On substitution tilings and Delone sets without finite local complexity

Abstract

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences.In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [29] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.

A dichotomy for bounded displacement equivalence of Delone sets

“…D(•, •) is a metric It is known that the function D(•, •) in (1.1) constitutes a metric on C (R d ) when it is capped by 1/ √ 2 instead of 1, see e.g. [LSo,§7]. We show that it is indeed a metric also when capped by 1.…”

Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings

“…where the closure is taken in the topology induced by the metric . For non-FLC tilings, we can consider 'local rubber topology' on the collection of tilings (Mü ller & Richard, 2013; Lenz & Stollmann, 2003;Baake & Lenz, 2004;Lee & Solomyak, 2019) and obtain X T as the completion of the orbit closure of T under this topology. For tilings with FLC, the two topologies coincide.…”

“…where a j , 1 j J, are given in (18). One can find an example of a non-FLC tiling that the rigidity property fails in (Frank & Robinson, 2008;Lee & Solomyak, 2019).…”

“…Proof. Under the assumption of pure discrete spectrum, we know that T has FLC from the work of Lee & Solomyak (2019) and fulfils the Pisot family condition (Lee & Solomyak, 2012). From Theorem 3.1, we know that T is rigid.…”

Cut-and-Project Schemes for Pisot Family Substitution Tilings

“…In this article, we consider mainly primitive substitution tilings in R d with pure point spectrum. It is proven in [7] that primitive substitution tilings with pure point spectrum always show finite local complexity (FLC). So, it is not necessary to make an assumption of FLC in the consideration of pure point spectrum.…”