Sharp Holder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

Abstract

We show that if the base frequency is Diophantine, then the Lyapunov exponent of a $C^k$ quasi-periodic $SL(2,\mathbb{R})$ cocycle is $1/2$-Hölder continuous in the almost reducible regime, if $k$ is large enough. As a consequence, we show that if the frequency is Diophantine, $k$ is large enough, and the potential is $C^k$ small, then the integrated density of states of the corresponding quasi-periodic Schrödinger operator is $1/2$-Hölder continuous.

References in corpus (6)

• Genericity of zero Lyapunov exponents
• Complex one-frequency cocycles
• The absolutely continuous spectrum of the almost Mathieu operator
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators
• Continuity of Lyapunov Exponents for Random 2D Matrices
• All couplings localization for quasiperiodic operators with Lipschitz monotone potentials

Cited by in corpus (2)

• Large deviation estimates and Hölder regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles
• Hölder continuity of the integrated density of states for Extended Harper's Model with Liouville frequency