Upper bound on the regularity of the Lyapunov exponent for random products of matrices

Abstract

We prove that if $\mu$ is a finitely supported measure on ${\text{SL}}_2(\mathbb{R})$ with positive Lyapunov exponent but not uniformly hyperbolic, then the Lyapunov exponent function is not $\alpha$-Hölder around $\mu$ for any $\alpha$ exceeding the Shannon entropy of $\mu$ over the Lyapunov exponent of $\mu$.