Uniform positivity and continuity of Lyapunov exponents for a class of C2C2 quasiperiodic Schrödinger cocycles

We show that for a class of C2C2 quasiperiodic potentials and for any fixed Diophantine   frequency, the Lyapunov exponent of the corresponding Schrödinger cocycles, as a function of energies, are uniformly positive and weakly Hölder continuous. As a corollary, we obtain that the corresponding integrated density of states is weakly Hölder continuous as well. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general SL(2,R)SL(2,R) cocycles, which in turn can be applied to get uniform positivity and continuity of Lyapunov exponents around unique nondegenerate extremal points of any smooth potential, and to a certain class of C2C2 Szegő cocycles.