Complex one-frequency cocycles

Avila, Artur, Jitomirskaya, Svetlana, and Sadel, Christian. "Complex one-frequency cocycles." Journal of the European Mathematical Society 016.9 (2014): 1915-1935. <http://eudml.org/doc/277261>.

@article{Avila2014,
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.},
author = {Avila, Artur, Jitomirskaya, Svetlana, Sadel, Christian},
journal = {Journal of the European Mathematical Society},
keywords = {analytic cocycles; dominated splittings; Lyapunov exponents; analytic cocycles; dominated splittings; Lyapunov exponents},
language = {eng},
number = {9},
pages = {1915-1935},
publisher = {European Mathematical Society Publishing House},
title = {Complex one-frequency cocycles},
url = {http://eudml.org/doc/277261},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Avila, Artur
AU - Jitomirskaya, Svetlana
AU - Sadel, Christian
TI - Complex one-frequency cocycles
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 9
SP - 1915
EP - 1935
AB - 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.
LA - eng
KW - analytic cocycles; dominated splittings; Lyapunov exponents; analytic cocycles; dominated splittings; Lyapunov exponents
UR - http://eudml.org/doc/277261
ER -