Properties of Lyapunov exponents for quasiperodic cocycles with singularities

Tao, Kai

Electronic journal of differential equations, Tome 2015 (2015)

Résumé

We consider the quasi-periodic cocycles $(\omega,A(x,E)): (x,v)\mapsto (x+\omega, A(x,E)v)$ 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.

Détail

Classification : 37C55, 37F10
Keywords: Lyapunov exponent, quasiperodic cocycles, holder exponent