H\"older continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps

TL;DR

This paper proves that Lyapunov exponents of certain hyperbolic cocycles depend H"older continuously on the cocycle, supported by large deviations estimates, and also establishes related statistical properties like CLT and large deviations.

Contribution

It demonstrates H"older continuity of Lyapunov exponents for hyperbolic cocycles and introduces uniform large deviations estimates, along with statistical properties.

Findings

Lyapunov exponents are H"older continuous near typical cocycles.

Established a large deviations principle for cocycle iterates.

Proved a central limit theorem for the cocycle dynamics.

Abstract

Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are H\"older continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are H\"older continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a central limit theorem and a large deviations principle.