Regularity of Lyapunov exponents at one-point Lyapunov spectra: the semisimple case

TL;DR

This paper proves that Lyapunov exponents are pointwise log-Hölder continuous for semisimple measures with one-point spectra, using a decomposition approach and probabilistic estimates.

Contribution

It establishes the regularity of Lyapunov exponents in a specific setting, extending understanding of their stability under measure perturbations.

Findings

Lyapunov exponents are pointwise log-Hölder continuous at semisimple measures.

The proof uses a decomposition into conformal subspaces and Berry-Esseen estimates.

Continuity is with respect to the Wasserstein distance.

Abstract

We study the regularity of Lyapunov exponents as functions on the space of compactly supported probability measures on $\mathrm{GL}(d,\mathbb{R})$. We prove that the Lyapunov exponents are pointwise log-H\"older continuous with respect to the Wasserstein distance, at semisimple probability measures with one-point Lyapunov spectrum. The proof relies on a decomposition of the action into virtually conformal subspaces and a Berry-Esseen type estimate for the random walk towards these subspaces.