Sufficient conditions on the continuous spectrum for ergodic Schrödinger operators

We investigate the spectral types of families of discrete one-dimensional Schr & ouml;dinger operators {H omega}omega is an element of Omega , where each H omega has a potential function V omega (n) = f(T n omega) for n is an element of Z , T is an ergodic homeomorphism on a compact space Omega and f:Omega -> R is a continuous function. The main analytical tool of the paper is Gordon's lemma, of which we provide a complete proof. We demonstrate that a generic operator H omega is an element of{H omega}omega is an element of Omega has purely continuous spectrum if {Tn alpha}n >= 0 is dense in Omega for a certain alpha is an element of Omega. Furthermore, we extend this result to the case where {Omega, T} satisfies topological repetition property, a concept introduced by Boshernitzan and Damanik (2008, [1]). Theorems presented in this paper weaken the hypotheses of previous research and enable us to reach the same conclusion as those authors. Our paper presents a comprehensive study of the topological and metric repetition properties, highlighting their binding role between the theory of dynamical systems and spectral theory. We also discuss the implications of these properties on the qualitative description of specific physical systems.