Bounded displacement non-equivalence in substitution tilings

Frettlöh D, Smilansky Y, Solomon Y (2021)
Journal of Combinatorial Theory, Series A 177: 105326.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Frettlöh, DirkUniBi; Smilansky, Yotam; Solomon, Yaar
Abstract / Bemerkung
In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets associated with tilings of Euclidean space. First we consider substitution tilings, and exhibit a substitution matrix associated with two distinct substitution rules. The first rule generates only periodic tilings, while the second generates tilings for which any associated Delone set is non-equivalent to any lattice in space. As an extension of this result, we introduce arbitrarily many distinct substitution rules associated with a single matrix, with the property that Delone sets generated by distinct rules are non-equivalent. We then turn to the study of mixed substitution tilings, and present a mixed substitution system that generates representatives of continuously many distinct BD equivalence classes. (C) 2020 Elsevier Inc. All rights reserved.
Stichworte
Bounded displacement; Substitution tilings; Delone sets; Quasicrystals; Mixed substitution
Erscheinungsjahr
2021
Zeitschriftentitel
Journal of Combinatorial Theory, Series A
Band
177
Art.-Nr.
105326
ISSN
0097-3165
eISSN
1096-0899
Page URI
https://pub.uni-bielefeld.de/record/2948623

Zitieren

Frettlöh D, Smilansky Y, Solomon Y. Bounded displacement non-equivalence in substitution tilings. Journal of Combinatorial Theory, Series A. 2021;177: 105326.
Frettlöh, D., Smilansky, Y., & Solomon, Y. (2021). Bounded displacement non-equivalence in substitution tilings. Journal of Combinatorial Theory, Series A, 177, 105326. doi:10.1016/j.jcta.2020.105326
Frettlöh, D., Smilansky, Y., and Solomon, Y. (2021). Bounded displacement non-equivalence in substitution tilings. Journal of Combinatorial Theory, Series A 177:105326.
Frettlöh, D., Smilansky, Y., & Solomon, Y., 2021. Bounded displacement non-equivalence in substitution tilings. Journal of Combinatorial Theory, Series A, 177: 105326.
D. Frettlöh, Y. Smilansky, and Y. Solomon, “Bounded displacement non-equivalence in substitution tilings”, Journal of Combinatorial Theory, Series A, vol. 177, 2021, : 105326.
Frettlöh, D., Smilansky, Y., Solomon, Y.: Bounded displacement non-equivalence in substitution tilings. Journal of Combinatorial Theory, Series A. 177, : 105326 (2021).
Frettlöh, Dirk, Smilansky, Yotam, and Solomon, Yaar. “Bounded displacement non-equivalence in substitution tilings”. Journal of Combinatorial Theory, Series A 177 (2021): 105326.

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arXiv: 1907.01597

Inspire: 1743116

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