We show that if the base frequency is Diophantine, then the Lyapunov exponent of a C k quasi-periodic SL(2, R) cocycle is 1/2-Hölder continuous in the almost reducible regime, if k is large enough. As a consequence, we show that if the frequency is Diophantine, k is large enough, and the potential is C k small, then the integrated density of states of the corresponding quasi-periodic Schrödinger operator is 1/2-Hölder continuous.

This paper is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H.Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density, of reducible cocycles near a constant. Some algebraic structure can also be preserved, by doubling the period if needed.Moreover, in dimension 2 or if G = GL(n, C) or U(n), Z ǫ ,Ā ǫ ,F ǫ are continuous on T d .

The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasi-periodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H.Eliasson which deal with the diophantine case so as to implement a Brjuno-Rüssmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Pöschel-Rüssmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.Quasiperiodic cocycles are the fundamental solutions of quasi-periodic linear systems

Let G be a linear Lie group. We define the G

: Quasi-periodic cocycles with a diophantine frequency and with values in SL(2,R) are shown to be almost reducible as long as they are close enough to a constant, in the topology of k times differentiable functions, with k great enough. Almost reducibility is obtained by analytic approximation after a loss of differentiability which only depends on the frequency and on the constant part. As in the analytic case, if their fibered rotation number is diophantine or rational with respect to the frequency, such cocycles are in fact reducible. This extends Eliasson's theorem on Schrödinger cocycles to the differentiable case.