Large deviation estimates and H\"older regularity of the Lyapunov exponents for quasi-periodic Schr\"odinger cocycles

Abstract

We consider one-dimensional quasi-periodic Schr\"odinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates which lead to optimal H\"older continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.