1. A dichotomy for bounded displacement equivalence of Delone sets
- Author
-
Yotam Smilansky and Yaar Solomon
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Substitution (algebra) ,Metric Geometry (math.MG) ,Dynamical Systems (math.DS) ,Natural topology ,Space (mathematics) ,01 natural sciences ,37B05, 37B52 ,010101 applied mathematics ,Combinatorics ,Compact space ,Mathematics - Metric Geometry ,Bounded function ,FOS: Mathematics ,Mathematics::Metric Geometry ,Mathematics - Dynamical Systems ,0101 mathematics ,Equivalence (measure theory) ,Topology (chemistry) ,Mathematics - 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., Comment: 16 pages. The title and abstract have been changed, the exposition is clearer, and an appendix was added. To appear in Ergo. Theo. Dynam. Sys
- Published
- 2021