Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential

TL;DR

This paper proves that the Lyapunov exponent for quasiperiodic operators with analytic potential is jointly continuous in frequency and energy at all irrational frequencies, regardless of Diophantine conditions.

Contribution

It establishes the joint continuity of the Lyapunov exponent without requiring Diophantine conditions on the frequency.

Findings

Lyapunov exponent is jointly continuous at irrational frequencies.

Continuity holds without Diophantine assumptions.

Results apply to quasiperiodic operators with analytic potential.

Abstract

We study regularity properties of the Lyapunov exponent L of quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove that L is jointly continuous, in frequency and energy, at every irrational frequency.