A dichotomy for bounded displacement equivalence of Delone sets

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.

Multiscale substitution tilings

“…Note that for polygonal prototiles (8.2) clearly holds with η = 1. In a subsequent work [47], we show that as a result of Theorem 8.2 and the minimality of the dynamical system (X F σ , R d ), the set of BD-equivalence classes that appear in X F σ has the cardinality of the continuum. We present two independent proofs of Theorem 8.2.…”

Discrepancy and rectifiability of almost linearly repetitive Delone sets

“…A new tile that arises as a rescaled copy of a prototile T j ∈ τ σ is said to be of type j. The patches {F t (T i ) : t 0} exhaust the space, and limits taken with respect to the natural topology on the space C (R d ) of closed subsets of R d , which is closely related to the Hausdorff distance described in (1.1) and is discussed in more detail for example in [FrRi,SS1,SS2], define a tiling space X σ of multiscale substitution tilings of R d . A multiscale substitution scheme σ is irreducible if for every 1 i, j n there exists t > 0 so that F t (T i ) contains a tile of type j.…”

“…This is due both to the various dynamical and geometric implications of linear repetitivity, as well as to the fact that several well-studied constructions in aperiodic order are known to have this property, including primitive self-similar tilings of finite local complexity (FLC) [bS] and certain cutand-project sets [HKoWa, KW]. In this paper we consider Delone sets and tilings of infinite local complexity, which have seen a surge of interest in recent years with examples including [Da,Fr,FrRo,FrS1,FrS2,FrRi,LS,Sa,SS1] and [SS2], and for which a suitable extension of the notion of linear repetitivity is required. Our study is motivated by the question of rectifiability of Delone sets of infinite local complexity, and in particular those defined by multiscale substitution tilings.…”

Discrepancy and rectifiability of almost linearly repetitive Delone sets

“…A new tile that arises as a rescaled copy of a prototile T j P τ σ is said to be of type j. The patches tF t pT i q : t ě 0u exhaust the space, and limits taken with respect to the natural topology on the space C pR d q of closed subsets of R d , which is closely related to the Hausdorff distance described in (1.1) and is discussed in more detail for example in [FrRi,SS1,SS2], define a tiling space X σ of multiscale substitution tilings of R d . A multiscale substitution scheme σ is irreducible if for every 1 ď i, j ď n there exists t ą 0 so that F t pT i q contains a tile of type j.…”

“…This is due both to the various dynamical and geometric implications of linear repetitivity, as well as to the fact that several well-studied constructions in aperiodic order are known to have this property, including primitive self-similar tilings of finite local complexity [bS] and certain cut-and-project sets [HKoWa, KW]. In this paper we consider Delone sets and tilings of infinite local complexity, which have seen a surge of interest in recent years with examples including [Da,Fr,FrRo,FrS1,FrS2,FrRi,LS,Sa,SS1] and [SS2], and for which a suitable extension of the notion of linear repetitivity is required.…”