 
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 ddimensional 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 FayadKrikorian [FK]. The key
technique is the analiticity of mfunctions (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 mfunctions for a larger class of systems including the
skewshift.
1. Introduction
Let d 1 be an integer. A (ddimensional) quasiperiodic SL(2, R) cocycle is a pair (, A)
Rd
× C0
(Rd
/Zd
, SL(2, R)) understood as a linear skewproduct:
(1.1) (x, w) (x + , A(x) · w).
The Lyapunov exponent is defined by
(1.2) L(, A) = lim
