Some Characterizations of Domination

TL;DR

This paper characterizes when cocycles have dominated splittings based on singular value gaps and extends these results to sets of matrices using invariant multicones, broadening previous 2D findings.

Contribution

It provides a new characterization of dominated splittings via singular values and extends the concept of invariant multicones to higher dimensions.

Findings

A cocycle has a dominated splitting iff a uniform exponential gap exists between singular values.

Sets of matrices with cocycles having dominated splittings are characterized by invariant multicones.

Examples show multicones may lack convexity properties.

Abstract

We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets $\Sigma$ in $GL(d,\mathbb{R})$ with the property that any cocycle with values in $\Sigma$ has a dominated splitting. We characterize these sets in terms of existence of invariant multicones, thus extending a 2-dimensional result by Avila, Bochi, and Yoccoz. We give an example showing how these multicones can fail to have convexity properties.