On the dynamics of quasi-periodic Schrödinger cocycles for positive measure sets of frequencies

Abstract

We consider the family of one-frequency quasi-periodic Schrödinger coycles $G_{\omega ,E}$, parametrized by the energy $E$. For potential functions $v(x)=\lambda v_0(x)$, where $v_0\in C^2(\mathbb{T},\mathbb{R})$ is a Morse function with finitely many critical points and $\lambda >0$ is large, we show that, for any value of $E\in \mathbb{R}$ and for any phase $x^{\ast }\in \mathbb{T}$ such that $|v_0(x^{\ast })-E/\lambda |$ is not too small, there exists a set of frequencies $\Omega =\Omega (E,x^{\ast })$ of positive measure such that the following hold: (1) for every $\omega \in \Omega$, the upper Lyapunov exponent of the cocycle $G_{\omega ,E}$ is $\gtrsim \log \lambda$ and $x^{\ast }$ is (essentially) a typical point in Oseledets’ theorem; (2) either $G_{\omega ,E}$ is uniformly hyperbolic, or there exists a phase $x_0\in \mathbb{T}$ such that $E$ is an eigenvalue of the corresponding discrete Schrödinger operator $H_{x_0,\omega }$.