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.

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We introduce the concept of mixed random -quasiperiodic linear cocycles. We characterize the ergodicity of the base dynamics and establish a large deviations type estimate for certain types of observables. For the fiber dynamics we prove the uniform upper semicontinuity of the maximal Lyapunov exponent. This paper is meant to introduce a model to be studied in depth in further projects.

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