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Equicontinuous Delone Dynamical Systems

Published online by Cambridge University Press:  20 November 2018

Johannes Kellendonk
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex, France, e-mail: kellendonk@math.univ-lyon1.fr
Daniel Lenz
Affiliation:
Mathematisches Institut, Friedrich-Schiller Universität Jena, Ernst-Abbé Platz 2, D-07743 Jena, Germany, e-mail: daniel.lenz@uni-jena.de
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Abstract

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

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2013

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