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We study the spectral types of the families of discrete one-dimensionalSchr\"odinger operators $\{H_\omega\}_{\omega\in\Omega}$, where the potentialof 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 ageneric operator $H_\omega\in \{H_\omega\}_{\omega\in\Omega}$ has purelycontinuous spectrum if $\{T^n\alpha\}_{n\geq0}$ is dense in $\Omega$ for acertain $\alpha\in\Omega$. We also show the former result assuming only that$\{\Omega, T\}$ satisfies topological repetition property ($TRP$), a conceptintroduced by Boshernitzan and Damanik (arXiv:0708.1263v1). Theorems presentedin this paper weaken the hypotheses of the cited research and allow us to reachthe same conclusion as those authors. We also provide a proof of Gordon'slemma, which is the main tool used in this work.

Sufficient conditions on the continuous spectrum for ergodic Schr\"odinger Operators