KAM, Lyapunov exponents, and the Spectral Dichotomy for typical one-frequency Schrodinger operators

We show that a one-frequency analytic SL(2,R) cocycle with Diophantine rotation vector is analytically linearizable if and only if the Lyapunov exponent is zero through a complex neighborhood of the circle. More generally, we show (without any arithmetic assumptions) that regularity implies almost reducibility, i.e., the range of validity of the perturbative analysis near constants is specified by a condition on the Lyapunov exponents. Together with our previous work, this establishes a Spectral Dichotomy for typical one-frequency Schrodinger operators: they can be written as a direct sum of large-like and small-like operators. In particular, the typical operator has no singular continuous spectrum.