Equicontinuous delone dynamical systems

Abstract

We characterize equicontinuous Delone dynamical systems as those coming from Delone sets with strongly almost periodic Dirac combs. Within the class of systems with nite local complexity the only equicontinuous systems are then shown to be the crystalline ones. On the other hand, within the class without nite local complexity, we exhibit examples of equicontinuous minimal Delone dynamical systems which are not crystalline. Our results solve the problem posed by Lagarias whether a Delone set whose Dirac comb is strongly almost periodic must be crystalline.