A dichotomy for bounded displacement equivalence of Delone sets

Yotam Smilansky, Yaar Solomon

Research output: Contribution to journalArticlepeer-review


We prove that in every compact space of Delone sets in Rd, 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 Rd . 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.

Original languageEnglish
Pages (from-to)2693-2710
Number of pages18
JournalErgodic Theory and Dynamical Systems
Issue number8
StatePublished - 18 Aug 2022


  • Delone sets
  • aperiodic order
  • bounded displacement equivalence
  • minimal spaces

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • General Mathematics


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