Second phase transition line (Q681624)

The almost Mathieu family is defined by \[ (H_{\lambda, \alpha, \theta}u)_n=u_{n+1}+u_{n-1}+2\lambda\cos 2\pi(n\alpha+\theta)u_n, \] where $\theta\in \mathbb R$, $\alpha\in \mathbb R\setminus \mathbb Q$ and $\lambda \in \mathbb R_+$. In the case $\lambda>1$, the main results of the paper are: \begin{itemize} \item[-] There exists a dense set of $\alpha$ such that $H_{\lambda, \alpha, \theta}$ has purely singular continuous spectrum for all $\theta$; \item[-] There exists a dense set of $\alpha$ such that $H_{\lambda, \alpha, \theta}$ has pure point spectrum for a.e. $\theta$.\end{itemize}