A dichotomy for bounded displacement equivalence of Delone sets

Abstract

We prove that in every compact space of Delone sets in ${\mathbb{R}}^d$ which is minimal with respect to the action by translations, either all Delone sets are uniformly spread, or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty--Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb{R}}^d$. This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

arXiv comment16 pages. The title and abstract have been changed, the exposition is clearer, and an appendix was added. To appear in Ergo. Theo. Dynam. Sys

References in corpus (8)

• Equivalence relations on separated nets arising from linear toral flows
• Spaces of closed subgroups of locally compact groups
• Equicontinuous delone dynamical systems
• Dynamical properties of almost repetitive Delone sets
• Some examples of non-rectifiable, repetitive Delone sets
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Cited by in corpus (3)

• Multiscale Substitution Tilings
• Discrepancy and rectifiability of almost linearly repetitive Delone sets
• Divergence of separated nets with respect to displacement equivalence