Discontinuity of Lyapunov exponent in spaces of quasiperiodic cocycles: Smoothness vs Arithmetic

TL;DR

This paper investigates how the Lyapunov exponent's continuity depends on the smoothness of quasiperiodic cocycles, revealing that certain smoothness spaces like Gevrey $G^2$ mark a transition point for continuity.

Contribution

It constructs explicit examples of Lyapunov exponent discontinuity in quasiperiodic cocycles, highlighting the role of smoothness and arithmetic properties of frequencies.

Findings

Gevrey $G^2$ space is the transition space for continuity.

Discontinuity can occur in less smooth spaces for certain frequencies.

Smoother spaces generally favor continuity, depending on frequency approximation difficulty.

Abstract

We construct examples of discontinuity of Lyapunov exponent in the spaces of quasiperiodic $\mathrm{SL}(2,\mathbb R)$-cocycles for fixed irrational frequencies. Especially, we prove that the Gevrey space $G^2$ is the transition space of continuity for all strong Diophantine frequencies. We also construct examples of discontinuity for other frequencies in less smooth spaces, which show that the more difficult it is to approximate the frequency with rational numbers, the more likely it is to exhibit discontinuity in smoother spaces.