Density of positive Lyapunov exponents for quasiperiodic
SL(2, R)-cocycles in arbitrary dimension
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  and Fayad-Krikorian . 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 .
Keywords: Lyapunov exponents, quasiperiodic, cocycles.
Received: September 2009; Revised: December 2009; Available Online: January 2010.