Semisimplicity of the Lyapunov spectrum for irreducible cocycles

TL;DR

This paper proves that strongly irreducible cocycles over semisimple Lie group actions are conjugate to block conformal cocycles, contributing to the understanding of Lyapunov spectrum structure in dynamical systems.

Contribution

It establishes the conjugacy of irreducible cocycles to block conformal form, advancing the classification of invariant measures in dynamical systems.

Findings

Cocycle conjugacy to block conformal form

Application to measure classification in moduli space

Extension of previous theoretical frameworks

Abstract

Let $G$ be a semisimple Lie group acting on a space $X$, let $\mu$ be a compactly supported measure on $G$, and let $A$ be a strongly irreducible linear cocycle over the action of $G$. We then have a random walk on $X$, and let $T$ be the associated shift map. We show that the cocycle $A$ over the action of $T$ is conjugate to a block conformal cocycle. This statement is used in the recent paper by Eskin-Mirzakhani on the classifications of invariant measures for the SL(2,R) action on moduli space. The ingredients of the proof are essentially contained in the papers of Guivarch and Raugi and also Goldsheid and Margulis.