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We show that for a class of $C^2$ quasiperiodic potentials and for anyDiophantine frequency, the Lyapunov exponents of the correspondingSchr\"odinger cocycles are uniformly positive and weak H\"older continuous asfunction of energies. As a corollary, we also obtain that the correspondingintegrated density of states (IDS) is weak H\"older continous. Our approach isof purely dynamical systems, which depends on a detailed analysis of asymptoticstable and unstable directions. We also apply it to more general$\mathrm{SL}(2,\mathbb R)$ cocycles, which in turn can be applied to getuniform positivity and continuity of Lyapuonv exponents around uniquenondegenerate extremal points of any smooth potential, and to a certain classof $C^2$ Szeg\H o cocycles.

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

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