Analyticity of the Lyapunov exponents of random products of quasi-periodic cocycles

TL;DR

This paper proves that the top Lyapunov exponent of certain random quasi-periodic cocycles varies real analytically with transition probabilities, especially when the spectrum is simple, extending the understanding of Lyapunov exponent regularity.

Contribution

It establishes the real analyticity of Lyapunov exponents in the context of random quasi-periodic cocycles under simplicity conditions.

Findings

Lyapunov exponents depend real analytically on transition probabilities when simple.

Analyticity holds for all Lyapunov exponents if the spectrum is simple.

Results extend the regularity theory of Lyapunov exponents in quasi-periodic systems.

Abstract

We show that the top Lyapunov exponent $\lambda_+(p)$ , $p = (p_1, \cdots, p_N)$ with $p_i >0$ for each $i$, associated with a random product of quasi-periodic cocycles depends real analytically on the transition probabilities $p$ whenever $\lambda_+(p)$ is simple. Moreover if the spectrum at $p$ is simple (all Lyapunov exponents having multiplicity one ) then all Lyapunov exponents depend real analytically on $p$.