Bounded displacement non-equivalence insubstitution tilings

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.

Automated summary

This study focuses on the bounded displacement (BD) equivalence relation on Delone sets and explores instances of non-equivalence. A general condition for two Delone sets to be BD non-equivalent is presented, with applications to tilings of Euclidean space. The results include investigations on substitution tilings and mixed substitution systems generating various BD equivalence classes.