It is known that the spectral type of the almost Mathieu operator depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya's conjecture in [28,30]. Together with [3], we give the sharp description of phase transitions for the almost Mathieu operator.Date: December 11, 2015.

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In this paper, we consider the higher dimensional nonlinear beam equations u tt + 2 u + σ u + f (u) = 0, with periodic boundary conditions, where the nonlinearity f (u) is a realanalytic function near u = 0 with f (0) = f (0) = 0 and σ is a real parameter in an interval I ≡ [σ 1 , σ 2 ]. It is proved that for 'most' positive parameters σ lying in the finite interval I, the above equations admit a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to a Cantor family of finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theorem, modified