A Characterization of Model Multi-colour Sets

Abstract

Model sets are always Meyer sets, but not vice-versa. This article is about characterizing model sets (general and regular) amongst the Meyer sets in terms of two associated dynamical systems. These two dynamical systems describe two very different topologies on point sets, one local and one global. In model sets these two are strongly interconnected and this connection is essentially definitive. The paper is set in the context of multi-colour sets, that is to say, point sets in which points come in a finite number of colours, that are loosely coupled together by finite local complexity.Comment: 23pages; to appear in Annales Henri Poincar

Characterization of model sets by dynamical systems

“…To make things clear, particularly the important ideas of irredundancy, which is not standard, and inter model sets, which are new [19], we have reworked this material and included it in the paper. However, our results require quite precise notions of what constitutes a CPS, which windows are permitted and how they relate to the CPS.…”

“…This approach is also taken in [19] in order to characterize multi-component model sets. This approach is also taken in [19] in order to characterize multi-component model sets.…”

Substitution Delone sets with pure point spectrum are inter-model sets

“…inter model multi-colour set) if each Γ i is a model set (resp. inter model set) with respect to the same CPS (see [27], [22] and [3] for more about model sets). One should note here that since π 2 need not be 1 − 1 on L in CPS, the notion of inter model set, which is hemmed in between two such sets differing only by points on the boundary of the window W , arises naturally.…”

“…Notice that the topology induced by this uniformity is not same with the topology (P ǫ -topology) induced by the metric ρ. We define A(Λ) as the closure of the orbit R d + Λ with the new uniformity in D (see [3]and [22] for more about A(Λ)).…”

“…The notion of inter model sets is introduced in [22] (under the name of model sets) and [3] as a model set satisfying a topological condition which is less restrictive than the boundary condition of a regular model set. In this paper we consider the inter model sets (see Def.…”