Characterization of hyperbolicity and generalized shadowing lemma

TL;DR

This paper extends Mather's characterization of uniform hyperbolicity to nonuniform hyperbolicity using operator invertibility on a Fréchet space and generalizes the Shadowing lemma in this broader context.

Contribution

It introduces a new operator-based criterion for nonuniform hyperbolicity and generalizes the Shadowing lemma for this setting.

Findings

Characterization of nonuniform hyperbolicity via operator invertibility.

Condition for boundary diffeomorphisms to be nonuniformly hyperbolic.

Generalization of the Shadowing lemma to nonuniform hyperbolic systems.

Abstract

J. Mather characterized uniform hyperbolicity of a discrete dynamical system as equivalent to invertibility of an operator on the set of all sequences bounded in norm in the tangent bundle of an orbit. We develop a similar characterization of nonuniform hyperbolicity and show that it is equivalent to invertibility of the same operator on a larger, Fr\'echet space. We apply it to obtain a condition for a diffeomorphism on the boundary of the set of Anosov diffeomorphisms to be nonuniformly hyperbolic. Finally we generalise the Shadowing lemma in the same context.