Paper detail

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

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 $λ$, there is a vector $\vec{t}_λ$ so that copies of this patch appear in the tilling "nearly" centered on $λF+\vec{t}_λ$ 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.