Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

For quasiperiodic Schrodinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schrodinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy. From spectral theory side, the Schrodinger conjecture and the Lasts intersection spectrum conjecture have been verified. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, see open problems by Fayad-Krikorian and Jitomirskaya-Mar. In this paper, we prove the above mentioned results for ultra-differentiable potentials.