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

Abstract

We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the Lyapunov exponents of the corresponding Schrödinger cocycles are uniformly positive and weak Hölder continuous as function of energies. As a corollary, we also obtain that the corresponding integrated density of states (IDS) is weak Hölder 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.

49 pages, 6 figures, a continuity result added

References in corpus (8)

• Complex one-frequency cocycles
• Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities
• Positive Lyapunov Exponents for Quasiperiodic Szego cocycles
• Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity
• Continuity of the Lyapunov exponents for quasiperiodic cocycles
• Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger Operators
• Regularity and convergence rates for the Lyapunov exponents of linear co-cycles
• The dynamics of a class of quasi-periodic Schrödinger cocycles

Cited by in corpus (4)

• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• Genericity of mode-locking for quasiperiodically forced circle maps
• Cantor spectrum for a class of $C^2$ quasiperiodic Schrödinger operators
• Abundance of mode-locking for quasiperiodically forced circle maps