Continuity of the Lyapunov exponents of random matrix products

TL;DR

This paper proves that Lyapunov exponents of random matrix products depend continuously on the underlying matrix coefficients and probability distributions, ensuring stability of these exponents under small perturbations.

Contribution

It establishes the continuity of Lyapunov exponents with respect to matrix coefficients and probability measures in a broad setting, extending previous results.

Findings

Lyapunov exponents depend continuously on matrix coefficients.

Lyapunov exponents depend continuously on probability distributions.

Continuity holds in the weak* topology and Hausdorff sense.

Abstract

We prove that the Lyapunov exponents of random products in a (real or complex) matrix group depends continuously on the matrix coefficients and probability weights. More generally, the Lyapunov exponents of the random product defined by any compactly supported probability distribution on $GL(d)$ vary continuously with the distribution, in a natural topology corresponding to weak$^*$-closeness of the distributions and Hausdorff-closeness of their supports.