Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles

Abstract

We study the regularity of the Lyapunov exponent for quasiperiodic cocycles $(T_{\omega },A)$ where $T_{\omega }$ is an irrational rotation $x\to x+2\pi \omega$ on ${\mathbb{S}}^1$ and $A\in {\mathcal{C}}^l({\mathbb{S}}^1,SL(2,\mathbb{R}\left)\right)$, $0\le l\le \infty$. For any fixed $l=0,1,2,\dots ,\infty$ and any fixed $\omega$ of bounded type, we construct $D_l\in {\mathcal{C}}^l({\mathbb{S}}^1,SL(2,\mathbb{R}\left)\right)$ such that the Lyapunov exponent is not continuous at $D_l$ in ${\mathcal{C}}^l$-topology. We also construct such examples in a smaller Schrödinger class.