Journal of Function Spaces
Volume 2017 (2017), Article ID 6707649, 8 pages
https://doi.org/10.1155/2017/6707649

Research Article

On Spectral Characterization of Nonuniform Hyperbolicity

Muna Abu Alhalawa¹ and Davor Dragičević2,3

¹Birzeit University, Birzeit, State of Palestine
²School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
³Department of Mathematics, University of Rijeka, Rijeka, Croatia

Correspondence should be addressed to Davor Dragičević; d.dragicevic@unsw.edu.au

Received 4 October 2017; Accepted 27 November 2017; Published 14 December 2017

Academic Editor: Yuri Latushkin

Copyright © 2017 Muna Abu Alhalawa and Davor Dragičević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We give a complete functional theoretic characterization of tempered exponential dichotomies in terms of the invertibility of certain linear operators acting on a suitable Frechét space. In sharp contrast to previous results, we consider noninvertible linear cocycles acting on infinite-dimensional spaces. The principal advantage of our results is that they avoid the use of Lyapunov norms.

1. Introduction

The problem of characterizing hyperbolic behaviour of dynamical systems in terms of the spectral properties of certain linear operators has a long history that goes back to the pioneering works of Perron [1] and Li [2]. More precisely, Perron [1] established a complete characterization of the exponential stability of a linear differential equation

in in terms of the solvability (in ) of the nonlinear equation

where and belong to suitable function spaces. Similar results for the discrete time dynamics were obtained by Li. The condition that (2) has a (unique) solution in some space for any choice of that belongs to some (possibly different) space is commonly referred to as admissibility condition. Clearly, this requirement can be formulated in terms of the linear operator

acting between suitable function spaces.

The major contribution to this line of the research is due to Massera and Schäffer [3]. Indeed, in contrast to the work of Perron, they have established complete characterization (in terms of admissibility) of the notion of (uniform) exponential dichotomy which includes the notion of exponential stability as a particular case. More precisely, rather than considering only the dynamics that exhibits stable behaviour, they have considered the case of dynamics with the property that the phase space splits into two complementary directions, where in one direction dynamics exhibits stable behaviour while in the complementary direction it possesses an unstable (chaotic) behaviour. In addition, they have developed an axiomatic approach to the problem of constructing all possible pairs of function spaces with the property that the corresponding admissibility condition is equivalent to the existence of exponential dichotomy.

To the best of our knowledge, the first results in this direction that deal with the infinite-dimensional dynamics are due to Dalećkiĭ and Kreĭn [4]. For more recent results devoted to continuous and discrete evolution families, we refer to [5–15] for those dealing with uniform exponential behaviour and to [16–20] for those that consider various concepts of nonuniform exponential behaviour.

In the context of smooth dynamical systems, first results are due to Mather [21] who proved that a smooth diffeomorphisms of a compact Riemannian manifold is Anosov if and only if the operator defined by

on the space of all continuous vector fields on is hyperbolic (see [22] for related results in the case of flows). Subsequent related results consider the general case of linear cocycles (or the so-called linear skew product flows) acting on Banach spaces. We refer to [23–29] and references therein. We stress that all of those works consider only uniform hyperbolic behaviour.

In the paper [30] devoted to the roughness property of nonuniform hyperbolicity (the so-called tempered dichotomy) for linear cocycles on Banach spaces, the authors posed a questions on whether it is possible to give functional theoretic characterization of nonuniform behaviour. It turns out that the answer to this question is positive and such characterization was developed in [31] (see also [32]) and applied to the above-mentioned roughness property of tempered dichotomies. However, the approach developed in [31] is far from satisfactory since the construction of appropriate spaces on which the Mather type of operator acts is given in terms of the so-called Lyapunov norms which transform nonuniform behaviour into the uniform. Thus, in order to use this characterization to detect nonuniform behaviour, one would first need to construct appropriate Lyapunov norms. Hence, the results in [31] are unfortunately only of limited applicability.

In the present paper, we propose an alternative functional theoretic characterization of tempered exponential dichotomies. More precisely, we show that the existence of tempered exponential dichotomy (under the assumptions of the multiplicative ergodic theorem) is equivalent to the invertibility of Mather-type operators acting on a certain Frechét space. Although dealing with Frechét instead of Banach spaces is in principle harder, we feel that nevertheless our results have an advantage over those in [31]. The reason for this is that our Frechét space is built in terms of the original norm and not in terms of Lyapunov norms. We stress that our approach is close in spirit to that developed in [33] for Lyapunov regular trajectories in the finite-dimensional setting, associated with a smooth diffeomorphism.

In order to formulate an explicit result, let be an invertible and ergodic measure preserving dynamical system. Moreover, let be a linear cocycle over this system which takes values in a family of compact and injective operators on some Banach space. We will construct a Frechét space as well as family of continuous operators , , such that the following result (which is a combination of Theorems 13 and 14) is valid.

Theorem 1. The cocycle admits a tempered dichotomy if and only if is an invertible operator on for -a.e. , where Id denotes the identity operator on .

We hope that our results will be applicable to the study of nonuniformly hyperbolic dynamical systems. We emphasize that since the landmark works of Oseledets [34] and in particular Pesin [35] this theory has become one of the central themes of the modern dynamical systems theory (see [36] for a detailed exposition). We refer to [37–39] and references therein for the discussions regarding the various extensions of this theory to the infinite-dimensional setting.

2. Preliminaries

In this section we recall some notions and collect previous auxiliary results that will be used in the following section.

2.1. Linear Cocycles and Tempered Exponential Dichotomy

Consider a probability space and assume that is an invertible, -preserving transformation which is ergodic. Furthermore, let be a separable Banach space and denote by the space of all bounded linear operators on . Finally, let .

We say that a map is a linear cocycle over if

(1) for ;(2) for and ;(3)

 where

(4) is a measurable map from to for each .

We recall that the map given by (6) is called the generator of a cocycle .

We also introduce the notion of a tempered random variable. We say that a measurable map is a tempered random variable if

The following well-known result (see [40] for example) will be useful in our arguments.

Proposition 2. Assume that is a tempered random variable. Then, for , there exists a measurable map such that

(1)For  -a.e. ,

(2)For  -a.e. and every , we have that

Finally, we recall the notion of a tempered exponential dichotomy [30, 31]. We say that the cocycle with generator as in (6) admits a tempered exponential dichotomy if there exist , a tempered random variable , and a family of projections , such that

(1) is a measurable map for each ;(2)for -a.e. ,

 and the map

 is invertible;(3)for -a.e. and every ,

 where

 is a well-defined linear map from to .

2.2. Multiplicative Ergodic Theorem

We now recall the version of the multiplicative ergodic theorem established by Lian and Lu [39]. For the sake of simplicity, we will restrict our attention to the case of compact cocycles.

Theorem 3. Assume that is a linear cocycle such that is a compact and injective operator for -a.e. , where is given by (6). Then either of the following is true.

(1)There is a finite sequence of numbers

 and a decomposition

 such that, for -a.e. ,

 Moreover, each , is a finite-dimensional subspace of .(2)There exists an infinite sequence of numbers

 and for each a decomposition

 such that, for -a.e. ,

 Moreover, each , , is a finite-dimensional subspace of .

We note that the numbers are called Lyapunov exponents of the cocycle . In addition, subspaces are called Oseledets subspaces. Now we are in position to state sufficient conditions for the existence of tempered exponential dichotomy. The following result was established by Lian and Lu [39].

Theorem 4. Assume that is a linear cocycle satisfying assumptions of Theorem 3. If all Lyapunov exponents of are nonzero, then admits a tempered exponential dichotomy.

2.3. Frechét Space

We now introduce our Frechét space that will play a central role in our arguments. Set

It is easy to verify that is a vector space. Furthermore, for each , set

where

It is straightforward to show that

(i) is a Banach space for each ;(ii)

(iii) for each

(iv) for every and .

It follows from the above properties that we can equip with the structure of the graded Frechét space by saying that the sequence converges to if and only if converges to in for each .

2.4. Mather-Type Operator

The other crucial ingredient in our characterization is the construction of appropriate linear operators. Let be a linear cocycle and consider its generator given by (6). For , we define

Obviously, is a linear map. Let us now establish several auxiliary results.

Lemma 5. For -a.e. ,

is a well-defined and bounded linear operator for each .

Proof. It follows from (5) that there exist and a tempered random variable such that

for -a.e. and every . Fix now and choose such that . Furthermore, let be given by Proposition 2 and let be such that and that (8), (9), and (28) hold for each . It follows that

for each , , and . We conclude that

for every and . Set . Then, and it follows readily from (30) that is a well-defined and bounded operator for each and .

As a direct consequence of Lemma 5, we obtain the following result.

Proposition 6. The operator is a well-defined and continuous operator for -a.e. .

Proof. By Lemma 5, there exists a full-measure set such that is a well-defined and bounded linear operator for each and .
We begin by noting that for each . Indeed, for any , we have that for each (see (24)). Hence, Lemma 5 implies that for every . We conclude that .
Let us now establish the continuity of . Take a sequence that converges to in . This implies that converges to in for each . By (30), we have that converges to in for every and therefore also in . The proof is completed.

3. Main Results

In this section we obtain the main results of this paper; that is, we establish the complete characterization of tempered exponential dichotomies in terms of the invertibility of operators on the space . Before we establish several auxiliary lemmas, we will introduce some additional notation. Assume that the cocycle admits a tempered dichotomy and let , , be the associated family of projections. For , we define linear operators , by

where .

Lemma 7. Assume that the cocycle admits a tempered dichotomy. Then, given by (31) is a well-defined and bounded linear operator for -a.e. and sufficiently large .

Proof. Take an arbitrary satisfying and choose . Furthermore, let be given by Proposition 2 (with respect to as in the notion of tempered dichotomy). It follows from (8), (9), and (12) that

for -a.e. and every and . Hence,

for -a.e. and every . The proof of the lemma is completed.

The following result follows from Lemma 7 in the same way Proposition 6 follows from Lemma 5.

Proposition 8. Assume that the cocycle admits a tempered dichotomy. Then, the operator given by (31) is a well-defined and continuous operator for -a.e. .

Proof. By Lemma 7, there exists a full-measure set such that is a well-defined and bounded linear operator for each sufficiently large and .
We begin by noting that for each . Indeed, for any , we have that for each (see (24)). Hence, Lemma 7 implies that for every sufficiently large. Hence, (24) and (25) imply that .
We now prove that is continuous. Take a sequence that converges to in . This implies that converges to in for each . By (34), we have that converges to in for every sufficiently large. This implies that converges to in .

Lemma 9. Assume that the cocycle admits a tempered dichotomy. Then, the operator given by (32) is well-defined and bounded for -a.e. and sufficiently large .

Proof. Using the same notation as in the proof of Lemma 7, it follows from (8), (9), and (12) that

for -a.e. and every and . Hence,

for -a.e. and every . This immediately yields the conclusion of the lemma.

Lemma 9 implies the following result whose proof is analogous to the proof of Proposition 8.

Proposition 10. Assume that the cocycle admits a tempered dichotomy. Then, the operator given by (32) is a well-defined and continuous operator for -a.e. .

Proposition 11. Assume that the cocycle admits a tempered dichotomy. Then, one has the following:

(1)the operator is well-defined and bounded for -a.e. and sufficiently large , where and are given by (31) and (32), respectively;(2)the operator is a well-defined and continuous operator for -a.e. .

Proof. The first assertion follows from Lemmas 7 and 9, while the second follows from Propositions 8 and 10.

The connection between and is given by the following result.

Lemma 12. Assume that the cocycle admits a tempered dichotomy. Then,

for -a.e. , sufficiently large, and every . In particular,

Proof. Observe that (recall that is given by (6))

Similarly, we have that

Hence,

which yields the first statement of the lemma. The second statement is a direct consequence of the first.

Theorem 13. Assume that the cocycle admits a tempered dichotomy. Then, is an invertible continuous operator on for -a.e. .

Proof. By Lemma 12, we have that

for -a.e. . Let us now prove that

for -a.e. . Observe that ( is again as in (6))

and thus

for -a.e. , every , and . Similarly,

and therefore

for -a.e. , every , and . Finally, we observe that (45) and (47) readily imply (43).

We now establish the converse of Theorem 13.

Theorem 14. Let be a linear cocycle with generator as in (6) such that is injective and compact operator for -a.e. . Furthermore, assume that is an invertible operator on for -a.e. . Then, the cocycle admits a tempered dichotomy.

Proof. In a view of Theorem 4, it is sufficient to show that all Lyapunov exponents of are nonzero. Assume the opposite, that is, zero is a Lyapunov exponent of and let denote the corresponding Oseledets subspace. Then, Theorem 3 implies that

Fix and consider a sequence defined by

By (48), . It is easy to verify that . Since , we obtain the contradiction with the invertibility of .

We note that Theorem 1 follows directly from Theorems 13 and 14.

Remark 15. We remark that Theorems 3 and 4 are in fact valid under weaker assumption that the cocycle is quasicompact and that given by (6) is injective for -a.e. . Consequently, all the results of this paper are also valid in this setting. We have decided to present our results for the particular case of compact cocycles in order to focus on the novelties of the present paper and to avoid introducing background on quasicompact cocycles (which can be found in [39, 41, 42]). We note that the notion of a quasicompact cocycle is not hard to introduce but it is quite challenging to verify in practice that the cocycle possesses this property.
Furthermore, in principle, the assumption on the injectivity of operators could be eliminated. Indeed, the most recent versions of the multiplicative ergodic theorem (see [41, 42]) require that the cocycle is quasicompact and there are no requirements on the injectivity of operators . Regarding Theorem 4, it is known that in the finite-dimensional setting it holds without injectivity assumptions (see [43, Theorem ]). We believe that those ideas, when combined with the tools developed in [38], could be extended to the general case of quasicompact cocycles on Banach spaces. However, we refrain from doing so since it would require many technical arguments that would completely overshadow main results of our paper.

Conflicts of Interest

The authors declare that there are no conflicts of interest related to this paper.

Acknowledgments

Davor Dragičević was supported in part by an Australian Research Council Discovery Project DP150100017 and by Croatian Science Foundation under the Project IP-2014-09-2285.

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