Analyticity of the Lyapunov exponents of random products of matrices

TL;DR

This paper proves that the maximal Lyapunov exponent varies analytically with transition probabilities for random matrix products on compact symbol spaces, extending previous finite-space results using spectral and holomorphic methods.

Contribution

It extends the analyticity results of Lyapunov exponents from finite to infinite compact symbol spaces for Markovian matrix products.

Findings

Lyapunov exponent is analytic in transition probabilities

Extension from finite to infinite symbol spaces

Uses spectral properties and holomorphic function theory

Abstract

This paper is concerned with the study of random (Bernoulli and Markovian) product of matrices on a compact space of symbols. We establish the analyticity of the maximal Lyapunov exponent as a function of the transition probabilities, thus extending the results and methods of Y. Peres from a finite to an infinite (but compact) space of symbols. Our approach combines the spectral properties of the associated Markov operator with the theory of holomorphic functions in Banach spaces.