Equicontinuous delone dynamical systems

TL;DR

This paper characterizes equicontinuous Delone dynamical systems, showing that with finite local complexity they are crystalline, while without it, non-crystalline examples exist, solving a problem posed by Lagarias.

Contribution

It provides a complete characterization of equicontinuous Delone systems, distinguishing between crystalline and non-crystalline cases based on local complexity.

Findings

Equicontinuous Delone systems with finite local complexity are crystalline.

Examples of non-crystalline equicontinuous systems exist without finite local complexity.

The paper resolves Lagarias' question about the nature of Delone sets with strongly almost periodic Dirac combs.

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.