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Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction

Volume 247 / 2019

Michael Baake, Natalie Priebe Frank, Uwe Grimm, E. Arthur Robinson Jr. Studia Mathematica 247 (2019), 109-154 MSC: Primary 37A30, 42A38, 37B50; Secondary 37H15, 52C23. DOI: 10.4064/sm170613-10-3 Published online: 23 November 2018


One of the simplest non-Pisot substitution rules is investigated in its geometric version as a tiling with intervals of natural length as prototiles. Via a detailed renormalisation analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely continuous component. This implies that the diffraction, apart from a trivial Bragg peak at the origin, is purely singular continuous. En route, we derive various geometric and algebraic properties of the underlying Delone dynamical system, which we expect to be relevant in other such systems as well.


  • Michael BaakeFakultät für Mathematik
    Universität Bielefeld
    Postfach 100131
    33501 Bielefeld, Germany
  • Natalie Priebe FrankDepartment of Mathematics and Statistics
    Vassar College
    Poughkeepsie, NY 12604, USA
  • Uwe GrimmSchool of Mathematics and Statistics
    The Open University
    Walton Hall
    Milton Keynes MK7 6AA, United Kingdom
  • E. Arthur Robinson Jr.Department of Mathematics
    George Washington University
    Washington, DC 20052, USA

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