Uniqueness Of Ergodic Optimization Of Top Lyapunov Exponent For Typical Matrix Cocycles

TL;DR

This paper proves the uniqueness of the maximizing measure for the top Lyapunov exponent in typical matrix cocycles and shows that Lyapunov-irregular points are common in certain dynamical systems.

Contribution

It establishes the uniqueness of the maximizing measure for the top Lyapunov exponent in typical matrix cocycles and explores the prevalence of Lyapunov-irregular points.

Findings

Unique maximizing measure for typical matrix cocycles.

Lyapunov-irregular points are typical in non-uniquely ergodic systems.

Results apply broadly to ergodic optimization in dynamical systems.

Abstract

In this article, we consider the ergodic optimization of the top Lyapunov exponent. We prove that there is a unique maximising measure of top Lyapunov expoent for typical matrix cocyles. By using the results we obtain, we prove that in any non-uniquely ergodic minimal dynamical system, the Lyapunov-irregular points are typical for typical matrix cocyles.