Dynamical properties of almost repetitive Delone sets

TL;DR

This paper studies the dynamical behavior of almost repetitive Delone sets, linking their geometric repetitivity properties to minimality and unique ergodicity of associated dynamical systems.

Contribution

It characterizes minimality via almost repetitivity and introduces linear versions that ensure unique ergodicity, with examples from various structured point sets.

Findings

Almost repetitivity characterizes minimality of the dynamical system.

Linear almost repetitivity implies unique ergodicity.

Examples include periodic sets with almost periodic modulations and substitution tilings.

Abstract

We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the point set. We also provide linear versions of almost repetitivity which lead to uniquely ergodic systems. Apart from linearly repetitive point sets, examples are given by periodic point sets with almost periodic modulations, and by point sets derived from primitive substitution tilings of finite local complexity with respect to the Euclidean group with dense tile orientations.