Uniform gap in Lyapunov exponents and dominated splitting for linear cocycles

TL;DR

This paper establishes that a uniform gap between Lyapunov exponents in linear cocycles guarantees the existence of a dominated splitting, linking spectral gaps to geometric structure in dynamical systems.

Contribution

It proves that a uniform gap in Lyapunov exponents implies a dominated splitting for linear cocycles over ergodic homeomorphisms.

Findings

Uniform gap in Lyapunov exponents leads to dominated splitting

Results apply to linear cocycles over ergodic systems

Provides a new criterion for dominated splitting existence

Abstract

Given a linear cocycle over an ergodic homeomorphism on a compact metric space, we show that the existence of a uniform gap between the $p$-th and $(p+1)$-th Lyapunov exponent on a $C^0$-neighbourhood implies the existence of a dominated splitting of index $p$.