Metrics on tiling spaces, local isomorphism and an application of Brown's Lemma

TL;DR

This paper applies a topological dynamics version of Brown's lemma to tiling theory, demonstrating how to find patches with near-repetitions across scales and introducing a new framework for tiling space analysis.

Contribution

It introduces a novel application of Brown's lemma to tiling patterns and develops a unifying setting for studying tiling spaces with general group actions.

Findings

Existence of nearly repeated patches at various scales

A new framework for analyzing tiling spaces and local isomorphism

Application of topological dynamics to tiling theory

Abstract

We give an application of a topological dynamics version of multidimensional Brown's lemma to tiling theory: given a tiling of an Euclidean space and a finite geometric pattern of points $F$, one can find a patch such that, for each scale factor $\lambda$, there is a vector $\vec{t}_\lambda$ so that copies of this patch appear in the tilling "nearly" centered on $\lambda F+\vec{t}_\lambda$ once we allow "bounded perturbations" in the structure of the homothetic copies of $F$. Furthermore, we introduce a new unifying setting for the study of tiling spaces which allows rather general group "actions" on patches and we discuss the local isomorphism property of tilings within this setting.