Admissibility and nonuniformly hyperbolic sets

TL;DR

This paper characterizes nonuniformly hyperbolic dynamics using admissibility properties, providing a novel approach to exponential bounds along stable and unstable directions without relying on standard test sequences.

Contribution

It introduces a new method for establishing exponential bounds in nonuniformly hyperbolic systems using admissibility, differing from traditional techniques.

Findings

Characterization of nonuniformly hyperbolic sets via admissibility

Single-step bounds for exponential dichotomies

Novel approach avoiding standard test sequences

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.