The precise regularity of the Lyapunov exponent for $C^2$ Cos-type quasiperiodic Schrödinger cocycles with large couplings

Abstract

In this paper, we study the regularity of the Lyapunov exponent for quasiperiodic Schrödinger cocycles with $C^2$ cos-type potentials, large coupling constants, and a fixed Diophantine frequency. We obtain the absolute continuity of the Lyapunov exponent. Moreover, we prove the Lyapunov exponent is $\frac{1}{2}$-Hölder continuous. Furthermore, for any given $r\in (\frac{1}{2},1)$, we can find some energy in the spectrum where the local regularity of the Lyapunov exponent is between $(r-\epsilon )$-Hölder continuity and $(r+\epsilon )$-Hölder continuity.

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