H\"older continuity of Lyapunov exponents for non-invertible and non-compact random cocycles

TL;DR

This paper proves that the top Lyapunov exponent for certain random linear cocycles is H"older continuous under specific conditions, with applications to Schr"odinger cocycles with unbounded potentials.

Contribution

It establishes H"older continuity of Lyapunov exponents for non-invertible, non-compact cocycles under natural conditions, extending previous regularity results.

Findings

H"older continuity of Lyapunov exponents proven under three conditions

Application to Schr"odinger cocycles with unbounded potentials

Results hold for cocycles driven by i.i.d. processes

Abstract

We study the regularity of Lyapunov exponents for random linear cocycles taking values in $\Mat_m(\R)$ and driven by i.i.d. processes. Under three natural conditions - finite exponential moments, a spectral gap between the top two Lyapunov exponents, and quasi-irreducibility of the associated semigroup - we prove that the top Lyapunov exponent is H\"older continuous with respect to the Wasserstein distance. In the final section, we apply the main results to Schr\"odinger cocycles with unbounded potentials.