TY - JOUR
T1 - Bounded displacement non-equivalence in substitution tilings
AU - Frettlöh, Dirk
AU - Smilansky, Yotam
AU - Solomon, Yaar
N1 - Funding Information:
We are happy to thank Dan Rust and Barak Weiss for helpful discussions, and the two anonymous reviewers for their thoughtful comments. We thank The Center For Advanced Studies In Mathematics in Ben-Gurion University and the Research Centre of Mathematical Modelling (RCM2) at Bielefeld University for supporting the visit of the first author in Israel. The second author is grateful for the support of the David and Rosa Orzen Endowment Fund via an Orzen Fellowship, and the ISF grant No. 1570/17.
Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2021/1/1
Y1 - 2021/1/1
N2 - 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.
AB - 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.
KW - Bounded displacement
KW - Delone sets
KW - Mixed substitution
KW - Quasicrystals
KW - Substitution tilings
UR - http://www.scopus.com/inward/record.url?scp=85090005981&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2020.105326
DO - 10.1016/j.jcta.2020.105326
M3 - Article
AN - SCOPUS:85090005981
VL - 177
JO - Journal of Combinatorial Theory - Series A
JF - Journal of Combinatorial Theory - Series A
SN - 0097-3165
M1 - 105326
ER -