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

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 $${\mathrm{\Phi }}_E(\theta ,r)=(\theta +\omega ,\lambda f(\theta )-E-1\mathrm{/}r),$$ acting on the “torus” $\mathbb{T}\times \widehat{\mathbb{R}}$. The map ${\mathrm{\Phi }}_E$ 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,$\mathbb{R}$)$isgivenby$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 ${\mathrm{\Phi }}_E$ is the map thus obtained. The analysis of ${\mathrm{\Phi }}_E$ 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 ${\mathrm{\Phi }}_E$ has exactly two ergodic probability measures for all $E\in $\mathbb{R}$$;(2)If$E$isontheedgeofanopengapinthespectrum$\sigma(H)$oftheassociatedSchr\ddot{o}dingeroperator$H_\theta$, then there exist a phase $\theta \in \mathbb{T}$ and a vector $u\in l^2(\mathbb{Z})$, exponentially decaying at $\pm \mathrm{\infty }$, such that $H_{\theta }u=Eu$; (3) The map ${\mathrm{\Phi }}_E$ is minimal iff $E\in \sigma (H)\setminus \{\text{edges of open gaps}\}$. In particular, ${\mathrm{\Phi }}_E$ is minimal for all $E$ for which the fibered rotation number $\alpha (E)$ associated to $(\omega ,A_E)$ is irrational with respect to $\omega$.