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DENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR QUASIPERIODIC SL(2, R) COCYCLES IN ARBITRARY DIMENSION
 

Summary: DENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR QUASIPERIODIC
SL(2, R) COCYCLES IN ARBITRARY DIMENSION
ARTUR AVILA
Abstract. We show that given a fixed irrational rotation of the d-dimensional torus, any analytic
SL(2, R) cocycle can be perturbed so that the Lyapunov exponent becomes positive. This result
strengthens and generalizes previous results of Krikorian [K] and Fayad-Krikorian [FK]. The key
technique is the analiticity of m-functions (under the hypothesis of stability of zero Lyapunov
exponents), first observed and used in the solution of the Ten Martini Problem [AJ]. In the
appendix, we discuss the smoothness of m-functions for a larger class of systems including the
skew-shift.
1. Introduction
Let d 1 be an integer. A (d-dimensional) quasiperiodic SL(2, R) cocycle is a pair (, A)
Rd
× C0
(Rd
/Zd
, SL(2, R)) understood as a linear skew-product:
(1.1) (x, w) (x + , A(x) · w).
The Lyapunov exponent is defined by
(1.2) L(, A) = lim

  

Source: Avila, Artur - Instituto Nacional de Matemática Pura e Aplicada (IMPA)

 

Collections: Mathematics