 
Summary: DENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR SL(2, R)
COCYCLES
ARTUR AVILA
Abstract. We show that SL(2, R) cocycles with a positive Lyapunov expo
nent are dense in all regularity classes and for all nonperiodic dynamical
systems. For Schr¨odinger cocycles, we show prevalence of potentials for which
the Lyapunov exponent is positive for a dense set of energies.
1. Introduction
The understanding of the frequency of hyperbolic behavior is one of the central
themes in dynamics. Precise questions can be posed in several levels, for instance,
in the context of areapreserving maps:
1. Under suitable smoothness assumptions, quasiperiodicity, and hence absence
of any kind of hyperbolicity, is nonnegligible in a measuretheoretical sense
[Kol] (under suitable smoothness assumptions),
2. In low regularity (C1
), failure of nonuniform hyperbolicity (which here can
be understood as positivity of the metric entropy) is a fairly robust phenom
enon in the topological sense [B].
On the other hand, very little is understood about the emergence of nonuniformly
hyperbolic areapreseving diffeomorphisms "in the wild": in fact all known examples
