Delone sets and dynamical systems

TL;DR

This paper explores the mathematical properties of Delone sets, their role in modeling quasicrystals, and the dynamics of related systems, emphasizing symmetries, substitutions, and tilings.

Contribution

It provides an overview of key topics connecting Delone sets with dynamical systems, highlighting recent developments and open questions in the field.

Findings

Delone sets effectively model quasicrystals

Inflation symmetries relate to expansion constants

Substitution Delone sets generate complex tilings

Abstract

In these expository notes we focus on selected topics around the themes: Delone sets as models for quasicrystals, inflation symmetries and expansion constants, substitution Delone sets and tilings, and associated dynamical systems.