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A dichotomy for bounded displacement equivalence of Delone sets

Part of: Topological dynamics Discrete geometry

Published online by Cambridge University Press:  18 June 2021

YOTAM SMILANSKY  and
YAAR SOLOMON Open the ORCID record for YAAR SOLOMON [Opens in a new window]
YOTAM SMILANSKY
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ, USA (e-mail: yotam.smilansky@rutgers.edu)
YAAR SOLOMON*
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, Beersheba, Israel
*
e-mail: yaars@bgu.ac.il
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Abstract

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We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

Keywords

bounded displacement equivalenceDelone setsminimal spacesaperiodic order

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 52C23: Quasicrystals, aperiodic tilings 52C25: Rigidity and flexibility of structures
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2693 - 2710
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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