Characterizing nonuniform hyperbolicity by Mather-type admissibility

TL;DR

This paper characterizes nonuniform hyperbolicity of linear cocycles on Banach spaces through Mather-type admissibility, providing a new perspective and a proof of robustness of tempered exponential dichotomies.

Contribution

It introduces an equivalent characterization of nonuniform hyperbolicity using Mather-type admissibility and proves the robustness of tempered exponential dichotomies under perturbations.

Findings

Nonuniform hyperbolicity characterized by Mather-type admissibility.

Equivalence between non-zero Lyapunov exponents and tempered exponential dichotomy.

Robustness of exponential dichotomies under small linear perturbations.

Abstract

We consider linear cocycles acting on Banach spaces which satisfy the assumptions of the multiplicative ergodic theorem. A cocycle is nonuniformly hyperbolic if all Lyapunov exponents are non-zero, which is equivalent to the existence of a tempered exponential dichotomy. We provide an equivalent characterization of nonuniform hyperbolicity in terms of a Mather-type admissibility of a pair of weighted function spaces. As an application we give a short proof of the robustness of tempered exponential dichotomies under small linear perturbation.