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 $\Sigma$ 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.

10 pages, 2 figures; acknowledgements added

References in corpus (3)

• Genericity of zero Lyapunov exponents
• Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals
• Generic Continuous Spectrum for Ergodic Schr"odinger Operators

Cited by in corpus (4)

• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• A geometric path from zero Lyapunov exponents to rotation cocycles
• Almost reduction and perturbation of matrix cocycles
• Density of mode-locking property for quasi-periodically forced Arnold circle maps