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ALMOST REDUCIBILITY AND ABSOLUTE CONTINUITY I ARTUR AVILA
 

Summary: ALMOST REDUCIBILITY AND ABSOLUTE CONTINUITY I
ARTUR AVILA
Abstract. We consider one-frequency 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 one-frequency Schr¨odinger operators.
1. Introduction
Here we consider one-frequency 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.

  

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

 

Collections: Mathematics