# Bounded Displacement Non-Equivalence In Substitution Tilings

## TL;DR

This paper investigates bounded displacement (BD) equivalence among Delone sets in tilings, providing conditions for non-equivalence and demonstrating the existence of infinitely many non-equivalent classes through substitution and mixed tilings.

## Contribution

It introduces a general criterion for BD non-equivalence and constructs examples of infinitely many non-equivalent Delone sets via substitution rules.

## Key findings

- Substitution rules can generate non-equivalent tilings.
- Existence of infinitely many BD classes in mixed substitution tilings.
- Delone sets associated with certain substitution rules are non-equivalent to lattices.

## Abstract

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.