Sufficient conditions on the continuous spectrum for ergodic Schrödinger Operators
We study the spectral types of the families of discrete one-dimensional Schrödinger operators $\{H_ω\}_{ω\inΩ}$, where the potential of each $H_ω$ is given by $V_ω(n)=f(T^nω)$ for $n\in\mathbb{Z}$, $T$ is an ergodic homeomorphism on a compact space $Ω$ and $f:Ω\rightarrow\mathbb{R}$ is a continuous function. We show that a generic operator $H_ω\in \{H_ω\}_{ω\inΩ}$ has purely continuous spectrum if $\{T^nα\}_{n\geq0}$ is dense in $Ω$ for a certain $α\inΩ$. We also show the former result assuming only that $\{Ω, T\}$ 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.