Electron. J. Diff. Equ., Vol. 2015 (2015), No. 67, pp. 1-16.

Properties of Lyapunov exponents for quasiperodic cocycles with singularities

Kai Tao

We consider the quasi-periodic cocycles with $\omega$ Diophantine. Let $M_2(\mathbb{C})$ be a normed space endowed with the matrix norm, whose elements are the $2\times 2$ matrices. Assume that $A:\mathbb{T}\times \mathcal{E}\to M_2(\mathbb{C})$ is jointly continuous, depends analytically on $x\in\mathbb{T}$ and is Holder continuous in $E\in\mathcal{E}$, where $\mathcal{E}$ is a compact metric space and $\mathbb{T}$ is the torus. We prove that if two Lyapunov exponents are distinct at one point $E_0\in\mathcal{E}$, then these two Lyapunov exponents are Holder continuous at any E in a ball central at $E_0$. Moreover, we will give the expressions of the radius of this ball and the Holder exponents of the two Lyapunov exponents.

Submitted October 10, 2014. Published March 20, 2015.
Math Subject Classifications: 37C55, 37F10.
Key Words: Lyapunov exponent; quasiperodic cocycles; Holder exponent.

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Kai Tao
College of Sciences, Hohai University
1 Xikang Road
Nanjing, Jiangsu 210098, China
email: ktao@hhu.edu.cn, tao.nju@gmail.com

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