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We show that a one-frequency analytic SL(2,R) cocycle with Diophantinerotation vector is analytically linearizable if and only if the Lyapunovexponent is zero through a complex neighborhood of the circle. More generally,we show (without any arithmetic assumptions) that regularity implies almostreducibility, i.e., the range of validity of the perturbative analysis nearconstants is specified by a condition on the Lyapunov exponents. Together withour previous work, this establishes a Spectral Dichotomy for typicalone-frequency Schrodinger operators: they can be written as a direct sum oflarge-like and small-like operators. In particular, the typical operator has nosingular continuous spectrum.

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