Large deviation type estimates for random cocycles

TL;DR

This paper establishes the continuity and regularity properties of Lyapunov exponents and Oseledets decomposition for irreducible cocycles over strongly mixing Markov shifts, extending previous results to a broader class.

Contribution

It generalizes Le Page's theorem by proving continuity and H"older regularity of Lyapunov exponents for a wider class of cocycles over Markov shifts.

Findings

Lyapunov exponents are continuous for irreducible cocycles over Markov shifts.

Gaps in the spectrum imply H"older continuity of Lyapunov exponents.

The results extend previous theorems to more general dynamical systems.

Abstract

In this paper we prove the continuity of all Lyapunov exponents, as well as the continuity of the Oseledets decomposition, for a class of irreducible cocycles over strongly mixing Markov shifts. Moreover, gaps in the Lyapunov spectrum lead to a H\"older modulus of continuity for these quantities. This result is an application of the abstract continuity theorems obtained in [4], and generalizes a theorem of E. Le Page on the H\"older continuity of the maximal LE for one-parameter families of strongly irreducible and contracting cocycles over a Bernoulli shift. This is a draft of a chapter in our forthcoming research monograph [4].