Avila's acceleration via zeros of determinants, and applications to Schr\"odinger cocycles

Abstract

In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove $\beta$-H\"older continuity of the integrated density of states for supercritical quasi-periodic Schr\"odinger operators restricted to the $\ell$-th stratum, for any $\beta<(2(\ell-1))^{-1}$ and $\ell\ge2$. We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.