Some Characterizations of Domination

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 $Σ$ in $GL(d,\mathbb{R})$ with the property that any cocycle with values in $Σ$ 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.

10 pages, 2 figures; acknowledgements added