Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schrödinger Cocycles

By Yiqian Wang, Zhenghe Zhang. 2013

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Abstract 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 Szegő cocycles.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0fd697d6aead46390c963a9a050fc0a047b74efc