Coexistence of measures with simple Lyapunov spectrum for fiber-bunched cocycles

TL;DR

This paper establishes conditions under which fiber-bunched cocycles over hyperbolic shifts have simple Lyapunov spectra, linking the spectral simplicity of all such measures to a specific irreducibility condition.

Contribution

It proves an equivalence between the existence of an ergodic measure with simple spectrum and all such measures sharing this property under certain irreducibility conditions.

Findings

Existence of an ergodic measure with simple spectrum under irreducibility.

Equivalence of simplicity of Lyapunov spectrum among all ergodic measures with full support.

Characterization of spectral simplicity in fiber-bunched cocycles over hyperbolic shifts.

Abstract

We prove that if a H\"older continuous fiber-bunched cocycle $\hat{A}$ over an invertible hyperbolic transitive shift $\hat{\Sigma}$ satisfies an appropriate strong irreducibility condition on Grassmannians, then $\hat{\Sigma}$ admits an ergodic measure $\hat{\mu}$ with full support and product structure with simple Lyapunov spectrum if and only if any other ergodic measure with full support and product structure also has simple Lyapunov spectrum.