Sufficient conditions on the continuous spectrum for ergodic Schrödinger Operators

We study the spectral types of the families of discrete one-dimensional Schrödinger operators $\{H_{\omega }{\}}_{\omega \in \mathrm{\Omega }}$, 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 $\mathrm{\Omega }$ and $f:\mathrm{\Omega }\to \mathbb{R}$ is a continuous function. We show that a generic operator $H_{\omega }\in \{H_{\omega }{\}}_{\omega \in \mathrm{\Omega }}$ has purely continuous spectrum if $\{T^n\alpha {\}}_{n\ge 0}$ is dense in $\mathrm{\Omega }$ for a certain $\alpha \in \mathrm{\Omega }$. We also show the former result assuming only that $\{\mathrm{\Omega },T\}$ 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.