 
Summary: ALMOST REDUCIBILITY AND ABSOLUTE CONTINUITY I
ARTUR AVILA
Abstract. We consider onefrequency analytic SL(2, R) cocycles. Our main
result establishes the Almost Reducibility Conjecture in the case of exponen
tially Liouville frequencies. Together with our earlier work, this implies that
all cocycles close to constant are almost reducible, independent of the fre
quency. In our forthcoming work, we discuss applications to the analysis of
the absolutely continuous spectrum of onefrequency Schr¨odinger operators.
1. Introduction
Here we consider onefrequency analytic SL(2, R) cocycles, that is, linear skew
products over an irrational rotation x x+ of the circle R/Z which have the form
(, A) : (x, w) (x + , A(x) · w) with A : R/Z SL(2, R). The iterates of the
cocycle have the form (, A)n
= (n, An) with An(x) = A(x + (n  1)) · · · A(x),
and the Lyapunov exponent is defined by
(1.1) L = lim
n
1
n
ln An(x) dx.
