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arXiv · 1311.4282

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

Yiqian Wang·Zhenghe Zhang
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Abstract

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 $\mathrm{SL}(2,\mathbb R)$ 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.

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Yiqian Wang, Zhenghe Zhang. 2013-11-18. Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schr\"odinger Cocycles . https://arxiv.org/abs/1311.4282

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