American Institute of Mathematical Sciences

October  2009, 3(4): 631-636. doi: 10.3934/jmd.2009.3.631

Density of positive Lyapunov exponents for quasiperiodic SL(2, R)-cocycles in arbitrary dimension

 1 CNRS UMR 7586, Institut de Mathématiques de Jussieu, 175, Rue du Chevaleret, , 75013–Paris, France

Received  September 2009 Revised  December 2009 Published  January 2010

We show that given a fixed irrational rotation of the $d$-dimensional torus, any analytic SL(2, R)-cocycle can be perturbed in such a way that the Lyapunov exponent becomes positive. This result strengthens and generalizes previous results of Krikorian [6] and Fayad-Krikorian [5]. The key technique is the analyticity of $m$-functions (under the hypothesis of stability of zero Lyapunov exponents), first observed and used in the solution of the Ten-Martini Problem [2].
Citation: Artur Avila. Density of positive Lyapunov exponents for quasiperiodic SL(2, R)-cocycles in arbitrary dimension. Journal of Modern Dynamics, 2009, 3 (4) : 631-636. doi: 10.3934/jmd.2009.3.631
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