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Singular analytic linear cocycles with negative infinite Lyapunov exponents

Published online by Cambridge University Press:  17 August 2017

CHRISTIAN SADEL  and
DISHENG XU
CHRISTIAN SADEL
Affiliation:
Institute of Science and Technology, 3400 Klosterneuburg, Austria and Facultad de Matemáticas, Pontificia Universidad Católica, Santiago de Chile, Chile email [email protected]
DISHENG XU
Affiliation:
Université Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Université Paris 06, F-75013, Paris, France email [email protected]
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Abstract

We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form. As a consequence, an arbitrarily small analytic perturbation leads to distinct Lyapunov exponents. Moreover, in the one-frequency case where the $k$th Lyapunov exponent is finite and the $(k+1)$st negative infinite, we obtain a simple criterion for domination in which case there is a splitting into a nilpotent part and an invertible part.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 4 , April 2019 , pp. 1082 - 1098
Copyright
© Cambridge University Press, 2017 

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