Uniform positivity and continuity of Lyapunov exponents for a class of C2 quasiperiodic Schrodinger cocycles

We show that for a class of C-2 quasiperiodic potentials and for any fixed Diophantine frequency, the Lyapunov exponent of the corresponding Schrodinger cocycles, as a function of energies, are uniformly positive and weakly Holder continuous. As a corollary, we obtain that the corresponding integrated density of states is weakly Holder 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) 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 C-2 Szego cocycles. (C) 2015 Elsevier Inc. All rights reserved.