Continuity of Lyapunov Exponents for Random 2D Matrices

TL;DR

This paper proves that the Lyapunov exponents and Oseledets decomposition for certain random matrix cocycles depend continuously on the cocycle and invariant measure, ensuring stability of these dynamical invariants.

Contribution

It establishes the continuity of Lyapunov exponents and Oseledets decomposition for locally constant GL(2,C)-cocycles over Bernoulli shifts, extending understanding of their stability.

Findings

Lyapunov exponents depend continuously on the cocycle and measure.

Oseledets decomposition varies continuously in measure.

Results apply to locally constant cocycles over Bernoulli shifts.

Abstract

The Lyapunov exponents of locally constant GL(2;C)-cocycles over Bernoulli shifts depend continuously on the cocycle and on the invariant probability. The Oseledets decomposition also depends continuously on the cocycle, in measure.