Examples of Discontinuity of Lyapunov Exponent in Smooth Quasi-Periodic Cocycles

TL;DR

This paper constructs examples of smooth quasi-periodic cocycles where the Lyapunov exponent exhibits discontinuity, highlighting irregular behavior in the regularity of Lyapunov exponents for certain dynamical systems.

Contribution

It demonstrates the existence of smooth cocycles with discontinuous Lyapunov exponents at specific points, advancing understanding of regularity issues in quasi-periodic systems.

Findings

Lyapunov exponent discontinuity in ${ m C}^l$ topology

Construction of examples in Schrödinger class

Discontinuity occurs for fixed irrational rotation of bounded-type

Abstract

We study the regularity of the Lyapunov exponent for quasi-periodic cocycles $(T_\omega, A)$ where $T_\omega$ is an irrational rotation $x\to x+ 2\pi\omega$ on $\SS^1$ and $A\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$, $0\le l\le \infty$. For any fixed $l=0, 1, 2, \cdots, \infty$ and any fixed $\omega$ of bounded-type, we construct $D_{l}\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$ such that the Lyapunov exponent is not continuous at $D_{l}$ in ${\cal C}^l$-topology. We also construct such examples in a smaller Schr\"odinger class.