Equivalence relations on separated nets arising from linear toral flows. Issue 5 (22nd July 2014) - Haynes, Alan; Kelly, Michael; Weiss, Barak
Abstract : In 1998, Burago–Kleiner and McMullen independently proved the existence of separated nets inR d 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 minimalR d ‐actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linearR d ‐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.