Almost reduction and perturbation of matrix cocycles

TL;DR

This paper proves that matrix cocycles with zero Lyapunov exponents can be perturbed to become orthogonal cocycles, extending previous results to more general dynamics and dimensions, and explores their structural implications.

Contribution

It extends the reduction and perturbation results for matrix cocycles to general base dynamics and higher dimensions, including fibered versions and applications to dominated splittings.

Findings

Zero Lyapunov exponents imply cocycles can be perturbed to orthogonal form.

Extension of previous results to broader dynamical systems and dimensions.

Application to the existence of conformal dominated splittings.

Abstract

In this note, we show that if all Lyapunov exponents of a matrix cocycle vanish, then it can be perturbed to become cohomologous to a cocycle taking values in the orthogonal group. This extends a result of Avila, Bochi and Damanik to general base dynamics and arbitrary dimension. We actually prove a fibered version of this result, and apply it to study the existence of dominated splittings into conformal subbundles of general matrix cocycles.