Avila's acceleration via zeros of determinants, and applications to Schrödinger 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 ${\hat{I}}^2$-Hölder continuity of the integrated density of states for supercritical quasi-periodic Schrödinger operators restricted to the $\ell$-th stratum, for any ${\hat{I}}^2<(2(\ell -1){)}^{-1}$ and $\ell \ge 2$. 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.

22 pages. Comments welcome