Generic simple cocycles over Markov maps

TL;DR

This paper proves that a specific condition ensuring simple Lyapunov exponents for linear cocycles over Markov maps is generic, meaning it holds for most such cocycles except for a very small set.

Contribution

It demonstrates that the criterion for simple Lyapunov exponents is generic among fiber bunched linear cocycles over Markov maps, extending previous explicit conditions.

Findings

The sufficient condition for multiplicity 1 Lyapunov exponents is generic.

Exceptional cocycles form a set of infinite codimension.

The result applies to fiber bunched linear cocycles over Markov maps.

Abstract

Avila and Viana exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity 1. Here, in terms of geometric perturbations, we prove that this sufficient criterion is generic in the space of all fiber bunched linear cocycles over Markov maps: the set of exceptional cocycles has infinite codimention, i.e. it is locally contained in finite unions of closed submanifolds with arbitrarily high codimension.