Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schr\"odinger Cocycles 

We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the Lyapunov exponents of the corresponding Schr"odinger cocycles are uniformly positive and weak H"older continuous as function of energies. As a corollary, we also obtain that the corresponding integrated density of states (IDS) is weak H"older continous. 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 $\mathrm{S}\mathrm{L}(2,\mathbb{R})$ cocycles, which in turn can be applied to get uniform positivity and continuity of Lyapuonv exponents around unique nondegenerate extremal points of any smooth potential, and to a certain class of $C^2$ Szeg\H o cocycles.