Some examples of non-rectifiable, repetitive Delone sets

Abstract

We construct examples of Delone sets of the plane (that is, discrete subsets that are uniformly separated and coarsely dense) that are repetitive (each patch of the set appears in every large-enough ball) though non-rectifiable (i.e. non bi-Lipschitz equivalent to the standard lattice). More generally, we construct such a set so that the translation action on the closure of its orbit has any prescribed Choquet simplex as its set of invariant probability measures (in particular, we provide uniquely ergodic examples). The construction relies on classical examples of Burago-Kleiner and McMullen, for which we give pure combinatorial (an effective) versions that have interest by themselves.

22 pages, 5 figures

Cited by in corpus (8)

• A dichotomy for bounded displacement equivalence of Delone sets
• Mapping $n$ grid points onto a square forces an arbitrarily large Lipschitz constant
• Multiscale Substitution Tilings
• Divergence of separated nets with respect to displacement equivalence
• Discrepancy and rectifiability of almost linearly repetitive Delone sets
• Statistical properties of Lorentz gases on aperiodic tilings, Part 1
• Planar Bilipschitz Extension from Separated Nets
• Highly irregular separated nets