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

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

2012-02-02

arxiv.org/abs/1202.0580v2

Mathematics

Dynamical Systems

31pages, 2 figures

Scientific paper

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 $A\in {\cal C}^l(\SS^1, SL(2,\RR))$, $0\le l\le \infty$. For any fixed $l=0, 1, 2,..., \infty$ and any fixed $\omega$ of bounded-type, we construct a $D_{l}\in {\cal C}^l(\SS^1, SL(2,\RR))$ such that the Lyapunov exponent is not continuous at $(T_\omega, D_{l})$ in ${\cal C}^l$-topology.

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