Equivalence relations on separated nets arising from linear toral flows

Alan Haynes, Michael Kelly, Barak Weiss

Research output: Contribution to journalArticlepeer-review


In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in ℝd which are not bi-Lipschitz equivalent (BL) to a lattice. A finer equivalence relation than BL is bounded displacement (BD). Separated nets arise naturally as return times to a section for minimal ℝd-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear ℝd-actions on tori. We show that generically these nets are BL to a lattice, and for some choices of dimensions and sections, they are generically BD to a lattice. We also show the existence of such nets which are not BD to a lattice.

Original languageEnglish
Pages (from-to)1203-1228
Number of pages26
JournalProceedings of the London Mathematical Society
Issue number5
StatePublished - 9 May 2013


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