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Equicontinuous Delone Dynamical Systems

  Published:2011-12-23
 Printed: Feb 2013
  • Johannes Kellendonk,
    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
  • Daniel Lenz,
    Mathematisches Institut, Friedrich-Schiller Universität Jena , Ernst-Abbé Platz~2, D-07743 Jena, Germany
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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 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.
Keywords: Delone sets, tilings, diffraction, topological dynamical systems, almost periodic systems Delone sets, tilings, diffraction, topological dynamical systems, almost periodic systems
MSC Classifications: 37B50 show english descriptions Multi-dimensional shifts of finite type, tiling dynamics 37B50 - Multi-dimensional shifts of finite type, tiling dynamics
 

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