Older Continuity of the Integrated Density of States for Quasiperiodic Schr Odinger Equations and Averages of Shifts of Subharmonic Functions

In this paper we consider various regularity results for discrete quasiperiodic Schrr odinger equations ? n+1 ? n?1 + V (+ n!)n = En with analytic potential V. We prove that on intervals of positivity for the Lyapunov exponent the integrated density of states is HH older continuous in the energy provided ! has a typical continued fraction expansion. The proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the \avalanche{principle". The latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks. In the multifrequency case the integrated density of states is shown to have a modulus of continuity of the form exp(?j log tj) for some 0 < < 1, but currently we do not obtain HH older continuity in the case of more than one frequency. We also present a mechanism for proving the positivity of the Lyapunov exponent for large disorders for a general class of equations. The only requirement for this approach is some weak form of a large deviation theorem for the Lyapunov exponents. In particular, we obtain an independent proof of the Herman{Sorets{Spencer theorem in the multifrequency case. The approach in this paper is related to the recent nonperturbative proof of Anderson localization in the quasiperiodic case by J. Bourgain and M. Goldstein.