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

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

2012-02-22

arxiv.org/abs/1202.4902v1

Mathematics

Dynamical Systems

Scientific paper

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 $\vec{t}$ so that copies of this patch appear in the tilling "nearly" centered on $\lambda F+\vec{t}$ up to "bounded perturbations". Furthermore, we introduce a new unifying setting for the study of tiling spaces which allows rather general group "actions" and we discuss the local isomorphism property of tilings within this setting.

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