t,,p)S(p))

c S(z)

(2.45)

and a linear isomorphism 6)(p, z) : TpM --* TvM such that [[6~ - I [ [ <_ fl, (9(p, z)(Dexp;l(y)D~(t2, z)U(z)) C U(p).

(2.46)

For p E M and a E (0, c) denote

Z~(p) = Z(p) N E~(p), S(p) = expp( Zc(p) ), S~(p) = eXPv( Z~(p) ). Obviously, S(p) is a smooth disk (with boundary) transverse to X(p) at p. We say below that a point p E M is of type (R) if Z(p) = TvM (in this case dimZ:v=dimM), and of type (O) if Z(p) 7~ TvM (in this case dimZ:v--dimM- 1). Let p be a point of type (O). The subspaces S(p), U(p) are defined by (2.38). Since St, Ul are continuous, their sum is transverse to X in Vt, and to _< To, we see that the angle between Z(p) and X(p) and the norm IX(p)[ are separated from zero (by a constant independent of p). Hence, the following two statements hold. L e m m a 2.2.13. There exist constants dl > 0, eo < 1, K1 >__ 1 such that if z is of type (0) and r(z,q) <_ da, then there is a unique scalar function f ( x ) in the dl-neighborhood of q such that ]f(x)] _< eo and

134

2. Topologically Stable, Structurally Stable, and Generic Systems

~ ( f ( x ) , x) 9 E(z). This function f is of class C 1 and If(x)l < K~dist(x, X(z)), r(z, ~ ( f ( x ) , x)) < g~r(x, z). Remark. This lemma is analogous to Lemma 1.5.2, we do not prove it here. The following statement is proved similarly, we leave details to the reader.

L a m i n a 2.2.14. There exist positive constants d2 <_ dl, b < c with the following properties. If z is of type (0), r(~(T, p), z) ~ d2, then there is a function T(x) of class C 1 on Zb(p) such that

r

:= S(v(x), x) 9 S(z), Iv(x) - TI _~ i.

If p is of type (0), r ( ~ ( - T , z ) , p ) C 1 on Xb(z) such that r

:=

< d2, then there is a function TI(X) of class

S(TI(X),X)9 ~(p),

Remark. Obviously, the norms of D r 1 6 2 below we assume that IIDr

IIDr

[Vl(X)+ T [ _ 1. are uniformly (in p,z) bounded,

~ K.

In addition, D r 1 6 2 are uniformly continuous in the following sense: given fl > 0 there is ~ > 0 (depending only on #) such that if Xl, z2 9 L'b(p), x* ----expp-l(xi), y, ----r and r(xl, x2) _~ ~, then

IlDexp~l(yl)Dr

- Dexp;l(y2)Dr

<

(and the same holds for Dr Let us prove this uniform continuity for De. In this proof, the existence of a uniform estimate for some value a means the following: there exists a number b such that for any point p 9 M we have the estimate of the type lal < C(b) in the b-neighborghood of p, where the function C depends only on such global properties of system (2.7) as max IX[, max IIDX(x)ll, and so on. Fix local coordinates in neighborghoods Wp, W, of the points p, z, respectively. We may assume that z is the origin of W~, and that in local coordinates y of W~ the disk ~P(z) belongs to the hyperplane yl -- 0. Without loss of generality, we may assume that Z ( z ) in W~ is given by the linear relation

2.2 Shadowing in Structurally Stable Systems

135

y = Ls, s E IR ~-1, where elements llj of the matrix L have the form l~i = 5(i,j + 1) for 1 < i < n, 1 < j <_ n - 1 (here 5(i,j) is the Kronecker symbol, 5(i,j) = 1 for i = j, 5(i,j) = 0 for i r j). Let us preserve the notation ~ for the flow in the chosen local coordinates. Denote p' = r E ~ , and let p' = ~(t0, p). Consider, for x E Wp, t E IR, and s E IR=-1, the function

F(x, t, s) = ~(t, x) - Ls. Obviously, a point y = Ls E Zz is the image of a point x E Wp under r if and only if there exists a number t close to to and such that F(x, t, s) = 0. Assume that p' = Ls'. Then F(p, to, s') = O. Consider the matrix OF

.

A - O-~,s)(p, to, s ) = [ X ( p ' ) , - L ] (we preserve the notation X for the vector field). The vector X(p') has nonzero angle with the hyperplane S ( z ) , hence detA # O. By the implicit function theorem, in a neighborghood W' of p there exist functions r(X), cr(X) of class C 1 such that T(p) = to, a(p) = S', and

F(x, T(X), a(X) ) = 0

(2.47)

in this neighboghood. Obviously, T(X) is the function given by Lernma 2.2.14, and r -- LiT(x) for x E W'. For x E W' denote A(x) = [ X ( r Denote by X1 the first component of the vector X in the local coordinates of Wz. If a basic set tDi is a rest point of ~ , then it follows from the construction of the families S, U before Lemma 2.2.10 that any point in the corresponding neighborhood Vi of ~2i is of type (R). Hence, there exist constants cl, c2 > 0 such that for any y in a cl-neighborghood of a point z of type (O) we have the estimate IX(y)l >_ c2. The angle between the vector X ( r (for x in a neighborhood of p) and the hyperplane Yl = 0 (containing Z(z)) is uniformly separated from 0, hence there is Ca > 0 such that for Xl(r the projection of X ( r to the Yl axis, the inequality Ixl(r _> c 3 1 x ( r _> c4 = c c3 holds. By the structure of the matrix L, IdetA(x)l = IXl(r the values of IIA-l(x)ll

> c4. Hence,

136

2. Topologically Stable, Structurally Stable, and Generic Systems

are uniformly bounded. The second derivatives of components of F with respect to t, sl are also uniformly bounded. It follows from the standard estimates in the implicit function theorem that the radiuses of W' can be chosen uniformly separated from 0. Set B(z) = A-l(x). Differentiate (2.47) to obtain

OF OF Or OF Oa 0---~+ -O-i-~z + Os Oz = o. This gives the equality

OF O~

= A(x)r[or G ' OxJ

(* means transposition). Thus, we can express

Oa -~x-

B~

OF

where B0 is a submatrix of B. It follows that D e ( x ) = - L Oa = -LBo(x)

Ox

The matrix

a(z)-

OF Oz"

OF Ox

coincides with D~.(v(x), x). Since the times v(x) are uniformly bounded, the matrix G(x) is uniformly continuous in the following sense: given r > 0 there exists a > 0 such that if r(Xl, x2) < c~, then IIa(~l) - C(x2)ll <

~.

Standard calculation shows that

OB_~x_

A-I~xA-I(x)"

Since the value

Oz is uniformly bounded, this obvously proves our remark (since the derivatives of exp, exp -1 are also uniformly bounded).

Given j3 > 0 there exists ~ > 0 such that, for any z,p E M, the following holds for T, T1, r r defined in Lemma 2.2.14. If z is of type (0) andr(.~(T,p),z) < a, then for anyv E S(p) with Ivl = 1 we have J(D~.(r(p),p) - Dr I < ~.

L e m m a 2.2.15.

2.2 Shadowing in Structurally Stable Systems

137

If p is of type ( 0 ) and r ( ~ ( - T , z ) , p ) < a, then for any v 9 U(z) with Ivl = 1 we have [(DZ(rl(z), z) - DOl(Z))V t < /3. Proof. We prove only the first statement, for the second one the proof is similar. Take v 9 S(p) with ]v I = 1. Let 7(s), s 9 [0,1], be a smooth curve such that 7(0) = p, ;/(0) = v. Denote TO = r(p),

~ ( s ) = 2 ( , 0 , ~(s)), ~2(s) = r

= z ( , ( ~ ( s ) ) , ~(s)).

There exists so > 0 such that if a < dl (see Lemma 2.2.13), then, for s 9 [0, so], the function T*(S) = f(71(S)) is defined. Denote va = D ~ ( r o , p ) v , v2 = Dr Obviously, d 731 : ~ s 1 (0) = ~,~(T0,~'(S))Is=0,

v2 =

(0) =

S(ro + T*(S), 7(S)) [8:o = X(q)--~-s (O) + vl,

(2.48) (2.49)

where q = ~(T0,p). Set v' =- Dexp-~l(q)vl. By Lemma 2.2.12, given/3 a > 0 there exists al > 0 such that if a _< al, then there is an isomorphism H ( p , z ) : T z M --+ T z M with

IlII(p,z) - I11 <_/31, H ( p , z ) v ' E S(z) C Z(z). Hence, given/32 > 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))
I~-*(s)l _< K1/33s, s e [0,Sl], and

(0) ~ K1/33.

It follows from (2.43) that

x(q)

(0) < KK1/3~.

Since/33 can be taken arbitrary, it remains to compare (2.48) with (2.49) to complete the proof of our lemma. []

138

2. Topologically Stable, Structurally Stable, and Generic Systems

Take v0 E (0,1) such that A = (1 + Vo)2# < 1. For this ~ and N (from Lemma 2.2.11) take the corresponding Nt (Theorem 1.3.1) and find ~ > 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 <_ 1 + u M ' Ivl

(2.51)

hold. We fix s such that if

r(z, q) < d', r(y,p) < d', then for the linear isomorphisms {9(p, z), II(p, z) (see Lemma 2.2.12) we have

IIH - / l l , IIO - Ill, II69-1 - 111 -< u.

(2.52)

Now we prove Theorem 2.2.3 for (d, T)-pseudotrajectories. Let # be a (d, T)pseudotrajectory, denote zk = # ( k T ) , k E 7/. Set Hk = Z(xk). Fix a pair xk, Xk+l. Denote p ---- X k ,

Z :

X k + 1.

We consider three possible cases. Case 1. z is of type (R). In this case, define a diffeomorphism ~ by ~(x) = F~(T, x) and let Ck = exp~-1 o ~ o expp. Obviously, we can take d' such that Ck is defined on Zx(p) and 4-1(z) C expp(E,(p)). It follows from (2.44) that

OCk(O) = Oexp-~l(q)D~(p) : Hk --~ Hk+l, where q = ~(p). Denote J = DCk(0). We can write Ck(~,) = J v + x(v) with x(O) = Ck(O), Dx(O) = O. Note that

ICk(0)l = l exp~-l(~(p))[ = r(~(p), z) < d.

2.2 Shadowing in Structurally Stable Systems

139

Since the derivatives of ~, expv are uniformly continuous, there exists d' such that

Ix(v) - x(v')l _< ~lv - v'l for v,

9

Zd,(p).

(2.53)

Set

ASk = II(p, z ) J P ( p ) , A~ = F O - l ( p , z)Q(p), where

F = D~(y)Dexpp(y*): Ilk -+ Hk+l, and y -- ~-l(z), y* = exppl(y). Below for I I ( p , z ) , O ( p , z ) given by Lemma 2.2.12 we write / / instead of H(p, z) etc. We represent Ck(v) in form (1.51), where

Ak = ASk + A~, Wk+l(V )

=

(J - Ak)v + X(V).

Take v s E S(p). By Lemmas 2.2.10 and 2.2.11, vl = D~(p)v ~ 9 S(q) and Iv~l < #lvq. By (2.51), for v~ = Dexp-;l(q)v~ we have the estimate ]v~l _< (1 + u)Iv;I. Since IIv~ 9 S(z) and (2.52) holds, we see that v~ = IIv~ = A~v ~ 9 S(z) and [v~l _< (1 + v)Iv~[ _< A[vq. Hence,

A~S(p) C S(z), [IA~ Is(p)II <-- A.

(2.54)

Now take v ~ 9 U(z) and let

U'(z) = D ( - l ( z ) U ( z ) and U"(z) = Dexpp-l(y)Ut(z). By Lemmas 2.2.10 and 2.2.11, v~' = D ( - l ( z ) v ~ 9 U'(z) C U(y) and Iv~'l < #lv~[. For v~ = Dexpp-l(y)v~ 9 U"(z) we have Ivy[ < (1 + u)lv~l (see (2.51)). Now (2.52) implies that for v~ = Ov~ 9 OU"(z) C U(p) we have Ivy] < A[v~l. Set Bk = OG, where

G = Dexp;l(y)D(-l(z):

Hk+l --~ Ilk.

Since v~ = Bkv ~, we see that

BkV(z) c U(p), IIBk Iu(=) II -< A.

(2.55)

Take w = v3~ 9 OU"(z), then Qw = w, O - l w = v2,~ and

sO-iT

= Fv'~ = D~(y)Dexpv(y*)v'~ = D~(y)v'~ = v",

hence

A~Bk

Iv(z)= I.

(2.56)

Since Ak Is(p)= A~, AkBk Iv(z)= A~Bk Iv(z), it follows from (2.54)-(2.56) that Ak, Bk satisfy conditions (1.63) and (1.64) of Theorem 1.3.1.

140

2. Topologically Stable, Structurally Stable, and Generic Systems Obviously, there exists d' such that if r(q, z) < d', then IID~(y)Dexp,(Y*) - Dexp2 a(q)D~(p)ll <_ u. For d < d' let us estimate [[J - A~[I = [[J(P + Q) - Ak]] _< ]]JR - ASk][ +

[IJQ -

A~]] =

= ]IJP - HJPI[ + IlJQ - F O - ' Q I I .

(2.57)

Since [[P[I < K,]]J][ <: (1 + u ) g < 2 K (see (2.51)), and [[H - I]] < u (see (2.52)), the first summand in (2.57) does not exceed 2K2u. Let us estimate the second summand, []JQ - FO-1QI[ <_ K l l J - FO-1]] : K I I F ( O - ' - I) + F - J]] < <_ 2K2u + K[ID~(y)Dexpp(y* ) - Dexp21(q)D~(p)]] <_ u K ( 2 K + 1). This gives ] ] J - akll _< 2K(2K + 1)u < a/2.

(2.58)

Now we deduce from (2.53) and (2.58) that if d _< d', then Ck satisfies conditions (1.63)-(1.65) of Theorem 1.3.1 with A = d' (of course, we identify Pk with P(p) etc). Case 2. p and z are of type (O). By Lemma 2.2.14, the functions % T1 and mappings r r are defined on ~b(P), ~b(Z), respectively. Denote q= r

y : el(Z), y" = e x p ; l ( y ) , q* = expzl(q), at : 3 ( T , p ) .

Set Ck = expz-1 o r o expp. Then Jo =DCk(0) = Dexp2a(q)Dr denote J = Dexp21(q)D~(T(p), p). By Lemma 2.2.13, the inequality r(z, q') < d < d' implies r(z, q) <_ Kid, hence ICk(0)l = Iz - q*l -< Kad. It follows from the properties of r (see the remark after Lemma 2.2.14) that we can take d' such that

r and for d < d' (2.53) holds. Set F = D ~ ( r ( y ) , y)Dexpp(y ), G = D exp'~l(y)DF.(rl(Z), z) and note that T ( y ) = - r l ( z ) , hence

2.2 S h a d o w i n g in S t r u c t u r a l l y S t a b l e S y s t e m s

141

F G = I.

(2.59)

We again take A~ = II(p, z ) J P ( p ) (O(p, z), II(p, z) are given by Lemma 2.2.12 for the points q, y fixed above). The same reasons as in case 1 show that (2.54) holds (note that D.~(r(p),p)S(p) C S(q)). Set Bk = OG and U' = BkU(z) C U(p). Repeat the proof of case 1 to show that (2.55) holds. Let U* be the orthogonal complement of U' in U(p). Represent Q(p) = Q' + Q*, where Q', Q* are the corresponding projectors. Obviously,

tIQ'II, IIQ*II -< K. Set

A'~ = FO-XQ' + JoQ*. For v 9 U(z) we have v' = Bkv = OGv 9 U ~, hence Q*v' = 0 and

A'~Bkv = F O - I O G v = v (we apply (2.59) here). This proves (2.56). Now we take Ak = A~ + A~ and represent Ck(v) in form (1.51) with wk+i(v) = (Jo - Ak)v + X(V). Apply Lemma 2.2.15 to find d3 = a corresponding to/3 = u. Take d4 > 0 so small that if d _< d4, then

IInr

exp,(q*) - Dexpp -1 (y)nr

_~ p,

(see the remark after Lemma 2.2.14). For d < d' < min(d3, d4) let us estimate I l J o - Akl[ _< IIA~ - JoPll + IIA's JoQJ[. Consider the first summand. By Lemma 2.2.15,

I I J P - & e l l < K(1 + v) --

max

" veS(p),lvl--1

I(D~(T(p),p) -- Dr

<_

< K(1 + v)v < 2Kv, hence

]IA~ - YoP]l = ][IIJP - YoPII ~

II//-

Ill. ][JPI[ + I]J P - yoP]] _<

<_ 2 K ( K + 1)u.

(2.60)

Now consider IIA~ - JoQll = IIFO-IQ' + JoQ* - Jo(Q' +

Q*)[I =

IIFO-IQ ' - JoQ'll <

_< IIF(O -1 - I)Q'l[ + I I F Q ' - JoQ'JJ.

(2.61)

The first summand in (2.61) does not exceed 2K~v. Let us estimate the second one. Take Vo 9 Z(p), Ivol = 1, let w = Q'vo, we have Iwl < Kivol = K. There

142

2. Topologically Stable, Structurally Stable, and Generic Systems

exists v E U(z) such that w = Bkv. Since v = F O - l w , we have Ivl _< 4Klw j < 4 K 2. We obtain

FQ I Vo - JoQ t Vo = F O G y - JoOGv = = F(o

- I)av - Jo(O - I)av +Fav

- JoGv.

Obviously,

IF(O - I)Gvl <_ 4 K 2 u H <_ 16K4u, IJo(O - I)Gv] <_ 16K4,. Since F G v = v, we come to the estimates I F a v - J o a v l = Iv - JoGv[ = [Jo(jolv - Gv)[ <

< IIJ0ll(J(yo ~ - Dexppl(y)Dr

< 2K[(Dr

+ [(Dexp-~l(y)Dr

-- G)v[) _<

- D exp~l(y)Dr

+2gI(Dexp;~(y)Dr

(2.62)

- Dexp;l(y)D3(7~(z),z))v[ <

(2.63)

< 6Kulv I < 24K3u. Note t h a t we apply the choice of d4 to e s t i m a t e (2.62) and the choice of d3 to e s t i m a t e (2.63). It follows from our estimates that IIAk - J0ll _< 4 K ( 8 K3 + 6 K2 + K + 1)v < a/2. Thus, for d < d', Ck satisfies conditions (1.63)-1.65) of T h e o r e m 1.3.1. Case 3. p is of type (R) and z is of type (O). Consider r and r defined on 22b(p) (see L e m m a 2.2.14). Take q, q*, Ck, J, Jo the same as in case 2. Represent Ck(v) = dov + X(v), we assume t h a t for d <_ d' (2.53) holds. Note t h a t similarly to case 2 we have ICd0)l - Kid. Denote r0 = r(p) and define a diffeomorphism ~ by ~(x) = ~ ( r o , x). Denote E' = ~-l(~b(z)),

v = ~ - l ( z ) , y* = e x p ; i ( y ) .

Obviously, p E Z", and for x e Z" we have ~(x) E Z'z. Hence, r ~ ' , and D~(p)w = Dr for w e TpZ'.

= ~(x) on (2.64)

Set

F = D((y)Dexpp(y*), G = Dexp-~l(g)D(-~(z). Consider O(p, z), II(p, z) given by L e m m a 2.2.12 for p, z, q, y. Define A~, Bk, U', I U * ,Q,Q*,A'~,Ak as in case 2. T h e same reasons as in case 2 show t h a t (2.59),(2.54)-(2.56) hold. Let us e s t i m a t e I IA~ - JoQIt. Since derivatives of exp, are uniformly b o u n d e d and It0 - T] _< 1 (see L e m m a 2.2.14), we can find d' such t h a t if d _< d', then the following two s t a t e m e n t s are true.

2.2 Shadowing in Structurally Stable Systems

143

If v E Ty2]' (or v 9 TpZ:'), then w = D e x p , - l ( y ) v (correspondingly, w = D e x p , ( y * ) v ) can be represented as w = w ' + w", where w' 9 Tp2?,' (correspondingly, w ' 9 TuS,' ) and Iw"l _< ~lwl. T h e inequality [[Dexp;-a(q)D~(p) - D~(y)Dexpp(y*)[[ < ,

(2.65)

holds. Take d _< s T h e first summand in (2.61) is estimated similarly to case 2. Now we estimate t [ F Q ' - JoQ'[[. Since D ~ - l ( z ) U ( z ) C U(y) ( L e m m a 2.2.10) and U ( z ) C Z ( z ) = T z Z z , for v C U(z) we have V1

D(-'(z)v

=

9 U(y) C T y S ' .

By our choice of d', v2 = Gv = D e x p p X ( y ) v l = v 2' + v 2" with v~.' 9 TpS', [v~[ < v[v2[.

Since (2.52) holds, v3 = O(p, z)v~ = Bkv = v~ + ~;, Iv;I _< -rv~l.

Obviously, ~ _< 1, hence it follows from (2.50) that u < 1/8. Since

Iv21(1 - ~,) _< Iv;I, we have Iv~l _< 21v;I. From the inequalities

Iv;I _< .Iv~l _< 2.1v;I

and IvY'+ v;[ _< 4.1v;I

we deduce that Bkv = v 2' + v 2"' with v 2' 9 TpS' and Iv~"l < 4,1v;I. Hence, any w 9 U' we can represent in the form w = w t + w " with w ' 9 TpS' and [w"[ _< 4ulw' I. Take Vo 9 Z ( p ) with Iv01 = 1, let w = Q'vo. We can write w = w' + w" with

w', W tt as above. Since Iwl _< g and I w ' l ( 1 - 4 . ) _< Iwl, we obtain the inequalities Iw'l _< 2K, Iw"l _< 8Ku. Let us write FQ'vo = D ~ ( y ) D e x p p ( y * ) w = = D ~ ( y ) D e x p p ( y * ) w ' + w~, [w~'[ < 2 g [ w " I < 16K2u; I

JoQ Vo = D e x p 2 1 ( q ) D r = Dexp2X(q)Dr

Since D r

=

w'2', [w~'] < 1 6 K 2 , .

(2.67)

= D ~ ( p ) w ' (see (2.64)), it follows from (2.65) that I D ( ( y ) D expv(y*)w' - D exp'~a(q)Dr

=

= I D ( ( y ) D expv(y*)w' - D e x p ~ l ( q ) D ( ( p ) w ' [ <_ u[w'].

Now we deduce from (2.66) and (2.67) that

(2.66)

144

2. Topologically Stable, Structurally Stable, and Generic Systems IFQ'vo - JoQ'vol <_

+ 32K2u <- 2K(16K + 1)t,.

Since the first summand in (2.61) does not exceed 2K~u, we come to the estimate IIA~ - JoQl] <- 2 K ( 1 7 K + 1)u. This gives ] l A k - J0ll -< 4 K ( 9 K + 1)u < a/2. Hence, in case 3 for d < d ~, bk also satisfies conditions (1.63)-1.65) of Theorem 1.3.1. Apply Theorem 1.3.1 to •k and find d~, L' (corresponding to A, N, A = d'). Take do = dlo/K1. Then for d _< do we have

I k(0)t _< Kid _< 4. It follows from Theorem 1.3.1 that there exist vectors vk E Hk such that [vk[ < L"d (with L" = KIL') and bk(vk) = Vk+l. The definitions of ~k imply that the points Pk = exp~ k(vk) have the following property: any Pk+l belongs to the trajectory of the flow ~" through Pk. Hence, these points belong to one trajectory of .~. Note that the inequalities

r(pk, xk) < L"d hold. Define numbers tk by pk+l = -~(tk,pk). In case 1, we have tk = T. In cases 2, 3, the inequalities

r(3(T, pk), z) < r(~(T,p), z) + r(~(T, pk), ~(T,p)) hold. The first summand does not exceed d, the second one does not exceed exp(s (recall that /~ is the Lipschitz constant of X we fixed above). It follows from Lemma 2.2.13 that Pk+l = ~(tk,pk) with Irk - T[ < Lid, where L1 = KI(1 + exp(s Set r 0 = 0 , r i c = t l + . . . + t k f o r k>0, andric=tk+...+t_l for k < 0 . Now we define a : IR ~ IR, for k E 7/ set a(Tk) = tic, and for t E [Tk, T(k + 1)] set + (t - T k ) .tk+l = ~ . The same reasons as in the proof of Theorem 1.5.1 show that a ERep(L2t), where L2 = L1/T, and that

Ioz(t) - Tic -- t~l ~_ Lid

(2.68)

for t e [Tk,T(k + 1)] and t' = t - Tk. Set p = p0. Obviously, F.(a(Tk),p) = ~,(Tk,p) = pic. For t E [Tk, T(k + 1)] we have

r(Z(a(t),p),g/(t)) <_r ( Z ( a ( t ) - ric,pk), ~(t',pk))+ +r(Z(t',pic), Z(t', xk)) + r(~(Tk + t'), .E(t', ~(Tk)))

2.2 Shadowing in Structurally Stable Systems

145

(note that ~ ( a ( t ) , p ) = ~ ( a ( t ) - rk,pk), xk = ~ ( T k ) , ~(t) = ~ ( T k + t')). The first term on the right does not exceed K L l d (we apply (2.68) and the inequality IX] < K). The second term does not exceed

exp(s

xk) < L"exp(f~T)d,

and the third term is not more than d (since ~ is a (d, T)-pseudotrajectory). This leads to the inequality

r ( X ( a ( t ) , p ) , ~(t)l < L3d, where L3 = KL1 + L" exp(t:T) + 1. Take L = max(L2, L3) to complete the proof of Theorem 2.2.3.

Remark. If J2' = O, then all xk are of type (R), hence tk = T for all k, and we can take a( t ) = t. 2.2.2 T h e C a s e o f a D i f f e o m o r p h i s m To introduce the notion of structural stability, we define the Cl-topology on the set of diffeomorphisms of M as follows. We again fix an embedding f : M --~ IR2~+1 (see the previous subsection), and consider M as a smooth submanifold of ]R2~+1. We preserve notation I.I for the norm in TpM, p E M, generated by the Euclidean norm of IR2~+1. For a diffeomorphism r of M and for a point p E M, the derivative De(p) is a linear mapping of a linear subspace TpM of ]R2~+1 into IR2~+1. Thus, for two diffeomorphisms r and r the value IIDr

- Dr

I=

max

veTpM,[v[=l

IDr

- Dr

I

is defined. Now we define the Cl-distance pl between two C 1 diffeomorphisms r r of M by the formula p1(r r

= p0(r r

+ max

pEM

IIDr

- OC(P)ll

(the C~ p0 was defined in the previous section). We denote by D i f f ' ( M ) the space of diffeomorphisms of class C 1 of M with the topology induced by pl. D e f i n i t i o n 2.7 We say that r E D i f f l ( M ) is "structurally stable" if given e > 0 there is ~ > 0 such that for any r E D i f f l ( M ) with P1(r162 < 5 there is a

homeomorphism h : M --~ M with the following properties:

146

2. Topologically Stable, Structurally Stable, and Generic Systems

(a) h o r 1 6 2 (b) r(x,h(x)) < e for x E M. Remark. One can give another definition of structural stability. Let us say that r EDiffl(M) is structurally stable if there exists 6 > 0 such that for any r EDiffl(M) with p1(r162 < 6 there is a homeomorphism h : M --~ M with property (a) of Definition 2.7. It follows from the main results of [Ma] and [Robil] that these two definitions are equivalent. Let r be a structurally stable diffeomorphism, and let 6 > 0 have the property formulated above. Take r such that the inequality P1(r r < 6 holds and let h be a homeomorphism with property (a). There exists 61 > 0 such that for any diffeomorphism X with Pl(X, r < 61 we have pa(r X) < 6, hence there is a homeomorphism hi such that ha o r = X o ha. Then we have h' o r = X o h', where h' -- ha o h -1. This shows that the set of structurally stable diffeomorphisms is an open subset of Diffa(M). Let f2(r be the set of nonwandering points for a diffeomorphism r (see Definition 1.10). It is easy to show t h a t / 2 ( r is a compact r set. Smale introduced in [Sm2] the following property of r A x i o m A. (a) ~2(r is a hyperbolic set; (b) the set of periodic points of r is dense in ,f2(r For a point p E ~ ( r

we define the stable manifold

W'(p) = {x E M : r(r162

--+ 0 as k ~ +c~}

and the unstable manifold

W=(p) = {x E M : r(r

ek(p)) --, 0 as k -+ - c r

One easily shows that if r satisfies Axiom A and p E ~2(r then W~(p) and W=(p) are related to the local stable and unstable manifolds described in Theorem 1.2.1 as follows:

W'(p) = {x E M : ek(x) E W~(p) for some k E 7]}, W~(p) = {x E M : ek(x) E W~(p) for some k E 7]}. In this case, the sets W'(p), W~(p) are the images of Euclidean spaces ]Rk under immersions lRk ~ M of class C ' (see [Pil]). Now we describe one more condition on r introduced by Smale [Sm2]. D e f i n i t i o n 2.8 We say that a diffeomorphism r satisfies the "geometric STC" (the "geometric strong transversality condition") if, for any p, q E ~2(r the

stable and unstable manifolds WS(p) and W~(q) are transverse.

2.2 Shadowing in Structurally Stable Systems

147

Necessary and sufficient conditions for structural stability are given by the following statement. T h e o r e m 2.2.4 [Robb, Robi2, Ma]. A diffeomorphism r is structurally stable if and only if r satisfies Axiom A and the geometric STC.

Remark. Robbin showed in [Robb] that the conditions of Theorem 2.2.4 are sufficient for the structural stability of a diffeomorphism r of class C 2, Robinson proved the analogous statement for diffeomorphisms of class C 1. Mafi~ established the necessity part. Nitecki showed in [Ni2] that the same conditions imply topological stability. T h e o r e m 2.2.5 [Ni2]. If a diffeomorphism r satisfies Axiom A and the geometric STC, then r is topologically stable. The following statement is a corollary of Theorems 2.1.1, 2.2.4, and 2.2.5. T h e o r e m 2.2.6. POTP.

If a diffeomorphism r is structurally stable, then r has the

Remark. This result was established independently by Robinson [Robi3], Sawada [Saw], and Morimoto [Mori2] (to be exact, they showed that Axiom A combined with the geometric STC implies the POTP). Indeed, a stronger statement holds. T h e o r e m 2.2.7. LpSP.

If a diffeomorphism r is structurally stable, then r has the

Remark. A proof of Theorem 2.2.7 was given in [Pi2], Appendix A. Here we describe another proof, close to the proof of Theorem 2.2.3 given in the previous subsection (note that the problem for flows is really more complicated than the one for diffeomorphisms). We omit all the auxiliary statements and formulate only the final result we need (Lemma 2.2.16), the details are left to the reader (note that an analog of Lemma 2.2.9 for a diffeomorphism was proved in [Robb]). Thus, we consider a structurally stable diffeomorphism r of M. There exist families of subspaces S(p),U(p) of TpM for p E M with the properties described in the lemma below (the corresponding statements are proved similarly to Lemmas 2.2.10, 2.2.11, 2.2.12). L e m m a 2.2.16. that

One can construct subspaces S(p), U(p) of TpM, p E M, such

148

2. Topologically Stable, Structurally Stable, and Generic Systems

(1) S(p) ~ U(p) = TpM;

(e) Dr

Dr

C S(r

C

U(r

(3) there exist constants C > 0, Aa E (0, 1) such that

IDr

<_CA~lvl for

v E S(p), k >_ O,

IDC-k(p)vl < CA lvl for v 9 V(p), k > o; (4) there exists N > 0 such that if P(p), Q(p) are the complementary projectors corresponding to S(p), U(p), then

IIP(p)ll, IIQ(p)II -< N; (5) given fl > 0 and a natural T there exists a > 0 such that if z,p E M, and r(z,q) < a, then there is a linear isomorphism II(p, z) : TzM --* TzM such that q = eT(p),y = r

[[/7 - 1[[ _
and a linear isomorphism O(p, z) : TvM ~ TpM such that [ [ 0 - I[[ < fl, O ( p , z ) ( V e x p p a ( y ) D r

C U(p).

Now we find T > 0 such that # = CA T < 1.

(2.69)

It follows from Lemma 1.1.3 that if r = eT has the LpSP, then the same holds for r To simplify notation, below we write r insread of r We will apply Lemma 2.2.16 to this new r with C = 1, A1 = # in (2), and with T = 1 in (5). Take u0 E (0,1) such that A = (1 + u0)2# < 1. For this A and N (from statement (4) of Lemma 2.2.16) take the corresponding N1 (Theorem 1.3.1) and find ~ > 0 such that (1.65) holds. Now we fix K > N such that [[Dr162

< K for p E M.

Find a number v E (0, v0) with the property 2 K ( 2 K + 1)v < a/2. In the previous subsection, a number c was fixed such that, for any p E M, the mapping expp is a diffeomorphism of the ball E2c(p) C TvM onto its image. We take a number d' < c such that, for any points x, y E M with r(x, y) < d' and for y ' = e x p ; ' ( y ) , inequalities (2.51) hold.

2.2 Shadowing in Structurally Stable Systems

149

We assume that d' has also the following property: if for p, z E M and q = r we have r(z, q) < d', then for the corresponding linear isomorphisms O(p, z), II(p, z) (see statement (5) of L e m m a 2.2.16) inequalities (2.52) are true. Now we take a sequence ~ = {Xk} such that < d

with d < d', and consider the spaces Ilk = T~kM. We fix two points xk,xk+x, and denote p :

Xk,

y = (~-l(z), y* ---- exp;l(y).

q= r

Z ~-- X k + l ,

Define Ck : Hk --* Hk+l by the formula Ck = exp~-1 o r o expv. It follows from our choice of d' that Ck is defined on Ed,(p) and that

ICk(O)l < d. Set J = DCk(0), we have S = Dexp~-1 (q)Dr Define an operator F = Dr

now we set

ASk = H(p, z ) J P ( p ) , A~ = FO-I(p, z)Q(p), and

Ak = A~ + A~. Let us write r

= Jv + X(V)

with X(0) = Ck(0), Dx(O) = O. Since the derivatives of r are uniformly continuous, we can choose d' so that (2.53) holds. Now we represent Ck(v) in form (1.51), where Wk+l(V) = (J - Ak)v + X(v). The same reasons as in the proof of Theorem 2.2.3 (case 1) show that relations (2.54)-(2.56) hold. To estimate the value IIAk - JI], let us note that ]]J]] < (1 + u ) g _< 2K, ]IF]] _< 2K, ]]P]I,I]Q]] -< g . Let us take s such that the inequality

z) =

xk+,) < d'

implies the inequality ] ] D e x p f (q)Dr

- Dr

< u

150

2. Topologically Stable, Structurally Stable, and Generic Systems

(obviously, we can find d' depending only on u and r Theorem 2.2.3, we show that if d < d', then

Similarly to the proof of /s

[]Ak- J[[ < 2K(2K + 1)u < ~-.

(2.70)

Now we fix A = d' and apply Theorem 1.3.1 to find numbers do, L (depending only on r such that if ICk(0)l < d < do, then there exists a sequence vk C Hk with the properties Ivkl <_ Ld and Ck(vk) = Vk+l. Set x = exP,0(vo ). Then ek(x) = exp~k (vk), and it follows that

r(xk, ek(x)) = Ivkl < Ld. This proves Theorem 2.2.7. Since the set of structurally stable diffeomorphisms is open in the C 1topology (see the remark after Definition 2.7), it follows from Theorem 2.2.7 that if a diffeomorphism r is structurally stable, then there exists a neighborhood W of r in Diffl(M) such that any diffeomorphism r in W has the LpSP. Below we show that one can take a neighborhood W such that the LpSP is uniform, i.e., the numbers do and L are the same for all r E W [Beg]. It was first noticed by Sakai [SAM] that the POTP is uniform in a neighborhood of a structurally stable diffeomorphism r (see statement (4) of Theorem 2.2.8 below). Later Sakai showed [Sak2] that the C 1 interior of the set of diffeomorphisms with the POTP consists of structurally stable diffeomorphisms (note that for dimM _< 2 this result was proved by Moriaysu [Moriy]). Let us sum up the information related to this problem. T h e o r e m 2.2.8. The following statements are equivalent: (i) a diffeomorphism r is structurally stable; (2) there is a neighborhood W of r such that any r E W has the POTP; (3) there is a neighborhood W of r such that any r E W has the LpSP; (4) there is a neighborhood W of r such that for any e > 0 one can find d > 0 with the following property: any d-pseudotrajectory of a r C W is (e, r shadowed by a point of M; (5) there is a neighborhood W of r and numbers L, do > 0 such that any r E W has the LpSP with constants L, do.

Proof. Obviously, the implications (5) =a (4) =~ (2) and (5) =~ (3) ~ (2) hold. We prove here that (1) =~ (5) and that (5) :a (1) (this establishes the equivalence of (1) and (5)), for the proof of the remaining part (i.e., of the implication (2) =~ (1)), we refer the reader to the original paper [Sak2].

2.2 Shadowing in Structurally Stable Systems

151

First we prove the implication (1) =~ (5). Let r be a structurally stable diffeomorphism. We fix families {S(p),U(p)} having the properties described in Lemma 2.2.16 (for r Take T such that (2.69) holds. There exists a neighborhood W0 of r in Diffl(M) such that all diffeomorphisms r in W have the same Lipschitz constant. Therefore, it follows from Lemma 1.3.1 that if all diffeomorhpisms cT, where r C W0, have the property described in statement (5) of our theorem, then all r C W0 have a similar property. We fix a number d' > 0 satisfying all the conditions imposed on A in the proof of Theorem 2.2.7. In addition, we assume that for a neighborhood W1 C W0 of r and for d' the following holds. If r E W1, and for points p, z C M the inequality r(r z) < d' is satisfied, then the mapping exp~-1 o ~b o expp

(2.71)

is a diffeomorphism from Ed,(p) into TzM (obviously, one can find Wx and d' with these properties). Now let us take r C Wx and a sequence ~ = {zk} such that

~(xk+l, r

< d

with d <: d'. Set Hk = T, kM, fix a pair of points p = xk, z = xk+x, and let Ck : gk ---* Hk+x be defined by (2.71). For r 9 Wx and for u 9 gk with lul ~ d' we set X ( r u) = D(exp~-x o r o expv)(u), so that

J' = DCk(0) = X ( r 0) and J = De'(0) = X(r 0),

(2.72)

where r = exp~-x o r o expp. Represent Ck(v) = g'v + X(r v). Take v,v' 9 Hk with Iv[, Iv'l < d' and let O(s),s 9 [0, 1], be a linear parametrization of the segment joining v and v'. Then we can apply the standard formula x ( ~ , v) - x ( r

v') =

x(r

o ( ~ ) ) d~ (v - v') - S ' ( v - v').

The uniform continuity of X(r u) and the second equality in (2.72) imply the existence of a number A < d' such that

/o x X ( r

- v') - J ( v -

v') < g l v -

v' I

for ]vl, [v'] < A. Hence, there exists a neighborhood W C Wx of r such that if r 9 W, then

[Ig -J'l[-<

~-

and Ix(r

- x(r

=

152

2. Topologically Stable Structurally Stable, and Generic Systems = "J01X ( r O(s)) ds (v - v') - J'(v - r

for

< -~ ~lv-r

Ivl, Iv'l ~ ~. Now we represent Ck(v) = Akv + Wk+l(V),

where Ak is constructed for r Theorem 2.2.7, and

and z by the same formulas as in the proof of

Wk+l(V) = (J - Ak)v + ( J ' - J)v + X(r Since the mappings Ak satisfy all the conditions of Theorem 1.3.1, and

Iwk+~(v) - wk+,(v')l < ~lv - v'l for Iv], Iv' I < A (here we refer to the estimates above and to (2.70)), we can apply Theorem 1.3.1 (in the same way as in the proof of Theorem 2.2.7) to show that that there exist constants do, L with the desired properties, and these constants do not depend on r This proves the implication (1) ==~ (5). To prove the implication (5) =~ (1), we first formulate some known results. Let us denote by GI(M) the set of diffeomorphisms r with the following property: r has a neighborhood W in Diffl(M) such that every periodic point of a diffeomorphism r 9 W is hyperbolic. T h e o r e m 2.2.9 [Hayl]. If r 9 ~ I ( M ) , then r satisfies Axiom A. Now we formulate some properties of diffeomorphisms satisfying Axiom A. The following three statements are analogs of Lemmas 2.2.1 - 2.2.3 for the case of a diffeomorphism r satisfying Axiom A. L e m m a 2.2.17 [Sm2]. There exists a unique decomposition of the nonwandering

set ~(r

~(r = ~1 U . . . U 9m, where ~i are compact disjoint invariant sets, and each ~2i contains a dense trajectory. The sets/21 are called basic sets. For a basic set f2i we define the stable and unstable sets WS(s

= {x 9 M :

dist(r

~2,) ~ 0 as k ~ +oo}

W"(f2~) = {x 9 M :

dist(r

f2~) ~ 0 as k ~ - c r

and

L e m m a 2.2.18 [Sm2].

2.2 S h a d o w i n g in S t r u c t u r a l l y Stable S y s t e m s

M =

U w'(~,)= U

l
w~(~,)

153

9

l
L e m m a 2.2.19 [Hirsl]. For a basic set f2~ we have

we(a,) = U we(p), w~

U W~(p).

pe~,

pe~

It follows from Lemmas 2.2.18 and 2.2.19 that if a diffeomorphism r satisfies Axiom A, then any point x 9 M belongs to the intersection

W'(p) n w~(q), where p, q E /2(r Below we will apply to hyperbolic sets for a diffeomorphism of a manifold an analog of Theorem 1.2.1 (we will simply refer to Theorem 1.2.1 in this case; the needed modifications are obvious). Let us note that if p 9 1 6 2 then we have

We(P)= U r

, W~,(p)= U Ck (w~(r

(W~(r

k>0

k>0

where A is given by Theorem 1.2.1 for the hyperbolic s e t / 2 ( r (these equalities are easy consequences of Theorem 1.2.1, we leave their proof to the reader). Take a point x E We(p) and find k0 > 0 such that x ' = ck~

E IntW~(r176

where the interior is taken with respect to the inner topology of the C 1 disk D = W~(r176 Take an open (with respect to the inner topology of D) C 1 disk Co in D containing x' and consider the C 1 disk

c = r176

c we(p).

Let us prove the following auxiliary statement (Sakai applied an analog of this statement for two-dimensional diffeomorphisms in [Sak3]). L e m m a 2.2.20. For any point x E W ' ( p ) and sets Co and C as above there exists e = e(x, C) such that if for a point y we have r(r162

< e

(2.73)

for all k >_ 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

<

154

2. Topologically Stable, Structurally Stable, and Generic Systems

for large k, hence there exists kl ~/co such that ek,(y) C Ws162 The set c'

=

=

- 0(C0)

contains a neighborhood of x" = ek, ( x ) i n W.~(r k' (p)). Since W~(r kl (p))is a C 1 disk, there exists 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')

<

min(d0, ~),

where do, L are from condition (5) of our theorem, and e is given by Lemma 2.2.20 for the disks C - , C +. If for a point x we have

r(r

Xk) <_ LH,

then it follows from Lemma 2.2.20 that x 6 C - N C q The obtained contradiction shows that case 2 is also impossible. This proves the implication (5) =~ (1). []

2.3 S h a d o w i n g

in T w o - D i m e n s i o n a l

Diffeomorphisms

It was shown in the previous sections of Chap. 2 that topological (structural) stability implies the P O T P (the LpSP, correspondingly). Without additional assumptions on the dynamical system, no necessary conditions for the P O T P are known. In this section, we assume that d i m M = 2 and that r is a diffeomorphism of class C ' satisfying Axiom A (see Subsect. 2.2.2). Below we call r an (A,2)-

diff eomorph ism.

2.3 Shadowing in Two-Dimensional Diffeomorphisms

159

We show that in this case necessary conditions under which r has the P O T P (or the LpSP) have very natural geometric structure. For an (1,2)diffeomorphism r the P O T P is equivalent to a sort of topological transversality condition [Sak3], and the LpSP is equivalent to the strong transversality condition (and hence to structural stability). We consider in this subsection also the so-called weak shadowing property (WSP) introduced in [Cor2]. Plamenevskaya [Pla2] studied this property for an (A,2)-diffeomorphism r with finite nonwandering set and with a nontransverse heteroclinic trajectory joining two saddle fixed points. In this case, necessary and sufficient conditions for the WSP are very delicate, they are connected with arithmetic properties of eigenvalues of D e at the saddle points (Theorem 2.3.4).

First we describe a result of Sakai [Sak3] giving necessary and sufficient conditions under which an (A,2)-diffeomorphism r has the POTP. Let us begin with some notation and definitions. We assume that an analog of Theorem 1.2.1 holds for the hyperbolic set f2(r take an arbitrary neighborhood U of this set, and fix the corresponding constants A and u. Take points x 9 M and p 9 ~ ( r such that x 9 W~'(p), where cr 9 {S, U}, and dimW~(p) = 1. We can find k 9 7I such that Ck(z) e IntW~(q~k(p)). Since the set

w:,(

)

is an embedded segment of class C 1, for a small a > 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

<

d

holds. Construct a sequence ~ = {xk : k E 7]} as follows:

9 ~ = Ck(z), k < l; ~k = C k - ~ - ~ ( ~ ) ,

k > 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

<

,, k > 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 <

do

and Ld < e, then there is a point x such that r(r

xk) ~ Ld

and x E C+MC -. Since x E C - , we have x = (y*, 0), and it follows from x E C + t h a t y* ~ (ym,y~). Now we see t h a t l Y o - y m I ~-lY0 - Y * I = r(xo, x) ~_ Ld = Lf~ig(yo)I <

< 2hLs

- Y,~I,

and this is impossible, since (2.82) holds. This completes the proof.

[]

Let us m e n t i o n a result by Sakai related to T h e o r e m s 2.3.1 and 2.3.2 (see [Sakh] for the details). 2.3.3. An (A,2)-diffeomorphism r belongs to the C 2 interior of the set of diffeomorphisms having the P O T P if and only if r is structurally stable.

Theorem

164

2. Topologically Stable, Structurally Stable, and Generic Systems

The following shadowing property was introduced in [Cor2] in connection with the problem of genericity of shadowing (see the next section). To define it, take a system r E Z(M) and let ~ = {xk :k E 7/} be a d-pseudotrajectory of

r

D e f i n i t i o n 2.10 We say that ~ is "weakly (e)-shadowed" by a trajectory O(p)

ff

C N~(O(p)).

(2.83)

D e f i n i t i o n 2.11 A system r E Z(M) has the "WSP" (the "weak shadowing

property") if given e > 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)

<

d3,

then Juk+l + 11
then

p-1 e

Ivk+ll < #_~_2_ + d3 =

2

+ da,

and we deduce from (2.86) that s

IVk+2t < -~ -t- #d3 q- d3 < e. In addition, if xk E N~(pl) and Ivkl > d3, then the signs of vk and Vk+l are opposite. Since (2.85) holds, this establishes the property of d3 formulated above.

168

2. Topologically Stable, Structurally Stable, and Generic Systems We assume, in addition, that d3 satisfies the conditions ~-ld2 + d3 < d2 and d3(A-1 + 1) <

similar to (2.86). The same reasons as above show that if d < d3, Xko E gd2(P2), and xkl • N~(p~) for some kl < k0, then there exists k E (kl, k0) such that

zk E W1. Consider the set F=

[(

l-e,

(

1 -t~------~e U l + - - 2 - - , l + e

)]

x{0}.

Since F C W'(pl), there exists d4 E (0, d3) and a number T2 > 0 such that for the set W = F x (-d4, d4) the inclusion

r (F) c Nd2(pl) holds. In addition, we take d4 so small that, for d < d4, the inclusion x0 E W implies the inclusion XT2 E Nd2(Pl), and the inequality r(xo, y) < 2d4 implies the inequalities r(xk, Ck(y)) < e for Ikl _< T=. Repeating the arguments applied to find d3, we can find ds E (0, d4) such that if d < ds, Xko E Nd,(P2), and Xkl ~ N~(p2) for some k~ > k0, then there exists k E (k0, kl) such that Xk E W. Set Q~ = Nd2(Pl) and Q2 = Nd4(P2). We claim that r has the LpSP on each of the sets Mi = T 2 \ Qi, i = 1, 2. Consider the sets

M~ = T 2 \ Nd~/2(p~), Ms = T 2 \ Nd4/2(p2). It is easy to see (the details are left to the reader) that an analog of Lemma 2.2.16 holds for each of the sets M[, i = 1, 2. This can be done so that C = 1 in statement (3), and all the statements are true for points p E Mi such that r r E M~'. Of course, we can find d > 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 <_ k < k2} C N~(p2) and

{xk : k2 + T2 <_ k < k3} C N,(pl).

(2.87)

Let Xk~ = (Ui, vi), i = 1, 2, 3. We can find d' > 0 such that, for the sets

(v3 - d', v3 + d')

X1 = ( - 1 - d2,-1 + d2) x and

X2 = (1 - d2, 1 + d2) x (vl - d', Vl + d'), the inclusions Xi C Wi,i = 1,2, hold. By the choice of d2, for any point

x E N~(p2) fq [ U CJ(X2)] LJ_>o

J

we have

{xk: k < kl} c

N:dO(x)).

Set

$2 = [ U CJ(X2)] f'l [(u2 -d4,u2 + d4) • (-d4, d4)]. L./_>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)

<

'71/2(1

_

'7,/2)

if sm < rj. None of these two inequalities is true. The obtained contradiction completes the proof of Theorem 2.3.4. []

172 2.4

2. Topologically Stable, Structurally Stable, and Generic Systems C~

of Shadowing

for Homeomorphisms

Let X be a topological space. A subset Y of X is called residual if Y contains a countable intersection of open and dense subsets of X. If P is a property of elements of X, we say that this property is generic if the set {x E X : x satisfies P} is residual. Sometimes in this case we say that a generic element of X satisfies P. The space X is called a Baire space if every its residual subset is dense in X. The classical theorem of Baire says that every complete metric space is a Baire space. We consider the space Z(M) of discrete dynamical systems with the metric P0 introduced in Sect. 2.1. It is an easy exercise for the reader to show that the space Z(M) is complete (and hence it is a Baire space). The main result of this section is the following statement [Pih]. T h e o r e m 2.4.1. A generic system in Z(M) has the POTP.

Remark. The genericity of the P O T P for M

S 1 was proved by Yano in [Y2]. Odani [Od] established the genericity of the P O T P in the case n = d i m M < 3. His proof was based on the possibility of approximation of an arbitrary system r e Z(M) by a diffeomorphism [Mu, Wh] and on the theorem of Shub [Shul] on the C~ of structurally stable diffeomorphisms. Unfortunately, in the case dimM > 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
1 fort 9

(3.5)

We claim t h a t Igik(x ') -- x~l < e for k 9 2z.

(3.6)

We prove (3.6) by induction (and consider only the case k > 0). For k = 0, (3.6) coincides with (3.3). Assume that (3.6) holds for some k _> 0. T h e n it follows from (3.5) t h a t

x~ + e + 1 > dhk(x ') + 1 > ~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<

' Xk+l

+l<

!

<xk+ l+l<x

Xk'

!

k+l+2e.

This means that r(xk,xk+l) < 2e on S 1. By the choice of e and d, we have r(xk, xk+ ) >

>

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 <_ al < bl <_ a2 < . . .

<_a2N < b2N <_ 1;

- the length of any segment [b~,a2],..., [b2N,a~] is less than e (note that some of these segments may be points); -O(t)-t#0on(an, bn),l
n I " = 0 for m r n.

C IntI" for n odd

and r

C IntF,~ for n even.

Find A C (0, 1) such that

r(r162

> A for x E In, 1 < n < 2N.

Now we take da > 0 such that . A dl < mln(e, ~-);

(3.8)

Nda(r

f3 I " = 0 for n ~ m;

(3.9)

Ndl(r

C IntI: for n odd;

(3.10)

Nd~(r

C IntI'n for n even.

(3.11)

Take d C (0, dl) and such that the inequalities r(r

2 x,+k) < dl for Ikl < Z

(3.12)

hold for any d-pseudotrajectory {xk} of r We claim that this d has the property described in Definition 1.3. Take a d-pseudotrajectory ~ = {xk} of r First, assume that xk E I" for n odd. Then the inequality r(xk+l, r < d < dl and inclusion (3.10) imply that xk+l E I ' . This shows that if xk E F for n odd, then xk+t 9 I: for l > 0.

3.1 One-Dimensional Systems

179

Similarly we prove that if zk E I~ for n even, then Zk+t E I'~ for I < 0. To do this, we apply inclusion (3.11) and the inequality

r(r

(3.13)


following from (3.12). Now assume that Xk E In with n odd, i.e., (a~, b~) is an r-interval. Then A zk+~ > r

and

- d~ > zk +

A Xk-1 < ~(xk) + da < xk - -~

It follows from these inequalities and from the properties of the corresponding segments I ~--1 ~ and I~ established above that there exist indices k_ and k+ such that 2 k _ k _ < k_ - - - A , and

Xl E I: for l > k+, xl E 4_1 for l < k_.

(3.14)

A segment I,~ with n even has a similar property. For the fixed d-pseudotrajectory ~, one of the following two cases is possible. Case 1. 2N

uro.

It follows from (3.9) that in this case there exists a segment I'~ such that ~ C I~. Since the length of I~ is less than ~, and this segment contains a fixed point b~ of r ~ is (e)-shadowed by b=. Case 2. There exists a segment I~ such that ~ f3 I,~ # 0. Assume that x0 E I~ and that n is odd. It was shown above that in this case there exist indices k+ > 0, k_ < 0 such that Ik+h Ik_l < 2/A, and relations (3.14) hold. It follows from our considerations that el(x0) E I'= for l > k+, r

E I~_1 for l < k_.

These inclusions and inequalities (3.12) imply that ~ is (r by the point x0. The case of n even is considered similarly. Thus, Theorem 3.1.1 is proved. []

Remark. The example of Yano [Y1] mentioned in Sect. 2.1 is based on the following construction. Set ak = 1/k for k E IN. Consider a homeomorphism r of S 1 such that any interval (ak+l, ak) with k odd is an r-interval, and any interval (ak+l,ak) with k even is an /-interval. By Theorem 3.1.1, r has the POTP. On the other hand, 0 is a nonisolated fixed point of r and it is easy to show that r is not topologically stable (see [Pi2] for details).

180

3. Systems with SpeciM Structure

Let us describe necessary and sufficient conditions of the LmSP for a class of homeomorphisms of S 1 [Plal]. T h e o r e m 3.1.2. A homeomorphism r preserving orientation and with nonempty set Fix(C) has the LmSP if and only if the set Fix(C) is nowhere dense, and 9 ( t ) - t changes sign on [0, 1).

Proof. It follows from L e m m a 3.1.1 crate arc, then r does not have the but ~(t) - t > 0 (or ~(t) - t < 0) can apply the construction from the = {xk} such that

that if the set Fix(C) contains a nondegenLmSP. If the set Fix(C) is nowhere dense at all t such that P(t) ~Fix(r then one proof of L e m m a 3.1.2 to obtain a sequence

r(xk+l, r

0, k - - ,

(3.15)

and ~ "rotates" around S 1 infinitely many times. Obviously, in this case the relation r(xk, Ck(p)) ~ 0, k ~ ~ , (3.16) cannot hold for a point p C S 1. This proves the necessity of the conditions of our theorem. Let us prove their sufficiency. Take a sequence ~ such that (3.15) holds. We claim that this sequence converges to a fixed point p of r Fix a sequence en > O, en ---+ 0 aS n---+ 00.

Let us show that for any n there exists a number a(n) and a segment J= of length less than e= such that xk E J= for k > a(n). Then the family {J~} has a unique common point p. Obviously, in this case we have r = p, and xk --* p. We apply a construction analogous to the one used in the proof of Theorem 3.1.1. Fix e = e~ (below we omit indices n). Find a system of r-intervals and /-intervals (al, b , ) , . . . , (aN, bN) such that it contains both r- and /-intervals, and the length of any segment [bin, am+l] is less than e (as previously, aN+l = al etc). Now we choose a,~', b~ f I such that a,~ < a Im < bIm < b~n, and the length of any [bin, am+l] is also less than e. There exists d having with respect to the system of segments

Im= [a~, bm], ' I~'

=

[b~, am+l] ,

the same properties as in the proof of Theorem 3.1.1. Find k0 such that r(ff(xk),Xk+l) < d for k > k0. If there exists m such that xk E 1" for k > 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 <_ g . Take a sequence ~ = {zk} such that (3.17) holds, and let sup I A z k - x k + l l = d. kE:E

We are looking for a sequence vk such that for the point x = x0 + v0 the relations

3.2 Linear and Linearly Induced Systems

A k x = xk + vk, k 9 7/,

185

(3.20)

and inequalities (3.18) are satisfied. We can write (3.20) in the form Xk+l + vk+i = A ( x k + vk) or vk+l = Av~ + ( A x k - xk+l), k 9 7/.

(3.21)

It follows from Theorem 1.3.1 (with Ak = A, Wk+l(V) = Axa - xk+l, Pk = P etc) that there exists a sequence {vk} satisfying (3.21) and such that sup Ivkl < Ld, kEZ

where L = L ( $ , N ) . This proves that r has the LpSP. The uniqueness of a sequence vk such that sup Ivkl < keT/

follows from Theorem 1.3.2. Obviosly, this implies the uniqueness of a point x with property (3.18). This completes the proof of Theorem 3.2.1. [] It is easy to see that a similar statement holds for a mapping r : C ~ ~ C ~ of the form r b9 Cn (and for a mapping of the same form in IR~, see [Morim3]). An analogous shadowing problem for a linear system of differential equations, ~: = A x ,

(3.22)

where the matrix A is constant, was studied by Omhach [Om3]. It is shown in [Om3] that system (3.22) has the Lipschitz shadowing property (analogous to the one established in Theorem 1.5.1) if and only if all eigenvalues of the matrix A have nonzero real parts. Now we pass to shadowing for linearly induced systems. Here we treat in detail one class of linearly induced systems, the so-called spherical linear transformations [Sas], later we mention known results for some other classes. Consider the space ]R~+1 with coordinates x = ( x 0 , . . . , x~). Let S ~ be the unit sphere, S ~ = {x :Ix I = 1}. We denote by r the canonical distance on S'L Fix a real nonsingular (n + 1) x (n + 1) matrix A, i.e., an element of G L ( n + 1, JR). We define the spherical linear transformation r corresponding to A by the formula Ax r ]Ax]" Obviously, r is a diffeomorphism of S ~. Denote by ~ 1 , . . . , ~+1 the eigenvalues of the matrix A. T h e o r e m 3.2.2. For the system r on S ~ the following statements are equivalent:

186

3. Systems with Special Structure

(1) r has the POTP; (2) r has the LpSP;

(s) lad r I Jl for i r j. Proof. If A, B E GL(n + 1, IR), and r r are the corresponding spherical linear transformations, then it is easy to see that the system r o r is the spherical linear transformation corresponding to the product AB. Since, for any matrix T E GL(n + 1, IR), the corresponding spherical transformation is a Lipschitz homeomorphism of S '~, it follows from Lemma 3.2.1 that we may consider the matrix A in its Jordan form. To prove the implication (1) =~ (3), we first establish two auxiliary statements. L e m m a 3.2.2. Assume that the matrix A has the form

o

where B E GL(m + 1,1R), m < n. Let ~ be the spherical linear transformation of S m generated by the matrix B. If r has the POTP, then ~ has the same property. Proof. For x = (Xo,...,x=) E m n§ (Xr~+l .. 9 X~). Consider Sm={xES~:x'=O}

we

set x' = (Xo,...,xm) and x" =

andP={xES=:x'=0}.

We define the projection r : S = \ P --+ S TM by X!

Ix'l Obviuosly, for the projection 7r, the inequality r(x, y) > r'(~r(x), y) holds for x E S ~ and y E S m, where r' is the distance on Sm. It follows from (3.23) that ~r o r = ~b o ~r(x) for x E S ~ \ P. Assume that r has the POTP. Take e > 0, we assume that e 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) _<

< e.

This proves our lemma.

L e m m a 3.2.3. Assume that the matrix A has one of the following forms:

3.2 L i n e a r a n d L i n e a r l y I n d u c e d

ax=( 0) 0

A

,A2=

(a b) b

a

Systems

187

'

or

A3 =

A 0 1X

... ...

0 0

0 0

:

:

"..

:

:

0

0

...

1A

Then the spherical linear transformation r generated by the matriz A does not have the P O T P . Proof. T h e first two matrices A1 and A2 generate diffeomorphisms of the circle S 1. In the case of A1, every point of S 1 is a fixed point of r hence r does not have the P O T P (see T h e o r e m 3.1.1). In the case of A2, r reduces to rotation of $1; if a E [0, 27r) is angular coordinate on S a, and if a + bi = pexp(ig), then r = a +/9 (mod 27r). If the n u m b e r # = 0/Tr is rational, then there exists a natural m such t h a t every point of S 1 is a fixed point of era, and the same reasons as above show t h a t r does not have the P O T P . If the n u m b e r # is irrational, then the rotation n u m b e r of r is also irrational, and r does not have the P O T P by T h e o r e m 3.1.1. It remains to consider the case of A3. We assume for definiteness t h a t A3 is an (n + 1) • (n + 1) matrix and t h a t ~ > 0. It is easy to see that, for c > 0, matrices A and cA induce the same spherical transformation. Since we can find a J o r d a n form of A3 such t h a t all nonzero off-diagonal terms equal A, we m a y assume t h a t A = 1. A simple calculation shows that for a vector x = ( x 0 , . . . , x = ) and for a natural m we have Y where y = (Yo,..., y,~), i

Yo = Xo, . . . , Yi = ~

n

J . . . . . , Y, = ~ Cmx,-3,

j=O

J Cmx~-3,

j=O

and C~ are the usual binomial coefficients, C~ -

rn! j!(m-j)!

Denote

s+n = {x 9 s - :~0 > 0 } , s_~ = {x 9 s n : ~ 0 < 0 } , it follows from the formulas above t h a t if x0 > 0, then era(x) 9 S~., and if x0 < 0, then era(x) 9 $2, and b o t h inclusions hold for m 9 7]. Take z + = (1, 0 , . . . , 0) and denote z +'m = era(z+). For natural m we have _

z+'m

-

ym

l y ' l ' where y'~ = (1, C ~ , . . . , C : ) .

188

3. Systems with Special Structure

Since each C~ is a polynomial in m of degree i, we see that z +'m - - * p = ( 0 , 0 , . . . , 1 ) as m--~ cr Take arbitrary d > 0. There exists m0 > 0 such that r(z+'m~ < d. Similarly one shows that there exists a point z - and a natural number rnl such that r ( z - , p ) < d and r = ( - 1 , 0 , . . . , 0 ) . Construct a sequence = {zk : k C 7/} as follows: z0 = z - , zk = Ck(z0) for k > 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

) _<<~r(Z_l,p) < ~d,

hence ~ is an (/: + 1)d-pseudotrajectory of r For any trajectory O+ C S~ we have d i s t ( O + , ( - 1 , 0 , . . . , 0 ) ) > 7r/2, for any trajectory O_ C S_~ we have d i s t ( O + , ( 1 , 0 , . . . , 0 ) ) _> 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 <

2d

(3.39)

for all j, n. Let ff = G(zj_~(n),..., zj_~+l(n)). It follows from (3.36) and from the choice of N that

]zj(n)l <_ g + d', Izj(n) + FI I <_ N + d', hence

(zj-s(n),...,zj+s-l(n), zj(n)), (zj-s(n),...,zj+s-l(n),zj(n)

+ El) 9 K.

Compare (3.38) with the equality F(zj_s(n),...

, zj+s_ 1 (rt),

r : zj(n)

and apply the second statement of Theorem 3.3.2 and estimate (3.39) to show that zj+,(n) = G(zj_,(n),..., zj+~-l(n)) + F~, IF21 < 2dco. (3.40) Now we fix n and introduce Zj

: ( z ! l ) , . . . , Z} 2s)) 9 ~2sk

setting z}" = zj_,(n),

=

zj+,_,(n)

We deduce from (3.35) that (zj) =

,

= (Zj+l-s(n),...,

=

G(zj_s(n),...,

zj+s-l(n))).

Note that ZJ+l = \ (Z j(1) +l,...,

57(2s)'~ -~ (ZJ+l--s(72), "'" , z/+~(n)), L~j+l]

so that IZj+l -

r

I = IG(zj_s(n),...,zj+s_l(72)) - zj+s(n)h

hence (3.40) implies that {Zj} is a 2dco-pseudotrajectory for r By condition (3.36), Zj E Ndo(V), and it follows from (3.37) that there exists a point x e ]R2'k such that IZj - CJ(x)l = = I ( z j - , ( n ) , . . . , zj+s-l(n)) - r

< 2coL'd = Ld for j e 27.

(3.41)

There is a logical possibility that the point z for which (3.41) holds depends on n. Denote it x(n). Apply the same procedure as above to find x(n + 1) with I(zj_~(n + 1 ) , . . . , z i + s _ , ( n + 1 ) ) - CJ(x(n + 1)) I < Ld, j 9 27. Let us estimate

196

3. Systems with Special Structure

ICJ(x(n +

1)) -

CJ(x(n))l

< I(zj_s(, + 1 ) , . . . , zj+s_l(n + 1)) - CJ(x(n + 1))1+

+l(zj-,(~ + 1),... ,zj+,_,(n + 1 ) ) - (zj_~(n),..., zs+,_,(n))l+

-[-[(Zj-s(n),..., Zj+s-l(n)) -- C J ( x ( n ) ) [ <_ (2L + 2s)d < b, j E 7] (we apply (3.31) to estimate the second term). Since dist(Zj,A) < do, [Zj - CJ(z(n))[ < Ldo (and a similar inequality holds for z(n + 1)), the inequality (L + 1)d0 < d' (see (3.37)) implies the inclusions CJ(x(n)), cJ(x(n + 1)) E U, hence it follows from the choice of b that x(n) = x(n + 1). This completes the proof. []

Remark. Of course, a statement similar to Theorem 3.3.3 holds if r is a diffeomorphism on a neigborhood of its hyperbolic set. E x a m p l e 3.1 Consider the following one-dimensional lattice dynamical system:

uj(n + 1) = uj(n) + g ( ~ j + l ( n ) - 2~j(~) + ~ j _ l ( n ) )

+

af(uj(n))

(3.42)

(it is called a system with diffusion coupling). Here ~ is the diffusion coefficient, a is a parameter, and f ( u ) = u(u - a)(1 - u) with 0 < a < 1. System (3.42) has form (3.25) with

F(Zl, z2, z3) = KZ1 + (1 -- 2~)Z2 + af(z2) + ~Z3, and it is easy to see that if ~ ~ 0, then F E GHC(2,1). It is shown in [Afl, Cho6] that, for certain values of t:, a, a, the corresponding Henon-type diffeomorphism

has a hyperbolic set (it acts like a "Smale horseshoe diffeomorphism" on some rectangle), hence Theorem 3.3.3 applies to system (3.42).

Shadowing of

d-travelling p s e u d o t r a j e c t o r i e s

Fix l, m E 7]. We define a "travelling wave" coordinate q = lj - ls + ran, then (3.28) can be rewritten in the form

v(q + ls + m) = F(v(q),v(q + l ) , . . . , v ( q + 2/s)). Let us begin with the case

(3.43)

3.3 Lattice Systems l > 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
N}.

Since F C CI(IR (2~+1)k, IRk), there is M > 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

<_ d(L1 + L2v)L, t > 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[ <

(3.58)

Mvd. By the definition of a d-pseudotrajectory, we can write Zu+I(/~ ~- 1) ~-- F ( z a ( ~ ) , Za+l(~) , za+2(/~)) -~-F2,

(3.59)

where IF21 < d. Inequality (3.32) implies that Iz~+2(/~) - z~+1(/3 + 1)[ < d, and we deduce from (3.59) that Zz+l( fl -~- 1) = F(z~(fl), z~+l(/~),

z~+l(fl + 1)) + Fa,

(3.60)

where [Fal < ( M + l ) d . Comparing (3.58), (3.60), and taking (3.55) into account, we can show that

l r z~+l(~ + 1)l < d(1 + M + My). Since z~

= z~(~,,+1)(~(2, t + 1)),

we see that ]z~+1(/~-4-1)- Z~+)I] < I v - l i d _ <

(2v + 1)d,

and finally we have IZt+l - r

= [Z~2) - (I <

d(L1 + L2v).

202

3. Systems with Special Structure []

This completes the proof.

E x a m p l e 3.2 Consider again system (3.42). Here s = 1. For n r 1 the conditions of the last theorem are satisfied. The corresponding system for (1, 1)travelling waves is generated by the following Henon-type diffeomorphism r ~

auk+ ~(1)1 =

T

(k),"~k+a ~(2)

~

iz~I)+

(1 - 2~)x~2) + a f ( x ~ 2)) 1-x

It is shown in [Af2, Cho6] that in the space of parameters (x, a, a) there is a nonempty region for which r has a hyperbolic set. Hence, Theorem 3.3.5 applies to system (3.42). Shadowing

of d-homogeneous

pseudotrajectories

Let us write Eq. (3.29) in the form v(n + 1) = G(v(n)),

(3.61)

where G(u) = F ( u , . . . , u). We assume that (3.61) satisfies conditions similar to the generalized Hadamard conditions, so that (3.61) defines a diffeomorphism r : IRk ~ 1Rk. T h e o r e m 3.3.6. Assume that the diffeomorphism r defined by (3.61) has a hyperbolic set A. Then there exist constants L, do such that if (a) {zj(n)} is a d-homogeneous d-pseudotrajectory o r e with d < do; (b) there exists j E 7] such that for all n > 0 we have dist(zj(n), A) < do, then there exists x E lit k such that Izj(n) - r

< Ld for n > O.

The proof of this theorem mostly repeats the proof of Theorem 3.3.3, and we do not give it here.

3.4 Global

Attractors

for Evolution

Systems

Theory of global attractors for infinite-dimensional dynamical systems was intensively developed in the last two decades (see [Bab, Hal, Ladl]). The main objects of application of this theory are partial differential equations. We discuss in this book applications of the shadowing theory to evolution systems generated by semilinear parabolic equations, the mostly studied class of evolution systems (see [Hel D .

3.4 Global Attractors for Evolution Systems

203

Note that the shadowing approach was applied to study qualitative behavior of discretizations for parabolic equations near hyperbolic fixed points by Larsson and Sanz-Serna ([Lar2, Lar3]). Their works developed methods of Beyn [Bey] who studied multi-step approximations of autonomous systems of ordinary differential equations near a hyperbolic rest point, and of Alouges-Debussche [Alo] who investigated pure time discretizations for parabolic equations. In this section, we apply analogs of shadowing results for structurally stable diffeomorphisms (Subsect. 2.2.2) to establish a kind of Lipschitz shadowing in a neighborhood of the global attractor for an evolution system generated by a parabolic equation [Lar4]. We show that a wide class of semilinear parabolic equations satisfy our conditions. Consider a semilinear parabolic equation u, = u ~ + f(u), u ~ n~, x C [0, 1],

(3.62)

with the Dirichlet boundary conditions u(O,t) = u(1,t) = O.

(3.63)

It is assumed that f C C 2. Let C~r be the set of functions v(x) of class C ~ on (0, 1) with compact support in (0, 1). We consider two standard Hilbert spaces, L 2, the closure of C ~ in the norm

,ivll _and H 1, the closure of C ~ in the norm

llVllH~ = llVllL2 + (fol lV,12 dz) 1/2 . Below we denote H0~ by 7-/, we write Ivl instead of IlvllHo~,and dist(g, X) is the corresponding distance between Y, X C 7-/. We assume that problem (3.62)-(3.63) generates a semigroup of operators S(t)uo, t > O, uo E 7-[, defining solutions of (3.62). It is well known (see [Hel]) that it is enough, for example, to assume that the following condition is satisfied: there is a constant C such that

uf(u) < C.

(3.64)

It is also known that under our conditions S(t)uo depends smoothly on u0 and that for a fixed t > 0 and a bounded set B C 7-/there exists C(t, B) such that IS(t)v - S(t)dl _< C(t, B)lv - v'l (3.65) for v, v' E B. Now we describe the following two properties (a) and (b) of S(t). Property (a): S(t) has a global attractor .,4 in 7-/, i.e., a compact subset of 7-I such that

204

3. Systems with Special Structure

(al) A is invariant, i.e., S(t)A = A for t 9 IR; (a2) ,4 is uniformly globally attractive, i.e., for any e > 0 and for any bounded set B in 7"l there exists T > 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<_i<_g

(every trajectory of S(t) tends to a fixed point);

(a6)

(3.67)

3.4 G l o b a l A t t r a c t o r s

A=

U

for E v o l u t i o n

W~(r,)

Systems

205

(3.68)

l
(the attractor .A is the union of unstable manifolds of the fixed points of S(t)); (a7) W~(rl) and W~(~rj) are transverse, i,j E { 1 , . . . , N}. It is known that, for a generic nonlinearity f , all fixed points of S(t) are hyperbolic. Let us explain the meaning of this statement. Fix an integer q > 0. We introduce the C q strong Whitney topology on the set of functions f : IR ~ IR of class C ~ as follows. For two functions f, g and for a compact set K C IR define the number

P~(.f,g)

~.. sup O~f av ~

~O~g - .

(3.69)

r=O vEK

The base of neighborhoods of a function f in the C q strong Whitney topology consists of the sets {g: Pqg,,(f,g) < e~}, where {K~} is a countable family of compact subsets of ]R such that any point of IR has a neighborhood that intersects a finite number of the sets K~, the equality [.JK~ = ]R n

holds, and {e,~} is a sequence of positive numbers. It is known [Hirsl] that every residual subset of the space of functions f : ]R -+ IR of class C q with the C q strong Whitney topology is dense in this space. Brunovsky and Chow [Bru] showed that there exists a residual subset of the set of functions of class C 2 on IR with the C 2 strong Whitney topology having the following two properties. (1) If f E G, then any solution p(x) of (3.66) is hyperbolic. (2) If f E ~, then the set of # > 0 for which the problem

p~ + #f(p) = 0, p(0) = p(1) = 0,

(3.70)

has nonhyperbolic solutions is countable. For a special class of functions f , it is possible to give an explicit description of the set {#} that correspond to nonhyperbolic solutions of problem (3.70). Chaffee and Infante [Cha] considered Eq. (3.62) with a nonlinearity # f E C 2 and f satisfying the conditions (1) f ( 0 ) = 0, f'(O) = 1; (2) uf"(u) < 0 for u # 0; (3)

lin~-..•

u

<_ O.

It is shown in [Hel] that problem (3.70) has nonhyperbolic solutions only if # = n27r2, n = 1,2, ....

206

3. Systems with Special Structure

Thus, for a C2-generic nonlinearity all fixed points of S(t) are hyperbolic. It follows from results of Henry [He2] and Angenent [Ang] that for hyperbolic fixed points their stable and unstable manifolds are always transverse. Hence, for a generic nonlinearity f in Eq. (3.62) satisfying condition (3.64), the global attractor .A has properties (a4) and (a7). It is easy to establish also properties (ab) and (a6). Denote a(u) = S(1)u, u E 7-/. Our goal is to study conditions under which the semi-dynamical system a has a variant of the LpSP+ on a neighborhood of the global attractor of S(t). Fix d > 0. Our proof of the shadowing property is based on the existence of a finitedimensional smooth inertial manifold constructed as follows. There exists an orthogonal projection P with finite-dimensional PT-/, and a Cl-mapping 9 : PT-/---* QT-/(where Q = I - P) such that 34, the graph of ~, has the following two properties: (ml) S(t)34 C 34, t >_ 0; (m2) 34 is exponentially attractive, i.e., for any bounded set B C 7-I there exist positive constants C, a (depending on B) such that

dist(S(t)u, 34) <_ C exp(-at)dist(u, 34)

(3.71)

for u E B and t > 0. Obviously, it follows from our assumptions that .A C A d . Let us fix an absorbing set 7"/0 for .A. It is known that there is a bounded set 7-/r such that S(t)7"lo C 7-/' for t _> 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 <_ dist(aT(zk), Zk+l) + dist(aT(zk), A/t) <_ < C2d + udist(zk, ~4) = C~d + ubk (we refer to (3.71) and (3.73) here). It follows that

bo < 2d < 2C2d, bl < C2d + ubo < 2C2d(1 + v), . . . , bn<2C2d(l+v+...+v

2C2d n) < 1 - v - C 3 d "

This completes the proof.

[]

L e m m a 3.4.2. Assume that for some u we have

~n(u) ~ 7~o, n >__o, and I~kr(u) - zkl _< L'd, k >_ O. Then Ink(u) - Uk[ < Chd, k > O, where C5 -=1+ C4 § ... + C T, C4 = C1L'. This lemma is proved similarly to statement (2) of Lemma 1.3.1. Lemmas 3.4.1 and 3.4.2 show that our problem of Lipschitz shadowing for a is reduced to the same problem for a T .

3.4 Global Attractors for Evolution Systems

209

We need below also the following statement.

Let the ball B = {]u I <_ R} be a subset ofT-to. For any d > 0 there exists a number N = N ( R , d) such that if {uk} C 7-lo, Uo E B, and

L e m m a 3.4.3.

I T(uk) -- Uk+ll --< d, then dist(u~, M ) < 2d for some n E [0, N]. Proof. Denote b~ ----dist(uk, A//) and set vl = v + 1/2. It follows from (3.73) that Vl < 1. Find N > 0 such that vNR < 2d. To obtain a contradiction, assume that bk > 2d for 0 < k < N. For 0 < k < N - 1 we have

bk+l " ( d i s t ( a T ( u k ) , M )

+ ]o'T(uk) -- Uk+l] ~ vbk + d.

By our assumption, bk > 2d, hence d < bk/2, and 1 bk+l <_ (u + ~)bk = ulbk for O < k < N - 1 . It follows that

bN <_ uln bo <_ UlNR < 2d. The obtained contradiction proves our lemma. We proceed with a sequence {zk} that satisfies the assumptions stated before Lemma 3.4.1. Take d 1 = d~ We consider below d < d 1. Denote z~, = Pzk and set vk = z~ + ~(z~) C M . We shall prove that {vk} satisfies - v +lf <_ C ' d

with C' independent of d. Since dist(zk,M) < C3d (Lemma 3.4.1), we can find wk E M such that

Iwk - zkl ~ C3d.

(3.77)

We deduce from the choice of d I that wk E 7-/o. Now it follows from (3.77) that

l a T ( w k ) - aT(zk)l <_ C6d,

(3.78)

where C6 = CTC3. From Lemma 3.4.1 and from (3.78) we conclude that IcrT(wk) -- Zk+ll <_ laT(zk) -- Zk+ll + I~rT(wk) -- aT(zk)l <-- CTd,

(3.79)

where Cr = C2 -4- C6. Denote w~ = Pwk. By our choice, zk; w k' E 171. It follows from the definition of r that

210

3. Systems with Special Structure p a T ( w k ) = Pr

= a~r(Pwk) = aT(Wk).* '

(3.80)

We want to estimate the value *

I

I

.

I

,

<_ I ~ ( z ~ ) -

I~T(zk) - zk§

I

,

I

~T(w~)l + laT(Wk) - PZk+ll.

(3.81)

Since Iz'k - w'kl = IPzk - Pwkl <_ C3d (see (3.77)), and K1 is a Lipschitz constant of a~ on HI, the first term in (3.81) does not exceed K1C3d. By (3.80), * ! laT(wk) -- Pzk+l I = IpaT(wk) -- Pzk+l I. Hence, the second term in (3.81) does not exceed IJ(w~)

- z~+x I <

CTd

(see (3.79)). Thus, we obtain the estimate *

I

I

laT(Zk) -- Zk+ll < Csd,

(3.82)

where Cs = K1 C3 + C7. Let K2 be a Lipschitz constant of 9 on HI. We obtain from (3.82) the estimate Ivk+x - r 2 4 7

_

'

- ~ (*z k ) l'

+ 1~(4§

< Cgd, - ~(aT(Zk))l * '

(3.83)

where C9 = (1 + K2)Cs. Denote by r the metric on A4 generated by the distance in 7-{. Obviously, there is a constant K3 > 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) <_ Ld, k > 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 <_ Ld, k ~ O.

211 (3.84)

Let us estimate By (3.77), the first term (and also the value ]z'k--w~l ) does not exceed C3d. Since Wk = w~k + +(w~), vk = z~k + 4i( gk), we estimate the second term by C3d(1 + g2). Now we obtain from (3.84) the inequality

Iz~ - Ck(v)l = I~k - ~ ( ~ ) 1

< (L + (C~(2 + g~)))d.

This proves our reduction statement. S h a d o w i n g on t h e inertial m a n i f o l d Properties (a4)-(aT) of the attractor A of a imply the following properties of r on .4. (a4)' .4 contains fixed points 7ra,..., 71"N of r and they are hyperbolic (we denote by W~(Tr~), W~(Tr~) their stable and unstable manifolds); (ah)' = @ W~(~,); (3.85) l_
(a6)' A=

[..J w~(Tri);

(3.86)

l <~i(N

(a7)' WS(0r,) and W~(~rj) are transverse, i , j E { 1 , . . . , g } . Let us prove only (aT) ~. Take a point

As previously, we denote by TpN the tangent space of N at p. Let

T~ = T=W"(~), T ; = T~W~(~:j).

By (a7), W~(r,) and W~(~rj) are transverse at x, i.e., T; + T~' = 7-/.

(3.87)

T; = T~:M N T;, z = s, u.

(3.88)

We claim that Note that the norm in TwA4,p E A/I, generated by the metric r on M coincides with the norm in TpA4 induced from 7"/. The equality T; = {w 9 Tp./t4 : ]Dr

--+ o, k ~ oo}

is proved in [Pil], Chap. 13. Similarly one proves that

212

3. Systems with Special Structure T~ = {w e ~ : ID~k(x)wl ~ o, k ~ ~ } .

Obviously, these equalities imply (3.88) for z = s. For z = u the proof is similar. Now it follows from (3.87) that

T; + T ~ = T , M , i.e., WS(~r,) and W~(Trj) are transverse at x. Since any fixed point 7ri of r is hyperbolic, there exist "stable" and "unstable" subspaces E~ and E~' of T ~ M with the standard properties, - E~, z = s, u, are Dr - E~ 9 E 7 = T~, M ; - there are Ki > 0, Ai E (0, 1) such that

[Dr IDr

[ <_ KiA?lv[, v E E~, m > 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~
_< Ao, IIDr

<_ Ao.

(3.91)

As previously, for a set X C A/[ we denote

0+(x)

o-(x)

= U r

k>o

= U r

o(x) = o+(x) u o-(x).

k
The following statement is proved similarly to Lemma 2.2.9. We denote by T M the tangent bundle of M . L e m m a 3.4.4. There exists A1 E (A0, 1), neigborhoods Zi of ~ri in M , and continuous subbundles Si, Ui of TAIl on Zi U O(Zi) such that (1) &(x) @ Ui(x) = T , M for x E Z,;

(e) &, v, are Dr

on O(Z,) (consequenav, th~ equalities from (1)

hold for x e O(Z,)); (S)

IIDr

~ )~1, IIDr

~ A~

for x E Z~;

(4)

s,(.) c &(.), uj(.) c v,(~)

for x e O+(Z,) n O-(Zj).

3.4 Global Attractors for Evolution Systems

213

It is shown in [Pi3] that there exists a neighborhood M2 C M1 of .4 such that r163 C M2 (the neighborhood M1 was introduced during the reduction of Theorem 3.4.1 to Theorem 3.4.2). We denote

z=Uz,. I
Let r be a Birkhoff constant for r on .M2, i.e., a positive number with the property card{k E 7/: Ck(x) C M2 \ Z} <: T for x E A42 (here card is the cardinality). Let A//3 = r Denote No = max (llDr IIDr xEJ~2

Take p E .M2, there exists m0 E [0,7"] such that q = Cm0(p) E Z~ and era(p) ~ Zt with i r l for 0 < m < m0. Define linear subspaces S(p), U(p) of Tp.M by

W(p) = Dr176

W = S, U.

Now we define a mapping ep : TpA~I --~ ,44 for p C ~4 by

ep(v) = P(p + v) + ~(P(p + v)), v e Tp.~. Since q5 is of class C 1, ep is also of class C 1. Straightforward calculation shows that

Dep(O) = I.

(3.92)

The following statement is proved similarly to Lemma 2.2.10. L e m m a 3.4.5. For p E .M2, the spaces S(p), U(p) have the properties

(1) S(p) @ U(p) = Tp.M; (2) there exists N1 > 0 such that, for QS(p),Q~(p), the projectors onto S(p), U(p) parallel to U(p), S(p), respectively, the inequalities

IIQS(p)ll, llQ~(p)ll ~ N1 hold; (s)

Dr

C S(r

Dr

C U(r

(the second inclusion holds if, in addition, p E .Ma); (4) given ~, al > 0 there exists a~ > 0 such that if p, z ~ r and r(z, r < a2, then there is a linear isomorphism H(p, z) : Tz.A4 --~ T~./M

with the properties IlH(p, z ) - Ill < al, H(p,z)[De-~l(q)DCU(p)S(P)] C S(z),

214

3. Systems with Special Structure

and a linear isomorphism O(p, z) : Tp./~ --+ TpJ~ with the properties ]]O(p,z) - I l l < el, O(p,z)[De;'(t)Dr where q = r

C U(p),

= r

It is easy to see that for v E Si(x) we have

IDCm(x)vl < g A ? l v l , m _> O, and for v E Ui(x) we have

[ncm(x)v] <_ KA~'~]v] if m < 0 and Ck(z) E ~ 2 for k E [m,0], where g = (No~A1) ~ (and T is the Birkhoff constant chosen above). Take # > T such that KAy' < Aa- Denote = r

Lemma 1.1.3 shows that r has the LpSP in a neighborhood of .2, if and only if ~ has. Thus, we work below with ~. Obviously, Lemma 3.4.5 remains true for ~ instead of r but in statement (3) .3,43 is to be replaced by JM4, where

M 4 = ~(M2). In addition, by the choice of #, the following inequalities hold:

ID~(p)v[ < AllVl, v E S(p), p c .A,42,

(3.93)

Io~-a(p)vl
(3.94)

For a > 0 we denote C~(p) = {v e T v M : Ivl < a}, 7~o(p) -- %(E~(p)). Obviously, there exists a neighborhood Ads C Ad4 of ,4 and a number c > 0 such that for p E Ads we have :De(p), ~(:Dc(p)), r162

C Ad4,

and % is a diffeomorphism of S~(p) onto :D~(p) with uniform estimates of

IlO%lh tiDe;all. Denote M = Ads- Take a sequence {vk : k > 0} C M such that

I~(vk) - vk+,l _< d. Below we denote by d' positive constants that depend on properties of ~ on M and do not depend on {vk}. At each step of the proof, we consider d that does not exceed the minimal d' previously chosen. Since we choose d' finitely many times, no generality is lost.

3.4 Global Attractors for Evolution Systems

215

Fix k > 0 and denote p = vk, z = Vk+l,Hk = T, k M . Take d' < e/2 and such that the inequality

r(z,~(p)) < d'

(3.95)

~-I(z) 9 ~l)c(p).

(3.96)

implies the inclusion Find 0 < b < c such that ~(Db(x)) C :Dd,(~(x)), x e M (obviously, b depends only on ~). It follows from (3.95) that

~(vb(p))

c

~c(z).

Thus, the mapping Ck : Cb(p) ~ Hk+l given by

Ck(w) = ~;1 o ~ o %(w) is properly defined. Let us introduce the following notation: q = ~(p), q' = e~-l(q), t = ~ - l ( z ) , t ' = e ; l ( t ) , D : VCk(0), V ' :

V~(t)V%(t'), G = Ve;l(t)D~-l(z).

Note that t', D', G are well-defined since (3.96) holds. Fix N2 > 0 such that

IlD~(z)ll, IID~-I(x)II

< g2, x 9 M.

Let N = max(N1, N2) (N1 is defined in Lemma 3.4.5). Find v0 9 (0, 1) such that A=(l+vo)2A1 < 1 and let

I+A

N~ = N1_-~.

Take ~ > 0 such that ~N1 < 1 and find u < uo with the property N

N(4N + 1)~, < -~.

(3.97)

D = De-~l(q)D~(p).

(3.98)

It follows from (3.92) that

Now we find d' such that inequality (3.95) implies the inequalities ID%(t')wL < (1 + u)lwt, w e TpM,

(3.99)

[ee-~l(q)w[ <_ (1 + ~)lwl, w e TqM,

(3.100)

216

3. Systems with Special Structure

IDe;l(t)wl <_ (1 +

~,)lwl, w 9 Tt.M

(3.101)

(we apply (3.92) and the uniform continuity of De~:, D e ; l ) ,

liD - D'tl <

(compare (3.98), the

v

(3.102)

definition of D', and take into account the previous argu-

ment), and IlU(p,z)-xfl,

llO(p,z)-llhllO-l(p,z)-Zll

<.

(3.103)

(see Lemma 3.4.5). Define Ak : Hk -+ Hk+l by

Ak = II(p, z)DqS(p) + D'O-l(p, z)q"(p). For w 9 S(p) we have w 8 = Akw = H(p, z ) D w 9 S(z) (see Lemma 3.4.5), hence

AkS(p) C S(z).

(3.104)

Since ID{(p)w I <_ A, Iw I by (3.93), it follows from (3.100) and (3.103) that Iwq _< (1 + u)=)q[wl = )qwl, hence

IlAkls(p)]l <_ A.

(3.105)

Now we consider a mapping Bk : U(z) ---+ Hk defined as follows: Bkw = O(p, z)Gw. Apply Lemma 3.4.5 to show that w ~ = Bkw 9 U(p), hence

BkU(z) C U(p),

(3.106)

and Iw~l < Alwh hence IIBklur

)~-

(3.107)

It follows from

Akw" = D'O-'(p, z)O(p, z ) G w = D'Gw = w that AkBklu(.) = I.

(3.108)

Represent

Ck(w) = Dw + p(w). Obviously there exists d' such that tr

Ip(w) - p 0 o ' ) l ~<

~lw - w'l

(3.109)

for I~1, lw'l ~< d'. Now we represent Ck(w) = A k w + x ( w ) , where X(w) = (D - Ak)w + p(w), x(O) = Ck(O).

(3.110)

3.4 Global Attractors for Evolution Systems

217

Let us estimate liD - Ak]], liD - Ak][ = IID(QS(p) + Q~(p)) - Ak[] < [[DQS(p) - II(p, z)DQ~(p)[[+

+][DQ~(p) - D'Q~(p)[[ + [[D'Q~(p) - D'O-l(p,z)Q~(p)[[.

(3.111)

Since [[Q'(p)[[ _< N, lID[[ _< ( l + v ) N _< 2N, and [[II(p,z)-I[[ < v (see (3.103)), we see that the first term on the right in (3.111) does not exceed 2N2~. The same reasons (and inequality (3.102)) show that the second term is estimated by N~. Since [[D']] < 2N, the third term is estimated by 2N2t,. This gives the inequality liD - Akll _< N(4N +

<_

(see (3.97)). Combined with (3.109), the last inequality shows that Ix(w) - X(w')[ _< ~[w - w'[ for [w[, [w'[ _< d'.

(3.112)

Since the derivatives De~(y), D e i l ( y ') are uniformly bounded for x e M, y 9 Co(x), y ' 9 :De(x),

(3.113)

there exists N* > 0 such that for x, y, yl that satisfy (3.113) we have

r(x,e~(y)) < g * ] x -

Yl, Ix -exl(y~')[--~ N*r(x, yl) 9

Since Ck(0) = e~-l(~(p)), it follows from r(~(p), z) < d that [r

< g*r(z,~(p)) < N*d.

(3.114)

Now it follows from Lemma 3.4.5 and from (3.104)-(3.108), (3.112), and (3.114) that ~bk satisfy all the conditions of Theorem 1.3.1 (with obvious change of notation). Hence, there exist d', L' > 0 (depending only on ~ and M) and a sequence wk 9 Hk such that Ck(wk) = wk+l and [wk[ _< L'd (if d _< d'). Set w~ = e.k(wk), then it follows from the definition of Ck that w~+1 = ~(w~), hence the sequence w~ is a ~-trajectory of v = w~, i.e., w~ = ~k(v). Now we deduce from the definition of N* that there exists d' such that if r(~(Vk) , Vk+l) < d < d t, then the inequalities

r(vk,~k(v)) = r(Vk,W~k) < g*[wkl = N*L'd = n d hold for k > 0. This completes the proof of Theorem 3.4.2.

4. Numerical Applications of Shadowing

As was mentioned, one of the goals of this book is to describe some "numerically oriented" applications of the shadowing theory. There is a lot of expository papers devoted to this topic. Let us mention, for example, the papers [Po, San] devoted to applications of shadowing in the study of chaotic dynamical systems and the review paper [Coo5], where the use of shadowing results in numerical computations is demonstrated.

4.1 Finite Shadowing Various methods were developed to establish the existence of a real trajectory near a computed one (see [Cool-Coo4, Ham, Saul-Sau2, V], and others). Here we prove two theorems of Chow and Palmer [Cho2, Cho3]. These results allow us to estimate how far a numerically computed finite trajectory is from a true one. Note that Hadeler [Had] applied the Newton-Kantorovich method in a similar situation. The first theorem we prove deals with one-dimensional semi-dynamical systems generated by mappings of a segment into itself. Let f : [0, 1] --~ [0, 1] be a function of class C 2. Consider a finite sequence X = { z 0 , . . . , Xy+l} C [0, 1] . We assume that Df(x,~) # 0 for 0 < n < N, and introduce the values

II~ = D f - l ( x n ) D f - l ( x n + l ) . . . D f - l ( x , O for n < m < g (note t h a t / / ~ = Df-l(xn) according to this notation). T h e o r e m 4.1.1 [Cho2]. Let

M = max "iz,,2rtxj,. ' ~ ~' =e[O,1]

Assume that, for the values N

=

max

Z

O
and

N

r = max I~--~ / / m ( x m + l - f(xm))l' O
rn=n

220

4. Numerical Applications of Shadowing

the inequality (4.1)

2M~rT < 1 holds. I f X C [e, 1 - e], where 2T e~

1 + x/1 - 2 M a r '

then there exists a point x 9 [0, 1] such that

r

1 + 2(1 + 4 1 - - - 2 M A T )

<-- o<.<<_NmaxIff(x) - x=[ _< e.

(4.2)

Proo]: Take a point x 9 [0, 1] and set z,~ = f " ( x ) - x,~, 0 <_ n < N . T h e n z~ satisfy the relations z,~+l = f(x,~) - Xn+l -~- l)f(x,~)z,~ + g.(z,~) for 0 < n < N - 1,

(4.3)

where g . ( z ) = f(x,~ + z) - f ( x . )

- Df(x,~)z.

We want to find a sequence {z,~} satisfying (4.3) and such t h a t Iz,~l < r Let S be the set of sequences 2 = {zn : Iznl < e,0 < n < N}. S can be identified with a c o m p a c t convex susbet of IRN+I. Define a m a p p i n g 7" : S ~ ]Rg + l as follows: for 2 = {z~} 9 S set N

(7-2),~ = - ~

H ' ~ ( f ( x m ) - xm+, + g , , ( z m ) ) , n = 0 , . . . , N.

Since x . 9 [e, 1 - e] and Iz~l S e, we have x~ + z~ 9 [0, 11, and it follows from the definition of g~ t h a t 7- is a continuous m a p p i n g and t h a t M

2

Ig.(z )l S TIz.I 9

(4.4)

Let us show t h a t 7- maps S into itself. Applying (4.4), we have, for 2 E 8 , rr~

X

Mo-~ 2


2

--e, 0
(we take into account t h a t e satisfies the equation M a c 2 - 2e + 2r = 0). Hence, 7- maps 8 into itself, and Brower's fixed point theorem implies the existence of a fixed point 2" of 7- in S. It follows from the definition of 7- t h a t

4.1 Finite Shadowing

221

N

z: = (rz*),

= -Df-l(x,)

~_,

II~(f(xm)

- Xm+l AV gm(Zm) ) -

m=narl

-Df-l(x,~)(f(x,~) = nf-i(xn)z~+l

- xn+l + g , ( z : ) ) =

- Df-l(x,~)(f(x,~)

- x~+i + g n ( z : ) ) ,

hence the sequence 2* is a solution of (4.3). To prove our theorem, it remains to establish the estimate from below in (4.2). Set m =

max

0
Iz l.

It follows from N

N

y~ m=n

that

U~(f(xm)

-

Xm+l) :

Z~ - - E Ilnmgrn(Z~n) rn,=n

Mam 2 Mo's < m + - - < T < m"4--2 -2

--

(we take into account that m < e) < (we apply (4.1))

( =

1+

m=

1 1 + 2(1 + ~/1 --2MAT)

) m.

This gives the desired estimate from below. In [Cho2], problems of estimation of the values a and r are discussed, and the result is applied to the quadratic mapping f ( x ) = a x ( 1 - x ) with a = 3.8 and x0 = .6. The finite shadowing result obtained in [Cho3] for the multidimensional case is based on similar ideas. Let f : ]IRk -+ IRk be a mapping of class C 2. We consider a finite sequence X = { x 0 , . . . , x N } , where x~ E IRk and the values If(x=) - Xnarl[ are small. For any sequence {h~ E IRk : 0 < n < N - 1}, the difference equation "gnarl =

is obviously solvable. Thus, the linear operator/~ : IRk(N+l) - + fi={u~EiRk:0
(4.5)

Df(x,~)u,~ + h,~, 0 < n < N - 1,

---- Unarl - -

IR kN

defined for

D f ( x , ~ ) u n , 0 < n < N - 1,

is onto. Hence, it has right inverses. We choose any such right inverse of s and denote it by s in the following theorem.

222

4. Numerical Applications of Shadowing

Remark. If we assume that the matrices D f ( x ~ ) are invertible, then we can solve Eq. (4.5) setting u n = 0 and consequently finding UN_

--Df-I(xN_I)hN_I

1 :

and so on. Obviously, this process gives a right inverse o f / : corresponding to the operator applied to define T in the proof of T h e o r e m 4.1.1 above.

Theorem

4.1.2 [Cho3].

Let M = sup I I D 2 f ( x ) l l

.

xE~ ~

Assume that Ix,~+l -

f(x.)l < d f o r 0 < n < N - 1

and the inequality

2MIIZ:-all~d _< 1 holds. Then there exists a point x E IRk such that

d

If=(x) - x=l ~ 2lie-all 1

Proof. T a k e a p o i n t satisfy the relations

forO < n < N.

/

+ ,u/ 1

-

2MIIZ:-alPd

x e IRk and set z~ = f ~ ( x ) - x ~ , O

z~+a = D f ( x , ) z ,

< n < N . T h e n z~

+ g,(z~) for 0 < n < N - 1,

(4.6)

where

g , ( z ) = f ( x n ) - x,+l + f ( x ~ + z) - f ( x , ) - D f ( x , ) z . We want to find a sequence {z,} satisfying (4.6) and such t h a t for0
Iz.I ~ ~ = 211z::-lll 1 + ~/1 - 2 M l l s

Let S be the set of sequences 2 = {z~ C 1Rk :]z~] < ~,0 < n < N}. S can be identified with a c o m p a c t convex susbet of IRk(N+0. We define g(2) E IRklV setting

(g(2))~=g~(z~), 0 < n < N write Eq. (4.6) in the form and introduce the operator T : S --~ IRk(N+1)

1,

4.2 Periodic Shadowing for Flows

223

=

Obviously, the operator 7- is continuous. Let us show that 7- maps S into itself. We first note that Ig,~(zn)t <_ Ig,~(O)l + If(x~ + z,~) - f ( x n ) - Df(x,~)z,~ I < d +

2

It follows from the definition of c that if 2 E 8, then

Hence, 7- maps S into itself, and Brower's theorem implies the existence of a fixed point 2* of ~r in S. A p p l y / ; from the left to the equality Z* : /:-lg(2") to show that 2* gives a solution of Eq. (4.6). It follows that the point x = Xo+Zo has the desired properties. ID The main attention of the authors in [Cho3] is paid to the choice of/~-1 with not very large I1s and to the influence of round-off errors in the process of computation.

4.2

Periodic

Shadowing

for Flows

In computer investigation of systems of ordinary differential equations arising in practice, it is important to be sure that a numerical solution reflects the real behavior of the system under investigation. It is shown in Sect. 1.5 that in a neighborhood of a hyperbolic set any approximate trajectory is shadowed by a real one. But from the practical viewpoint, it is a difficult problem to check the existence of a hyperbolic set. Thus, one needs other tools to shadow numerically computed orbits. One of the basic problems in qualitative theory of differential equations is the search of closed trajectories, i.e., the trajectories of nonconstant periodic solutions. A lot of computer investigation of specific systems was connected with this problem, let us mention the papers of De Gregorio [DG], Franke and Selgrade [Fr2], Schwartz [Sc], and Sinai and Vul [Si]. We devote this section to a method of periodic shadowing developed by Coomes, Kodak, and Palmer [Coo2]. This method provides a possibility to establish the existence of a real closed trajectory near an approximate one and gives error bounds for the distance between the true and the approximate closed trajectories in terms of computable quantities associated with the variational flow along the approximate trajectory. The method of periodic shadowing we describe is a part of the theory of "practical" shadowing for ordinary differential equations created by Coomes, Kor and Palmer (see also [Cool-Coo6]).

224

4. Numerical Applications of Shadowing

Let us consider system (1.178) assuming that the vector field X is of class C 2 in IR~. We again denote by ~(t,x) the trajectory of system (1.178) such that ~(0, x) = x, and by D ~ ( t , x) the corresponding variational trajectory. Fix d > 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
Let U be a convex b o u n d e d open set containing the set N U ~,([0, hmax], Yk). k=O

For this set U, we define M0 = sup IX(x)l, M1 = sup IIDX(x)ll, M2 = sup IID2X(x)ll xEU

xEU

~:EU

(here D 2 X is the second derivative of X ) . Find a n u m b e r eo 9 (0, hmjn) with the following property: if Ix - yk] < e0, then the solution .~(t, x) is defined for [0, X], and ~ ( t , x ) 9 U for t 9 [0, X], where X = hmax+eO.

Finally, we define A

--

min 0
l~r/

,

\i

0

II~I/ II

max

0
..,.""". ,,

T h e o r e m 4.2.1. Assume that (Y, h) is a pd(d, h)-pseudotrajectory for system (1.178) such that the matrix

L = I - A N . . . Ao is invertible. Let

226

4. Numerical Applications of Shadowing

C1 = 0
(

II&_,...AolIIIL-111 C = max

(N

1+ ~ IIAN...Amll + ~ II&-l...Amll m=l

m--1

)

(OCl(Xnadl)~l,c1 1~--~1 ) A vfff_~_dl.J[_ A _ I ~

dK = C

d(M1 q- ~/1 + dl) -I- 3d1 1 _ di(1 q- A - 2 ) / '

and -M = MoM1 + 2M1 exp(Mix)~/1 + dl -t- M2 exp(2M1x)(1 + dl). Assume that for the introduced quantities the inequalities (i) (1 + A-2)dl < 1;

(ii) dK < 1; (iii) 2Cd(1 - dK)-~x/1 + d I < eo; (iv) 2MC2d(1 - d K ) -2 < 1 hold. Then there exists a point xo and numbers tk, O <_ k <_ N, such that the trajectory of Xo is closed, and ttk - hkl, Ixk - Ykl <- Lod for 0 < k < N, w h e r e Xk+ 1 :

(4.13)

~(tk, Xk), 2C ~/1 + dl. Lo - 1 ---dK

To prove T h e o r e m 4.2.1, we first establish a technical statement. 4.2.1. Let A" and y be finite-dimensional vector spaces of the same dimension, let B be an open subset of X, and let ~ : B --~ Y be a mapping of class C 2 having the following properties. (1) The derivative DG(vo),Vo E B, has an inverse E. (2) The set B contains the closed ball Xo centered at Vo with radius

Lemma

= 211~:111~(vo)l.

(3)

The inequality 2MI[/CII21G(Vo)I < 1

holds, where M = max IID2~(v)ll. veXo

Then the equation a(v) = o

has a unique solution in the ball 2(o.

(4.14)

4.2 Periodic Shadowing for Flows

227

Proof. Consider the mapping T ( v ) = Vo - lC(~(v) - DG(vo)(V - vo) ). Obviously, a point v E A'0 is a fixed point of T if and only if v is a solution of (4.14). Hence, to prove our lemma it is enough to show that T maps the ball A'0 into itself and contracts on it. Take v E X0 and estimate

I7(v) - v01 < II~:ll(IG(v0)l + I~(v) - G(v0) M

DG(vo)(,

-

v0)[) <

1

_< IIX:ll (l~(vo)l + y I v - vol 2) < ~(~ + MllX:lld) < < ~ + MllX:ll2l~(vo)le < e. This shows that T ( X o ) C Xo. For v, C E Xo we have

IT(v) - T(v')l ___ II~Cll IF(v) - ~(v') -

OG(vo)(V

-

v')l _<

_< I I ~ l M l v - v'l ~ < IlX:llMdv - v'l = = 211X:ll=Ml~(vo)l Iv - v'h and it follows from condition (3) that T contracts on Xo. This completes the proof. [] Now let us prove Theorem 4.2.1. Consider the hyperplanes Hk in IR~ which are the images of ]R'~-1 under the mappings z ~-* Yk + Skz. These hyperplanes are approximately orthogonal to the vectors X ( y k ) . We want to find the sequence of points xk described in our theorem so that Xk E Hk. Thus, we are looking for a sequence of times {tk : 0 < k < N} and a sequence of points {zk E lR~-1 : 0 < k < N} such that yk+l + Sk+lZk+l = ~ ( t k , Yk + SkZk), 0 < k < N (let us recall that, by our convention, YN+I = Yo etc). We introduce the space X ~-- ]R N+I x (]Rn-1) N+I

with the norm I{sk E IR : O < k < N } , {wk e lR ~-1: 0 < k < N } l =

\O
and the space

(4.15)

228

4. Numerical Applications of Shadowing y : IRn(N+I)

with the norm

I{gk 9 ~ :

0 < k < g ) l = max bkl. 0<_k
Let B be the open set in X defined as follows: v = ({sk}, {wk}) is in B if and only if G0

[8k -- hk[ < s and [wk[ < -l + -d l '

0
Now we define a mapping G : B --* y by (G(v))k = yk+l + Sk+lWk+l -- S(sk, yk + Skwk), 0 < k < N.

(4.16)

It follows from (4.11) that

IIS~ll < {1 + d,, hence the choice of e0 implies that the mapping G is well-defined and belongs to the class C 2. Since for Xk = Yk + Skwk we have

Ix~ - y~l < ~/1 + dllw~h to prove Theorem 4.2.1 it is enough to find a solution v = ({tk}, {Zk}) of Eq. (4.14) in the closed ball of radius (Lod)/x/1 + d~ centered at

vo = ({h~), 0). Thus, it remains to show that G satisfies the conditions of L e m m a 4.2.1. Condition (1). The main technical problem is to construct the operator K: inverse to D~(vo). Note that, for u = ({Tk}, {~k}) 9 X, the value of D~(vo)u is given by

(D~(vo)u)k = - T k X ( ~ ( h k , Yk)) + Sk+l~k+l

-

-

V--(hk, yk)Sk~k, 0 < k < N.

We will approximate DG(vo) by an operator T. To define T, we first prove the invertibility of the operator

Jk= [ X(yk)* ] We consider this operator acting from ]R~ equipped with the Euclidean norm to IR x IR'~-1 equipped with the norm I(s,w)l = max(18 h Iwl). Note that

[ x(yk) ]

J~ [ix-G~)12, s~ -- I

-

(4.17)

H,

where

H =

_ S , x(y~)

k ix(yk)12

I - S~Sk

"

4.2 Periodic Shadowing for Flows

229

We consider H as an operator from ]R x ]R~-1 into itself with the norm defined above. Condition (i) of our theorem implies that [[HI[ _< dl(1 + A -2) < 1, hence Lemma 1.2.3 shows that the operator I - H is invertible, and 1

I1(I- H)-III <

1 - dl(1 + A - 2 ) "

Now it follows from equality (4.17) that the operator Jk is invertible, and

[ x(yk) ]

j[l= [ 1 ~ 2 , s k ( / _ H ) - I . This leads to the following estimate: (

1

)

IIJk-lll -< II&l[ + [X(yk)------~ 1 - dl(1 + A -2) < < -

x/1 + dl + A -1 1 - dx(1 -1- Z~-2)

.

(4.18)

We define the operator T mentioned above for u 6 X by

j-1 (

(Tu)k = k+l

--Tk[X(yk+l)l2- X(yk+l)*Yk~k(k (k+l -- Ak(k

)

0 '

< k < N.

For the operator T one easily obtains the explicit expression of the inverse operator T as follows. Take g 6 Y and define v = ({rk}, {(k}) = Tg by the formulas 1

rk -

iX(yk+x)12X(yk+l)*(YkSk(k + gk), 0 < k < N,

N ~0 =

* L - 1 (S~gN + ~ AN... AreS*gin-I), m=l

and k

~k = Ak-1...Ao(o + ~_, Ak-1...A,,S*g,,-1, 0 < k < N. "m,= l

Let us show that for any g we have TTg = g. Obviously, it is enough to check the equalities (TTg)k = gk which are equivalent to

(--rk[X(Yk+J'2--X(Yk+l)*YkSk(k) = J k + lAk(k gk'O
(4.19)

We consider only the case k # 0 (it is useful for the reader to consider the remaining case). The "first component" on the left in (4.19) is equal to

230

4. Numerical Applications of Shadowing x(u~+l)'{(~&~k

+ gk) -

YkSk~}

: X

(Y~+,)* gk.

The "second component" is equal to k+l

Ak . . . Ao~o + ~_, Ak . . . A m S ~ g m _ a -

-Ak.

k 9 Ao~o Ak ~_, . . . . Ak-1

A m S * g , ~ - a = S*k+lgk"

m=l

These equalities prove (4.19). It follows from the formulas defining 7" that we can estimate IITII _< C,

(4.20)

where the constant C is introduced in the statement of Theorem 4.2.1. Now we use 7- to construct the inverse t: of DG(vo). We claim that ] l T ( D g ( v o ) - T)I I _< dK.

(4.21)

If inequality (4.21) holds, then we deduce from condition (ii) of our theorem and from Lemma 1.2.3 that the operator K = ( I + 7-(DG(vo)

-

T)) -1

exists. In this case, it follows from the relations (gT)DG(vo)

= (I + T(DG(vo) - T))-aTDG(vo)

= (I + TD~(vo) - I)-aTDG(vo)

=

= I

that K: = KT" is the inverse of D ~ ( v o ) . Now Lemma 1.2.3 and inequalities (4.20), (4.21) imply the estimate C -

(4.22)

-

I1~11 < i - dK"

To establish (4.21): we introduce an auxiliary operator To : 2d --* y by (Tou)k = - - r k X ( y k + a ) + Sk.-[-l~k-t-1 -- YkSk~k, 0 < k < N .

Now we can estimate the left-hand side of (4.21) as follows: IIT(DG(vo) - T)II _< IlTIl(llDG(vo) - To)ll + IITo - TII). For u E X and 0 < k < N we have ] ( ( D g ( v o ) - To)u)k] <_

< (IX(~(hk, yk)) - X(yk+l)l +

] l ( D ~ , ( h k , Y k ) --

Yk)Skll)lul _<

(4.23)

4.2 Periodic Shadowing for Flows

231

< _<

+ l r;70 lul

This shows that

We can write ((To - T)u)k = g;~lJk+a (--rkX(yk+l) + Sk+l~k+l -- r k & ~ k ) - (Tu)k =

--1 ( = Ji+x

X(yk+l)'Sk+l{k+l (Ak -- S~+lYkSk){k - S;+lX(Yk+a)rk - ( I

) -

S;+lSk+l)~k+l

'

hence it follows from (4.11), (4.12), and (4.18) that liT0 - TI[ < 3d,

( 2 q- dl -t- Z~-I _ dl(1 -I- A-2)"

(4.25)

Combining estimates (4.20), (4.23), (4.24), and (4.25), we obtain the desired inequality (4.21). This shows that the operator K: is the inverse of DG(vo), and inequality (4.22) holds. Thus, condition (1) of Lemma 4.2.1 is satisfied. Condition (2). Take = 2INll IG(vo)l. It follows from (4.7) and (4.16) that IG(v0)l _< d. Now we deduce from (4.22) and from condition (iii) of our theorem that Lod 2C d e < x/1-t-d------~- 1 - d ~

< ~

eo '

(4.26)

hence the closed ball X0 of radius e centered at Vo is contained in the open set B. Thus, to prove our theorem, it remains to check condition (3) of Lemma 4.2.1. Condition (3). Take v = ({ski, {wk}), u = ({~k}, {~k}), u' = ({~}, {~}) e X. Direct calculation shows that (D2G(v)uu')k =

-~-~,-kDX(-(~k, Yk + &wk))X(Z(~k, ~k + & ~ ) ) -rkDX(S(sk,

yk + S k w k ) ) D ~ ( s k , Yk + Skwk)Sk~'k--

-r~DX(~(sk,

Yk + S k w k ) ) D ~ ( s k , Yk + Skwk)Sk~k--

232

4. Numerical Applications of Shadowing It follows from our assumptions that, for v E A'0, the inclusion

~(t, yk + Skwk) E U holds for t E [0, Sk], hence all the arguments in the expression for D2G(v) above are in U. Since O(t) = D~.(t, Yk + Skwk) satisfies a system analogous to (4.10), we immediately obtain from the Gronwall lemma that

liD-(t, yk + &wk)[I _< exp(Mlt)

(4.27)

for t C [0, sk]. Differentiating (4.10), we see that ~(t) = D2~(t, yk + Skwk) is a solution of the following initial-value problem: d - ~ = DX(~.(t, yk + Skwk))~ + D2X(~.(t, Yk + Skwk))O(t)O(t), ~(0) = O, hence

9 (t) =

O(t - s)D2X(~,(s, yk + Skwk))O(s)O(s)ds

for t E [0, sk]. Taking (4.27) into account, we obtain the estimate

IlD2~(t, Yk + & ~ ) l l -< < ~ot exp(Ml(t - s))M2 exp(2Mxs)ds = M2 exp(Mat) exp(Mlt) M1 - 1 < -

< tM2 exp(2Mlt) (we apply the elementary inequality exp(s) -- 1 < sexp(s) for s _> 0 in the last estimate). These estimates and the expression for D2~(v) above show that

IID~(v)ll ~ ~ for v ~ Xo. Thus, to complete the proof of Theorem 4.2.1 it remains to apply Lemma 4.2.1 having in mind estimates (4.22), I~(v0)l < d, and condition (iv) of the theorem. [] In [Coo2], the described method is applied to rigorously establish the existence of one asymptotically stable and one hyperbolic "saddle" periodic orbit of the Lorenz system (1.177) for certain values of the parameters a, r, and ft.

4.3 Approximations of Spectral Characteristics

233

4.3 Approximations of Spectral Characteristics In the previous sections, the shadowing approach was applied to study "finitetime" problems. In the study of infinite-time behavior of a dynamical system, its "spectral" characteristics (for example, Lyapunov exponents of trajectories) are of great interest. We investigate two types of spectral characteristics in this section. In Subsect. 4.3.1, we show that application of numerical methods to evaluate upper Lyapunov exponents near a hyperbolic set leads to resulting errors proportional to the errors of the method and to round-off errors [Corl]. Note that the problem of evaluation of Lyapunov exponents is also discussed in

[Coo5]. In Subsect. 4.3.2, we introduce symbolic images of a dynamical system generated by partitions of its phase space. It is shown that this method allows us to approximate the Morse spectrum of the investigated dynamical system [Os3]. 4.3.1 E v a l u a t i o n of Upper Lyapunov Exponents Let r be a diffeomorphism of ]Rn of class C 1. Fix a point x 9 IR'~. We define the upper Lyapunov exponent of the positive semi-trajectory O+(x) by the usual formula 1 #(O+(x)) = max limm~oo--log [DCm(x)v[ (4.28) vE~. n ,[v[=l

m

(here lim denotes limsup). Below we often write #(x) instead of #(O + (z)). Note that the value [DCm(x)vt can be written as 1Dr162m-l(x))Dr162m-2 ( z ) ) . . .

Dr

I.

(4.29)

Even if we know the semi-trajectory O+(x) exactly, due to round-off errors we really compute not the value (4.29) but an approximate value IF( r

)F( r

x) ) . . . F ( x ) v l ,

(4.30)

where F(x) is an approximation of De(x). It is known that Lyapunov exponents are not always stable with respect to small perurbations of the linear operators De(x) [By], so that if we substitute (4.30) in (4.28) instead of (4.29), this may lead to a significant change of the value #(x) (even if the values IlF(x) - DC(x)[[ are uniformly small). Below we describe conditions [Corl] under which approximate values of upper Lyapunov exponents are close to the exact ones. Fix d > 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

<_ IF(xm)w(m) - Dr < L4d,

< (4.39)

where L4 = ( M -4- L2)L3 + L2 (we apply (4.36) to estimate the second term). Take an arbitrary point q e A, fix a unit vector v E U(q), and set a(q) = IDr Since U(q) contains two unit vectors, v and - v , the function a(q) is properly defined. The unstable spaces U(q) of the hyperbolic structure are continuous with respect to q E A (see Subsect. 1.2.1), hence the function a is

238

4. Numerical Applications of Shadowing

continuous. Since the set A is compact, there exist numbers a_ and a+ such that a_ <_a(q) <_a+ for q E A. Take do E (0, d2) such that 2L4d0 < a_. Since 2/a_ is a Lipschitz constant of the function logt on [a_/2,2a+], it follows from (4.39) that if d < do, then for m :> 0 we have I log am log a=l < Lod, -

-

where Lo = 2a_L4. Hence, we obtain the inequalities 1 1 log(am-a.., ao) - log(a~m_l.., a~o)l <_Lod, m

and the desired inequality (4.33) follows.

[]

Remark. We applied the assumption r E C 2 to obtain estimate (4.36). If r is of class C 1, then for given e > 0 one can find do such that for d < do the estimate IIH~]] < e holds. Hence, for a diffeomorphism r of class C 1 one can prove an analog of Theorem 4.3.1 of the following form: given e > 0 there exists do > 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
and p(~D) = min {dist(r

: 1 < i < s,j ~ c(i)}.

A sequence z = {zk : k E 7/} of vertices of the graph G is called a path if Zk+l E c(zk) for all k. We are mostly interested in periodic paths. If z = {Zk} is a periodic path of period p, we write it as z = { z l , . . . , zv}. A periodic path { Z l , . . . , zp} is called simple if zl # zj for 1 < i < j < p. A vertex of G is called recurrent if it belongs to a periodic path. Two recurrent vertices, i and j, are called equivalent if there is a periodic path containing both i and j. It is easy to see that the set of recurrent vertices decomposes into classes {T/k} of equivalent vertices, and each periodic path z determines a unique class 7-/k = 7-/(z) it belongs to.

Remark.

An approach based on the concept of symbolic image was applied by Osipenko [Os2] to approximate the set of periodic points of a dynamical system. A close method for investigation of dynamical systems was developed by Diamond, Kloeden, and Pokrovskii [D].

Approximation process. Now we fix a covering (4.41) of the set TA. For any pair of sets/)(i) and ~D(j) with j E c(i) (what is the same, for any edge i -o j of the corresponding graph ~), we fix a point x(i,j) = (x(i,j),v(i,j)) E ~D(i) such that q~(x(i,j)) E ~(j), and introduce the number A( i, j ) = a(x( i, j ) ). For a periodic path z = { z l , . . . , zv} of period p we introduce the number

4.3 Approximations of Spectral Characteristics 1 .(z)

=

241

P =

log A(zk, Zk+l).

(4.42)

It is easy to see t h a t if a sequence z = {zk} is a periodic p a t h with two different periods p' and p", then the corresponding numbers #(z) obtained by formula (4.42) with p = p' and p = p" coincide. Let 7-I be a class of equivalent recurrent vertices of G and let z l , . . . , z q be all simple periodic paths in 7-/. Set #~(~)

= m i n { # ( z i ) : 1 < i < q),

~max(~'~) ---- m a x {#(zi) : 1 < i < q}. L e m m a 4.3.1.

For any periodic path z E ~ , #(Z)

e [~tmin(~t~), # m a x ( ' ] ' ~ ) ] .

Proof. Consider a periodic p a t h z = { Z l , . . . ,zv} of period p. If the p a t h z is not simple, there exist different indices i and j , 1 < i , j < p, such t h a t zi = zj. Since z is a periodic path, we m a y assume t h a t i = 1. Introduce two sets of vertices of 6, z ' = { Z l , . . . , Z s - a } and z " = { z y , . . . , z p } . It follows from Zj_ 1 ---+ Zj

= Z 1 and zv --+ Zl = zj

that z' and z" are periodic paths of periods p' = j - 1 and p" = p - j + 1, respectively. In this case, we write z = z' + z". Repeating this process, we finally represent the p a t h z in the form Z = /~1 z l 7!- . . . "4-

I~qZq,

where ,r are natural numbers, z l , . . . , z q are simple periodic paths of periods p l , . . . , P q , and ~.1pl + . . . + '%pq = p. Let z j = { z ~ , . . . , z ~ } . Set r j = ,r We can write

~t(Z)

1 p

1 q

pJ

=

"=

k=l

Pk~l=llOgA(zk, Zk+l) ~ P j ~ l ~ J y ~ l o g A ( z ~ , 1 P j=l

~jpjlz(zj)

Zk+ j 1 ) ~-

ETrjlz(zj)" j=l

It follows that, for any periodic path z, the n u m b e r #(z) is a linear combination of numbers #(zJ), where z j are simple periodic paths. Since 7rl + . . . + 7rq = 1, our l e m m a is proved. [] Now we consider the set

242

4. Numerical Applications of Shadowing r(v)

=

7-/ where the union is taken over all classes 7~ of equivalent recurrent vertices of the graph ~. For b > 0 we define the value 7/(b) = sup {la(x, v) - a(x', v')[: (x, v), (x', v') E TA, r((z, v), (x', v')) _< b}. Since the mapping a is continuous and the set T A is compact, we have ~(b) --~ 0 as b --* O. The values of a are positive, hence we can find a number u > 0 such that 1 -= b'

min

a(x,v).

(x,v)ETA

Denote ~ = ~(:D). Consider a finite sequence ~ = {hk : 1 < k < p} C TA, let hk = (xk,vk). Find for this sequence the value A(~) by formula (4.40). Let us assume that there exist vertices z l , . . . , z p of G such that hk E 7:)(zk) for 1 < k < p, and Z 1 --~ Z 2 --~ ... " - r Z p "-'4 Zl, (4.43) i.e., z = { z i , . . . ,zp} is a periodic path of period p. L e m m a 4.3.2.

The inequality l~(r

#(z)l < u,(~)

(4.44)

holds. Proof. Since the points hk and Hk = X(Zk, Zk+l) (chosen above to define the values A(zk, zk+l)) belong to the same set 7)(zk), we have r(hk, Ilk) <_ diamD(zk) < 6(D) = 6. Since a(z, v) > 1/u for (x, v) e TA, the inequalities I log a(x, v) -- log a(x', v')l < ula(z,

v) - a(x', v')l

hold for (x, v), (x', v') 9 TA. Hence, t log A(zk, zk+l) - log a(xk, vk)l _< ula(Hk) -- a(hk)t <_ u~?($). Obviously, this inequality proves (4.44).

T h e o r e m 4.3.3. For any covering (4.41) we have

[]

4.3 Approximations of Spectral Characteristics

(65)

243

(4.45)

c

where 5 = 5('D). Proof. Fix ~ E ~U(65) = Ep(65). By definition, of periodic dm-pseudotrajectories of 65 such that m -~ co. Find m0 such that d m < p(7)) for m m>~20. Let ~'~ = { h ~ , . . . , h ~ } . Take vertices zk,1 < 7)(zk). Since

r(65(h ), h,7+1) <

there exists a sequence ~m ~(~'~) --* ~ and dm --~ 0 as ~_ too, below we work with k < p, of ~ such that h~ C

<

we have dist(65(V(zk)), D(zk+l)) < p(:D), and it follows that the graph G contains an edge zk --~ Zk+l. This means that, for z = { z l , . . . , zv}, (4.43) is satisfied, i.e., z is a periodic path of period p, and we can apply L e m m a 4.3.2. By inequality (4.44),

l~(,f") - #(z)l < "7('~). By L e m m a 4.3.1, #(z) E Z ( / ) ) , and it follows that dist(~(~m), ~U(D)) < ur/(5). Passing to the limit as m ~ co in this inequality, we see that dist(~, Z'(D)) < u~(5) < 2u~(5), and our theorem is proved. Let us recall the definition of the Hausdorff distance for a metric space ( X , r ) . Take two sets X1,X2 C X and set

D(X1,)(2) = inf dist(x,X2), xEX1

R ( X I , X2) ~- max(D(X1, X:), D(X1, X2)). The number R ( X 1 , X 2 ) is called the Hausdorff distance between the sets XI and )(2. T h e o r e m 4.3.4. Let l) m be a sequence of coverings (4.41) such that 6,~ = 6(l) '~) ~ 0 as m ---* co. Then

R(S(Vm), e(65)) --,

o as m

Proof. It follows from Theorem 4.3.3 that D(~(65), ,U(7)m)) -* 0 as m -~ co,

244

4. N u m e r i c a l

Applications

of Shadowing

and it remains to show that D(Z(Vm), S ( ~ ) ) ~ 0 as m --+ oc. To obtain a contradiction, assume that there exists c > 0 and points #.~ E Z(~) m) such that dist(#m, ~7(45)) 3, c, m >_ 0. Since 27(79m) is a finite union of closed segments, we may assume that the points #m are endpoints of the corresponding segments. Hence, there exist periodic paths z '~ in the graphs GTM (constructed by :Dm) such that dist(#(z~), S(45)) > c, m _> 0.

(4.46)

Let z m = { z ' ~ , . . . , z ' ~ } . Take H ~ = X(z'~,z'~+l) C 79(z'~) chosen to define A(z'~,z'~+l). Since # ( H ~ ) E :D(zp+l) and diam79(z~+~) G 6m, a sequence ~m = {h~}, where hk~+tp = H ~ for 1 G k G p and l C 7/, is a periodic 26m-pseudotrajectory of 9 of period p. The equalities a(h'~) = A(z'~, Z~+l) imply that A(~m) = #(zm). Obviously, the sequence {A(~m)} is bounded, let ,k be its limit point. By the definition of 27(#), ,k e 27(#). We see that the relations #(z,~) --* A and (4.46) are contradictory. This proves our theorem. [] Theorems 4.3.3 and 4.3.4 form a base of a numerical method for approximation of the Morse spectrum S ( ~ ) [Os3].

4.4

Discretizations

of PDEs

In this section, we study shadowing properties of discretizations of a parabolic equation. Consider a parabolic equation (3.62) with the Dirichlet boundary conditions (3.63). The following semi-implicit discretization of (3.62) is studied. Fix a natural N, let D = 1 / ( N + 1), and let h > 0 be the time step. We approximate the values u ( m D , nh) of a solution of (3.62) by v,~,n > O, m = 0 , . . . , N + 1, given by the system of equations A v '~+1 = A v '~+1 + f(vn), n > 0

(4.47)

where v ~ = ( v [ , . . . , v~v) e IRN, _f(v) = ( f ( v l ) , . . . , f ( v g ) ) , and A . O n + I __ V n + l - - 'On

h

'

(Av)m = vm+l

- - 2 V m -I- Vm--1 D 2

with v0 = vg+l = 0. System (4.47) defines a mapping r = r such that v ~+1 = r by

: IR N - , IRN

4.4 Discretizations of PDEs r

245 (4.48)

where J = I - hA, and I is the unit matrix. In Subsect. 4.4.1, we study the finite-dimensional diffeomorphism r and show that, for a generic nonlinearity f , it has Morse-Smale structure on its global attractor [Eil]. This allows us to estimate the influence of round-off errors. In Subsect. 4.4.2, shadowing results of Sect. 3.4 are applied to estimate differences between approximate and exact solutions on infinite time intervals in terms of h and D [Lar4] 4.4.1 S h a d o w i n g in Discretizations The dynamical system r defined by (4.48) was investigated by Oliva, Kuhn, and Magalhs in lOll. It was shown there that if f E C ~, If'(u)] <_ M, and h M < 1,

(4.49)

then r is a diffeomorphism of class C 1. We assume everywhere in this subsection that conditions (4.49) are satisfied. Due to round-off errors, we get from a computer not the sequence {v n} generated by (4.47) but a sequence {w ~ E IRg : n > 0} such that the inequalities t/~W n+l --

(4.50)

A w n+l - f_(Wn)] < d

hold with some d > 0 (we refine the choice of a norm in IRN below). L e r n m a 4.4.1. There exists l = l ( h , N ) such that any sequence {w ~} which satisfies (4.50) is an ld-pseudotrajectory of r = r Proof. Write (4.50) as A w ~+1 = A w ~+1 + f(w ~) + z ~,

where Iz~l < d. Since W n+m ~ .

J - l ( w ~ + hf_(w~)) + h J - l z ~ = r

n) + h J - l z ~,

our lemma is proved with l = hllJ-1]l.

[]

We study the problem of shadowing of sequences {w ~} satisfying (4.50) near global attractors ,4 = .Ah,N of diffeomorphisms r It follows from Theorem 4.4.5 below that, for a generic nonlinearity f, r has the LpSP+ on a neighborhood of .A. Hence, for such f and for sequences {w" } satisfying (4.50) with small d, there exist trajectories {v ~} of our discretization such that Iv ~ - w~l < L ( h , N ) d for n > 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) <_ ao + a, lvl ~

(4.52)

holds for any N _> 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) <_ ]v'~l2 + 2aoh, therefore we have ]v'~l~ < (1 + 2bh)-'~lv~ 2 + 2aoh f i (1 + 2bh) -m, m=l

and this gives oo

lim Iv'~t2 < 2aoh ~_, (1 + 2bh0) -m = ao We take f12 = ao/b, this completes the proof.

[]

Thus, condition (4.51) guarantees the existence of the global attractor

A(h, N) of Ch,g. We will consider the set of functions f : IR ---+ IR of class C q, q > 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
m= 1,...,N

(here a,m is the ~rm-component of a). Since (4.60) holds, l is fixed, and e is arbitrary, fl can be found in an arbitrary ~'q-neighborhood of f . Set O l ( v ) = A v + fl(v).

(4.62)

250

4. Numerical Applications of Shadowing

Obviously, for v E R0,.,,, ~ i ( v ) = ~ ( v ) - a, ng~l(V) =

D~(v).

(4.63)

Hence, if v E Ro,,,~, and ~ , ( v ) = 0, then ~ ( v ) = a. It follows from (4.61) and (4.63) t h a t v is a simple fixed point of r

--~ J-l(v

+ h_fl(v)).

We see t h a t the set 7"/o,~,~ is dense. Since there exists a countable family of sets 7-/o,.,~ such that their union contains Ko, the set 7-/o = N7/o,.,, (we take the intersection corresponding to the family) is residual in ~-q, and, for f E 7/o, all fixed points of r in Ko are simple. Now consider a decomposition of { 1 , . . . , N } into disjoint subsets A1,. 9 9 Am n u m b e r e d according to the lexicographical order, t h a t is ml <

Am1) < min{i E Am~}.

rn2 iff min{i E

Let A = { A 1 , . . . , Am}. Define the set IRN = {v E IR N : vi = vj iff there exists k with

i,j E Ak}.

T h e set ]R~ defined earlier corresponds to A = {{1},...,{g}). Let zr = ( ~ r l , . . . , ~'m) be a p e r m u t a t i o n of { 1 , . . . , rn}. Take 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
- -

i
"

"

"

,m},

and ~A,-,u = { f E ~'q : fixed points of r in

RA,,~,u are simple).

Obviously, any ~A,,~,u is open in ~'q. Let us show t h a t any 7/A,~,u is dense. Denote by PA a projection of IRN onto the linear subspace 7~A = ]RA N = {v E IR N : vi = vj if there exists k with Note t h a t for v E T~A we have f(v) E 7~a. Consider the m a p p i n g

i,j E Ak}.

4.4 Discretizations of PDEs

251

~ a = p, o ~l~a which is obviously of class C 1. It follows from Sard's t h e o r e m (we take into account t h a t 4i a acts from 7~A into 7~a) t h a t m e s a e a ( S(qSa) ) = 0 (here mesa is Lebesgue m e a s u r e in ~A). Given e > 0 there exists a E T~A such that laI < e and a

Consider fl E ~'q such t h a t f l ( u ) -- f ( u ) - aj for u E [li - l, ll + l] i f j E A.(0, i = 1 , . . . , m . We see that, for v E T~A, fi(v) = f(v) - a. T h e s a m e reasons as in the case of Ro,~,u show t h a t for v E T~a,~,u with ~il,A(V) = 0 we have rankD#l,A(V) = d i m n A , (4.64) where ~i,A

:

p a ( A v + _fi(v)).

Therefore, the corresponding r has a finite n u m b e r of fixed points in Ra,,,u, and for these fixed points (4.64) holds. Obviously, fl can be found in an a r b i t r a r y ~-q neighborhood of f . Now we show t h a t these fixed points can be done simple. For simplicity of notation we assume t h a t r itself has a finite n u m b e r of fixed points, w l , . . . , w ~, in Ra.,,u, and t h a t (4.64) holds for # instead of 4)l.A. It follows t h a t given a small neighborhood Vi of w i (in 7~a) we can find a neighborhood W~ of f in ~-q such t h a t if fl E Wi and fl(W/m) = f ( w mi ) , m = 1 , . . . , N ,

(4.65)

then w i is the unique fixed point of r in Vi. Take fl such t h a t (4.65) holds for wX,... ,w s, and t k f i ( w , , ) = f ' ( w ~ ) + #, k = l , . . . , s ,

m = l,...,N,

where # E IR. Then, for any w k, p~v(/~) :-- d e t ( D r

k) - I) --

= d e t [ g - a ( I + h diag(f'(w~) + # , . . . , f'(w~N) + #)) -- I], and easy c o m p u t a t i o n shows t h a t hU # u p k ( # ) _ det-----~+ lower-degree terms.

(4.66)

252

4. Numerical Applications of Shadowing

Take small disjoint neighborhoods Vk of points w k, find the corresponding neighborhoods W~ of f in P . There exists a neighborhood W0 of f in Yq such that, for fl E W0, the corresponding diffeomorphism r has no fixed points in

\ (Vi U . . . U Thus, if we take fl E W0 such that (4.65) and (4.66) hold, then the set of fixed points of r in RA,~,~, coincides with { w l , . . . ,wS}. let #0 be the least positive root of the polynomials P~v,..., P~v. Since the set of coordinates of w l , . . . , w s is finite, we can find # E (0, #0) and a corresponding fl that satisfies (4.65) and (4.66) and belongs to a given neighborhood of f in 9v~. For this fl, all fixed points of r in RA,~,, are simple. Hence, the set ~A,~,, is dense. The given compact set K is a subset of a countable union of sets RA,r,~. Note that if A = {A1},

i.e., if 7~ A =

{ V : V1 . . . . .

VN} ,

then 7~A M K belongs to one set RA,~,~,. Hence, the set 7-/ = MT-/A,,,, is residual in ~'q. Since both ~-~ and Y~ are Baire spaces [Hirs2], 7-/is dense in ~'q. Obviously, the set 7-/q(K) is open. Let us prove that 7-/q(K) is dense. Take f C ~-q. Since 7-/is dense, we may take f E 7"/. Let Ul,. 9 us be all the coordinates of fixed points of r in K. It was shown earlier that for f E 7-/there exists a neighborhood W in 9vq such that if fl E W and f~(u,) = f(u,), i = 1 , . . . , s , (4.67) then the set of fixed points of r in K coincides with that of r Fix ~ > 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
It follows that, for the point v ~ = r176 the following inequalities are satisfied: ICk(v~ - wkl < Lad for k >_ 0.

4.4.2 D i s c r e t i z a t i o n Errors on U n b o u n d e d T i m e I n t e r v a l s

We fix two natural numbers N and K and consider the semi-implicit discretization given by (4.47) with space step D = 1 / ( g + 1) and time step h = 1 / K . Denote by ~1"~g the subspace of 7-/consisting of continuous functions v(x) linear on any segment [iD,(i + 1)D], i = 0 , . . . , N . Our discretization generates a dynamical system ~bK,N on 7-/N in the following natural way. Take a function v E 7-/N, it defines a vector u = ( v ( D ) , . . . , v ( N D ) ) e IRN. Let r be as in (4.48), consider the vector

256

4. Numerical Applications of Shadowing W = ( W l , . . . , w N ) ~- O(tt)

and define a function v' E ~ g such that v~(iD) = wi, i = 1 , . . . , N . Now we set r

= v'.

Fix a neighborhood W of the global attractor ,4 for the semigroup S(t) generated by (3.62). Let T be the integer number introduced for S(t) in Sect. 3.4. Define T(V) = aT(v) = S ( T ) v . Find a bounded neighborhood W of ,4 and numbers do, L0 such that the statement of Theorem 3.4.1 holds with these W, do, L0 for r instead of cr (this is possible due to Lemmas 3.4.1 and 3.4.2). Since .A is Lyapunov stable, we can find neighborhoods W1, W2 of ,4 and a positive number A such that

N~(W:) c W1 and S(t)W1 C W for t > 0. Applying the results of [Larl], one can prove the following statement. L e m m a 4.4.3 There exists a neigborhood Wo of.4 in TI and positive numbers Co, Ko, No such that if K >_ Ko and N > No, then (I) r ~ W~ Sot v ~ Wo and n >_ O, (2) if u E W, v E W M TIN, and the inclusions 2T S(t)u E W, 0 < t < 2T, ~K,N(v) E W, 0 < n < --h-, hold, then

IS(nh)u - 45~,N(V)I _< C0(lu - vl + D + h) for T <_ nh < 2T (here ].l is the norm of T"l = Ha).

Below we assume that the numbers Ko and No satisfy the inequality Co ( K - - - ~ - t - N o ~ ) < min (do, ~---~) .

(4.71)

Now we fix K > 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 <_ L(D + h) for n >_ 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] <_ Lod prove the s t a t e m e n t of our t h e o r e m for K T < n < 2 K T , and so on.

Remark. It follows from the proof of L e m m a 3.4.4 that one can take m0 = log(,+1/2)

2C(D + h) R

where R is the radius of a ball that contains W.

+ 1,

[3

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[Aol] [Ao2] [Bab] [Bar]

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[Ge]

[Gr] [Gul] [Gu2] [Had]

[Hall [Ham] [Har] [Hayl]

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[Moriml] [Morim2] [Morim3] [Moriy]

[Mu] [ia] [Nil] [Ni2] [Ni3]

[Nu] [Oil [Od] [Oml] [Om2] [Om3]

[Osl] [Os2] [Os3] [Pali] [Palm 1] [Palm2] [Palm3] [Par]

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[Sak5] [San]

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[Saw] [Sc] [Shul]

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[si] IS1] [Sml] [Sm2] [Stel]

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Index

(A,2)-diffeomorphism 158 absorbing set 204 adapted norm 10 Anosov diffeomorphism 31 asymptotic pseudotrajectory 84 Axiom A 146 Axiom A' 111 Baire space 172 basic set (for a diffeomorphism) 152 basic set (for a flow) 113 Birkhoff constant 129 C o transversality condition 159 C 1 A-disk 13 chain recurrent set 238 critical point 249 d-pseudosolution 89 d-pseudotrajectory 1 (d, T)-pseudotrajectory 89 d-homogeneous pseudotrajectory 192 d-static pseudotrajectory 192 d-travelling pseudotrajectory 192 dissipative diffeomorphism 246 (~)-shadowing 2 (e, r 1 (e, r 1 equivalent recurrent vertices 241 expansion constant 85 expansive system 5 expansivity constant 5 exponential dichotomy 72 finite shadowing property 3 generalized Hadamard condition 192 generic property 172

270

Index

geometric STC (geometric strong transversality condition) for diffeomorphisms 146 geometric STC (geometric strong transversality condition) for flows 115 global attractor 203 Hadamard condition 192 Hausdorff distance 243 homoclinic point 20 hyperbolic matrix 182 hyperbolic set (for a diffeomorphism) 10 hyperbolic set (for a flow) 91 hyperbolic structure (for a diffeomorphism) 10 hyperbolic structure (for a flow) 92 hyperbolicity constants (for a diffeomorphism) 10 hyperbolicity constants (for a flow) 92 hyperbolicity on a segment 54 inertial manifold 206 invariantly connected set 73 isolated invariant set 22 /-interval 174 /:p-shadowing 68 /:e,p-shadowing 72 lattice dynamical system 190 lift 173 LmSP (limit shadowing property) 64 LpSP (Lipschitz shadowing property) 3 LpSP+ 3 local stable and unstable manifolds 12 locally maximal invariant set 22 lower semicontinuous family 118 Lyapunov norm 10 monotonous norm 35 Morse spectrum 239 Morse-Smale structure 204 nonwandering point (for a flow) 111 nonwandering point (for a homeomorphism) 26 path 241 periodic discrete (d, h)-pseudotrajectory 224 periodic Morse spectrum 239 piecewise hyperbolic family 55 property of weakly parametrized shadowing 91 property of strongly parametrized shadowing 91 POTP (pseudoorbit tracing property) 2 POTP+ 2 r-interval 174 (R, g, d)-ball 56

Index recurrent vertex 241 reparametrization 91 residual subset 172 rotation number 173 Sacker-Sell spectrum 72 SUP (shadowing uniqueness property) 5 simple fixed point 248 simple periodic path 241 Smale space 31 spatially-homogeneous solution 191 spherical linear transformation 185 stable and unstable manifolds (for diffeomorphisms) 146 stable and unstable manifolds (for flows) 112 steady-state solution 191 Steinlein-Walther hyperbolic set 48 stochastic stability 2 strong shadowing property 70 strong Whitney topology 205 structural stability (for diffeomorphisms) 145 structural stability (for flows) 111 symbolic image 240 topological equivalence 111 topological stability 103 topologically Anosov homeomorphism 6 topology of uniform convergence on compact sets 248 transverse homoclinic point 21 travelling wave solution 191 upper Lyapunov exponent 233 weak (c)-shadowing 164 WSP (weak shadowing property) 164

271