The dynamics of a class of quasi-periodic Schrödinger 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" $\mathbb{T}\times \hat{\mathbb{R}}$. The map ${\Phi }_E$ is intimately related to the quasi-periodic Schrödinger 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 $A_E(\theta )=\left(\begin{array}{cc}0& 1\\ -1& \lambda f(\theta )-E\end{array}\right),E\in \mathbb{R}.$ More precisely, $(\omega ,A_E)$ naturally acts on the space $\mathbb{T}\times \hat{\mathbb{R}}$, 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 Schrödinger 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 Schrödinger 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 $\omega$.

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• Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schrödinger Cocycles 
• On some generalizations of skew-shifts on ${\mathbb{T}}^2$