Complex one-frequency cocycles

TL;DR

This paper demonstrates that for a dense set of analytic one-frequency complex cocycles, the Oseledets filtration is either dominated or trivial, with Lyapunov exponents depending continuously on the cocycle, unlike the continuous case.

Contribution

It establishes the dichotomy of domination or triviality for a broad class of cocycles and links Lyapunov spectrum gaps with regularity of exponent sums.

Findings

Oseledets filtration is either dominated or trivial on a dense open set.

Lyapunov exponents depend continuously on the cocycle.

Domination characterized by regularity of Lyapunov exponent sums in presence of spectral gaps.

Abstract

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents.