ON SPECTRAL CHARACTERIZATION OF NONUNIFORM HYPERBOLICITY

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.


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 R  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  1 for any choice of  that belongs to some (possibly different) space  2 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 ( 1 ,  2 ) of function spaces with the property that the corresponding admissibility condition is equivalent to the existence of exponential dichotomy.
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 Γ defined by on the space of all continuous vector fields V 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][24][25][26][27][28][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 (Ω, F, P, ) be an invertible and ergodic measure preserving dynamical system.Moreover, let A 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 M  :  → ,  ∈ Ω, such that the following result (which is a combination of Theorems 13 and 14) is valid.

Theorem 1.
The cocycle A admits a tempered dichotomy if and only if  − M  is an invertible operator on  for P-a.e.  ∈ Ω, 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][38][39] and references therein for the discussions regarding the various extensions of this theory to the infinite-dimensional setting.

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

Linear Cocycles and Tempered Exponential Dichotomy.
Consider a probability space (Ω, F, P) and assume that  : Ω → Ω is an invertible, P-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 N 0 = {0, 1, 2, . ..}.
We recall that the map  given by ( 6) is called the generator of a cocycle A.
We also introduce the notion of a tempered random variable.We say that a measurable map The following well-known result (see [40] for example) will be useful in our arguments.

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 A is a linear cocycle such that () is a compact and injective operator for P-a.e.  ∈ Ω, where () is given by (6).Then either of the following is true.
(2) There exists an infinite sequence of numbers and for each  ∈ N a decomposition Moreover, each   (),  ̸ = ∞, is a finite-dimensional subspace of .
We note that the numbers   are called Lyapunov exponents of the cocycle A. 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].It is easy to verify that  is a vector space.Furthermore, for each  ∈ N, set where It is straightforward to show that (iii) for each  ∈ N (iv) ‖x‖  ≤ ‖x‖ +1 for every x ∈  and  ∈ N.
It follows from the above properties that we can equip  with the structure of the graded Frechét space by saying that the sequence (x  ) ∈N ⊂  converges to x ∈  if and only if (x  ) ∈N converges to x in   for each  ∈ N.

Mather-Type
Operator.The other crucial ingredient in our characterization is the construction of appropriate linear operators.Let A be a linear cocycle and consider its generator  given by (6).For  ∈ Ω, we define Obviously, M  is a linear map.Let us now establish several auxiliary results.
Lemma 5.For P-a.e.  ∈ Ω, is a well-defined and bounded linear operator for each  ∈ N.
As a direct consequence of Lemma 5, we obtain the following result.Proposition 6.The operator M  :  →  is a well-defined and continuous operator for P-a.e.  ∈ Ω.
Proof.By Lemma 5, there exists a full-measure set Ω  ⊂ Ω such that M  :  +1 →   is a well-defined and bounded linear operator for each  ∈ N and  ∈ Ω  .
We begin by noting that M   ⊂  for each  ∈ Ω  .Indeed, for any x ∈ , we have that x ∈  +1 for each  ∈ N (see (24)).Hence, Lemma 5 implies that M  x ∈   for every  ∈ N. We conclude that M  x ∈ .
Let us now establish the continuity of M  .Take a sequence (x  )  that converges to x in .This implies that (x  )  converges to x in  +1 for each  ∈ N. By (30), we have that (M  x  )  converges to M  x in   for every  ∈ N and therefore also in .The proof is completed.

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 Id − M  on the space .Before we establish several auxiliary lemmas, we will introduce some additional notation.Assume that the cocycle A admits a tempered dichotomy and let (),  ∈ Ω, be the associated family of projections.For  ∈ Ω, we define linear operators Γ   ,  = 1, 2 by where x = (  ) ∈Z .
Lemma 7. Assume that the cocycle A admits a tempered dichotomy.Then, Γ 1  :  +1 →   given by ( 31) is a well-defined and bounded linear operator for P-a.e.  ∈ Ω and sufficiently large  ∈ N.
The following result follows from Lemma 7 in the same way Proposition 6 follows from Lemma 5. Proposition 8. Assume that the cocycle A admits a tempered dichotomy.Then, the operator Γ 1  :  →  given by ( 31) is a well-defined and continuous operator for P-a.e.  ∈ Ω.
Proof.By Lemma 7, there exists a full-measure set Ω  ⊂ Ω such that Γ 1  :  +1 →   is a well-defined and bounded linear operator for each  ∈ N sufficiently large and  ∈ Ω  .
We begin by noting that Γ 1   ⊂  for each  ∈ Ω  .Indeed, for any x ∈ , we have that x ∈  +1 for each  ∈ N (see (24)).Hence, Lemma 7 implies that Γ 1  x ∈   for every  ∈ N sufficiently large.Hence, ( 24) and ( 25) imply that Γ 1  x ∈ .We now prove that Γ 1  is continuous.Take a sequence (x  )  that converges to x in .This implies that (x  )  converges to x in  +1 for each  ∈ N. By (34), we have that (Γ 1  x  )  converges to Γ 1  x in   for every  ∈ N sufficiently large.This implies that (Γ 1  x  )  converges to Γ 1  x in .
Lemma 9. Assume that the cocycle A admits a tempered dichotomy.Then, the operator Γ 2  :  +1 →   given by ( 32) is well-defined and bounded for P-a.e.  ∈ Ω and sufficiently large  ∈ N.
Lemma 9 implies the following result whose proof is analogous to the proof of Proposition 8.
Proposition 10.Assume that the cocycle A admits a tempered dichotomy.Then, the operator Γ 2  :  →  given by ( 32) is a well-defined and continuous operator for P-a.e.  ∈ Ω.
Proof.The first assertion follows from Lemmas 7 and 9, while the second follows from Propositions 8 and 10.
The connection between Γ  and M  is given by the following result.
Lemma 12. Assume that the cocycle A admits a tempered dichotomy.Then, (Γ  ( − M  )) x = x, (37) for P-a.e.  ∈ Ω,  sufficiently large, and every x ∈  +1 .In particular, Proof.Observe that (recall that  is given by ( 6)) 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 A admits a tempered dichotomy.Then,  − M  is an invertible continuous operator on  for P-a.e.  ∈ Ω.
We now establish the converse of Theorem 13.

Theorem 14.
Let A be a linear cocycle with generator  as in (6) such that () is injective and compact operator for P-a.e.  ∈ Ω.Furthermore, assume that  − M  is an invertible operator on  for P-a.e.  ∈ Ω.Then, the cocycle A admits a tempered dichotomy.
Proof.In a view of Theorem 4, it is sufficient to show that all Lyapunov exponents of A are nonzero.Assume the opposite, that is, zero is a Lyapunov exponent of A and let  0 () denote the corresponding Oseledets subspace.Then, Theorem 3 implies that lim  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 A is quasicompact and that () given by ( 6) is injective for P-a.e.  ∈ Ω.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 A is quasicompact and there are no requirements on the injectivity of operators ().Regarding Theorem 4, it is known that in the finitedimensional setting it holds without injectivity assumptions (see [43,Theorem 2]).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.