Aperiodic order and spherical diffraction, I: Auto-correlation of model sets

TL;DR

This paper generalizes the concept of model sets and auto-correlation to arbitrary locally compact second countable groups, extending mathematical models of quasicrystals beyond abelian and amenable groups, and establishes unique invariant measures.

Contribution

It introduces a generalized auto-correlation for model sets in lcsc groups and proves the existence and uniqueness of invariant measures for their hulls, even in non-amenable cases.

Findings

Auto-correlation formula for regular model sets in lcsc groups

Existence of unique invariant probability measure for the hulls

Extension of Hof's auto-correlation to non-amenable groups

Abstract

We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical models of quasi-crystals. We then define a notion of auto-correlation for subsets of finite local complexitiy in arbitrary lcsc groups, which generalizes Hof's classical definition beyond the class of amenable groups, and provide a formula for the auto-correlation of a regular model set. Along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure, even in the case where the punctured hull is non-compact and the group is non-amenable. In fact this measure is also the unique stationary measure with respect to any admissible probability measure.