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

TL;DR

This paper establishes an upper bound on the regularity of the Lyapunov exponent for certain random matrix products, linking it to the measure's entropy and hyperbolicity properties.

Contribution

It provides a theoretical bound on the Hölder continuity of the Lyapunov exponent function for non-uniformly hyperbolic measures with finite support.

Findings

Lyapunov exponent function is not Hölder continuous beyond a certain threshold

The threshold is determined by the Shannon entropy over the Lyapunov exponent

Results apply to finitely supported measures on SL(2,R) with positive Lyapunov exponent

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 $\alpha$-H\"older around $\mu$ for any $\alpha$ exceeding the Shannon entropy of $\mu$ over the Lyapunov exponent of $\mu$.