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.