Smooth quasiperiodic SL(2,\R)-cocycles (I)-Global rigidity results for rotations reducibility and Last's intersection spectrum conjecture.

Abstract: For quasiperiodic Schr\"odinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schr\"odinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy by Avila-Fayad-Krikorian. From spectral theory side, the ``Schr\"odinger conjecture" has been verified by Avila-Fayad-Krikorian and the ``Last's intersection spectrum conjecture" has been proved by Jitomirskaya-Marx. The proofs of above results crucially depend on the analyticity of the potentials. Is analyticity essential for those problems? Some open problems in this aspect were raised by Fayad-Krikorian and Jitomirskaya-Marx. In this paper, we prove the above mentioned results for ultra-differentiable potentials.

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UCI Mathematical Physics