Characterization of uniform hyperbolicity for fiber-bunched cocycles

TL;DR

This paper establishes a new criterion linking the existence of a uniform gap in Lyapunov exponents to uniform hyperbolicity in fiber-bunched cocycles, with additional examples illustrating the distinction.

Contribution

It provides a novel characterization of uniform hyperbolicity for fiber-bunched cocycles based on Lyapunov exponents, and constructs examples showing the gap does not always imply hyperbolicity.

Findings

A uniform gap in Lyapunov exponents implies uniform hyperbolicity.

Existence of an $ ext{α}$-Hölder cocycle with a gap but no hyperbolicity.

New criteria for hyperbolicity in fiber-bunched cocycles.

Abstract

We prove a new characterization of uniform hyperbolicity for fiber-bunched cocycles. Specifically, we show that the existence of a uniform gap between the Lyapunov exponents of a fiber-bunched $SL(2,\mathbb{R})$-cocycle defined over a subshift of finite type or an Anosov diffeomorphism implies uniform hyperbolicity. In addition, we construct an $\alpha$-H\"{o}lder cocycle which has uniform gap between the Lyapunov exponents, however it is not uniformly hyperbolic.