Self-Similarities and Invariant Densities for Model Sets

TL;DR

This paper explores the self-similarities of model sets, introducing invariant densities and averaging operators, and establishing their existence and properties related to diffraction and measures.

Contribution

It introduces the concept of self-similarities as a replacement for translation groups in model sets and studies their invariant densities and measures.

Findings

Invariant densities exist for model sets.

Invariant densities produce absolutely continuous measures.

Connections to diffraction and refinement operators are established.

Abstract

Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the self-similarities of model sets are a natural replacement for the group of translations of a lattice. This leads us to the concept of averaging operators and invariant densities on model sets. We prove that invariant densities exist and that they produce absolutely continuous invariant measures in internal space. We study the invariant densities and their relationships to diffraction, continuous refinement operators, and Hutchinson measures.