Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians

Abstract

We show that geometric disorder leads to some purely singular continuous spectrum generically. The main input is a result of Simon known as the Wonderland theorem in [17, Section 2]. Here we provide an alternative approach and actually a slight strengthening by showing that various sets of measures defined by regularity properties are generic in the set of all measures on a locally compact metric space. As a byproduct, we obtain the fact that a generic measure on euclidean space is singular continuous