Linear cocycles over Lorenz-like flows

TL;DR

This paper proves that for most fiber bunched linear cocycles over Lorenz-like flows, the Lyapunov exponents are simple, with only a negligible set of exceptions having higher multiplicity.

Contribution

It establishes that the set of cocycles with non-simple Lyapunov exponents is extremely small, being contained in finite unions of high-codimension submanifolds.

Findings

Lyapunov exponents are simple for typical fiber bunched cocycles

Exceptional cocycles form a set of infinite codimension

Results apply to Lorenz-like flows

Abstract

We prove that the Lyapunov exponents of typical fiber bunched linear cocycles over Lorenz-like flows have multiplicity one: the set of exceptional cocycles has infinite codimention, i.e. it is locally contained in finite unions of closed submanifolds with arbitrarily high codimension.