Journal of the European Mathematical Society

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Volume 16, Issue 9, 2014, pp. 1915–1935
DOI: 10.4171/JEMS/479

Published online: 2014-10-22

Complex one-frequency cocycles

Artur Avila[1], Svetlana Jitomirskaya[2] and Christian Sadel[3]

(1) Université Pierre et Marie Curie, Paris, France
(2) University of California, Irvine, United States
(3) Pontificia Universidad Católica de Chile, Santiago de Chile, Chile

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.

Keywords: Analytic cocycles, dominated splittings, Lyapunov exponents

Avila Artur, Jitomirskaya Svetlana, Sadel Christian: Complex one-frequency cocycles. J. Eur. Math. Soc. 16 (2014), 1915-1935. doi: 10.4171/JEMS/479