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

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.

Cited by in corpus (1)

• Statistical Stability for Multi-Substitution Tiling Spaces