The dynamics of a class of quasi-periodic Schr\"odinger cocycles

Abstract

Let $f:\mathbb{T}\to\mathbb{R}$ be a Morse function of class $C^2$ with exactly two critical points, let $\omega\in\mathbb{T}$ be Diophantine, and let $\lambda>0$ be sufficiently large (depending on $f$ and $\omega$). For any value of the parameter $E\in \mathbb{R}$ we make a careful analysis of the dynamics of the skew-product map $$\Phi_E(\theta,r)=\left(\theta+\omega,\lambda f(\theta)-E-1/r\right),$$ acting on the "torus" $\mathbb{T}\times\widehat{\mathbb{R}}$. The map $\Phi_E$ is intimately related to the quasi-periodic Schr\"odinger cocycle $(\omega,A_E): \mathbb{T}\times \mathbb{R}^2 \to \mathbb{T}\times \mathbb{R}^2$, $(\theta,x)\mapsto (\theta+\omega, A_E(\theta)\cdot x)$, where $A_E:\mathbb{T}\to \text{SL}(2,\mathbb{R})$ is given by $A_{E}(\theta)=\left(\begin{matrix}0 & 1 \\ -1 & \lambda f(\theta)-E \end{matrix} \right), \quad E\in \mathbb{R}. $ More precisely, $(\omega,A_E)$ naturally acts on the space $\mathbb{T}\times\widehat{\mathbb{R}}$, and $\Phi_E$ is the map thus obtained. The analysis of $\Phi_E$ allows us to derive three main results: (1) The (maximal) Lyapunov exponent of the Schr\"odinger cocycle $(\omega,A_E)$ is $\gtrsim \log \lambda$, uniformly in $E\in \mathbb{R}$. This implies that the map $\Phi_E$ has exactly two ergodic probability measures for all $E\in \mathbb{R}$; (2) If $E$ is on the edge of an open gap in the spectrum $\sigma(H)$ of the associated Schr\"odinger operator $H_\theta$, then there exist a phase $\theta\in\mathbb{T}$ and a vector $u\in l^2(\mathbb{Z})$, exponentially decaying at $\pm\infty$, such that $H_\theta u=Eu$; (3) The map $\Phi_E$ is minimal iff $E\in \sigma(H)\setminus\{\text{edges of open gaps}\}$. In particular, $\Phi_E$ is minimal for all $E$ for which the fibered rotation number $\alpha(E)$ associated to $(\omega,A_E)$ is irrational with respect to $\omega$.