H\"older continuity of the Lyapunov exponent for Markov cocycles via Furstenberg's Formula

TL;DR

This paper proves that the maximal Lyapunov exponent varies in a H"older continuous manner with respect to the cocycle and transition kernel for Markov cocycles, using Furstenberg's formula, near irreducible cocycles.

Contribution

It establishes the joint H"older continuity of the Lyapunov exponent for Markov cocycles, providing a more computable H"older exponent via Furstenberg's formula.

Findings

H"older continuity of Lyapunov exponent established

Dependence on stationary measure shown to be H"older continuous

Provides explicit H"older exponent for the dependence

Abstract

This paper is concerned with the study of linear cocycles over uniformly ergodic Markov shifts on a compact space of symbols. We establish the joint H\"older continuity of the maximal Lyapunov exponent as a function of the cocycle and the transition kernel in the vicinity of any irreducible cocycle with simple maximal Lyapunov exponent. Our approach, via Furstenberg's formula, shows the H\"older continuous dependence on the data of the stationary measure of the projective cocycle and in particular provides a more computable H\"older exponent.