Abstract
We obtain a characterization of two classes of dynamics with nonuniformly hyperbolic behavior in terms of an admissibility property. Namely, we consider exponential dichotomies with respect to a sequence of norms and nonuniformly hyperbolic sets. We note that the approach to establishing exponential bounds along the stable and the unstable directions differs from the standard technique of substituting test sequences. Moreover, we obtain the bounds in a single step.
Highlights
Our main objective is to obtain a characterization of two classes of dynamics with nonuniformly hyperbolic behavior in terms of an admissibility property
We consider the class of exponential dichotomies with respect to a sequence of norms and the class of nonuniformly hyperbolic sets
Most of the work in the literature related to admissibility has been devoted to the study of uniform exponential dichotomies
Summary
Our main objective is to obtain a characterization of two classes of dynamics with nonuniformly hyperbolic behavior in terms of an admissibility property. We consider the class of exponential dichotomies with respect to a sequence of norms and the class of nonuniformly hyperbolic sets. In the first part of the paper we consider a nonautonomous dynamics with discrete time obtained from a sequence of linear operators on a Banach space and we characterize the notion of an exponential dichotomy with respect to a sequence of norms. We characterize exponential dichotomies with respect to a sequence of norms in terms of the admissibility of a large family of Banach spaces (the particular case of lp spaces was considered in [1]). There are substantial differences between our approach and that in [5], which provides a characterization of ergodic invariant measures with nonzero Lyapunov exponents and not of nonuniformly hyperbolic sets
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