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

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 ${\mathcal{C}}^l$-topology. We also construct such examples in a smaller Schrödinger class.

43pages, 2 figures

References in corpus (2)

• Continuity of the Lyapunov Exponent for analytic quasi-perodic cocycles with singularities
• Positive Lyapunov Exponents for Quasiperiodic Szego cocycles

Cited by in corpus (5)

• Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
• Continuity of the measure of the spectrum for quasiperiodic Schrodinger operators with rough potentials
• Furstenberg Theory of Mixed Random-Quasiperiodic Cocycles
• Dynamical bounds for quasiperiodic Schrödinger operators with rough potentials
• On the spectral radius of compact operator cocycles