Continuity of the Lyapunov exponents of random matrix products

Artur Avila, Alex Eskin, Marcelo Viana

Published 2023-05-10Version 1

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.