Sufficient conditions on the continuous spectrum for ergodic Schrödinger Operators

Abstract: We study the spectral types of the families of discrete one-dimensional Schr\"odinger operators ${H_\omega}{\omega\in\Omega}$, where the potential of each $H\omega$ is given by $V_\omega(n)=f(T^n\omega)$ for $n\in\mathbb{Z}$, $T$ is an ergodic homeomorphism on a compact space $\Omega$ and $f:\Omega\rightarrow\mathbb{R}$ is a continuous function. We show that a generic operator $H_\omega\in {H_\omega}{\omega\in\Omega}$ has purely continuous spectrum if ${T^n\alpha}{n\geq0}$ is dense in $\Omega$ for a certain $\alpha\in\Omega$. We also show the former result assuming only that ${\Omega, T}$ satisfies topological repetition property ($TRP$), a concept introduced by Boshernitzan and Damanik (arXiv:0708.1263v1). Theorems presented in this paper weaken the hypotheses of the cited research and allow us to reach the same conclusion as those authors. We also provide a proof of Gordon's lemma, which is the main tool used in this work.