Journal of Modern Dynamics (JMD)

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

Pages: 631 - 636, Issue 4, October 2009      doi:10.3934/jmd.2009.3.631

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Artur Avila - CNRS UMR 7586, Institut de Mathématiques de Jussieu, 175, Rue du Chevaleret, , 75013–Paris, France (email)

Abstract: 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].

Keywords:  Lyapunov exponents, quasiperiodic, cocycles.
Mathematics Subject Classification:  Primary: 34C20; Secondary: 37Cxx.

Received: September 2009;      Revised: December 2009;      Available Online: January 2010.