Examples of Discontinuity of Lyapunov Exponent in Smooth Quasi-Periodic Cocycles
Examples of Discontinuity of Lyapunov Exponent in Smooth Quasi-Periodic Cocycles
复制标题
DOI：
10.1215/00127094-2371528
复制
发表时间：
2012-02
期刊：
影响因子：
--
通讯作者：
Yiqian Wang;J. You
中科院分区：
文献类型：
--
作者：
Yiqian Wang;J. You
我们研究了拟周期共环$(T_\omega, A)$的Lyapunov指数的规律性，其中$T_\omega$是$\SS^1$和$A\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$, $0\le l\le \infty$上的不合理旋转$x\to x+ 2\pi\omega$。对于任意固定的$l=0, 1, 2, \cdots, \infty$和任意固定的有界型$\omega$，我们构造$D_{l}\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$使得在${\cal C}^l$ -拓扑中Lyapunov指数在$D_{l}$处不连续。我们还在一个较小的Schrödinger类中构造这样的示例。
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.