A General Framework for tilings, Delone sets, functions and measures, and their interrelation

TL;DR

This paper introduces a broad framework connecting tilings, Delone sets, functions, and measures through local derivability, enabling a unified understanding of their interrelations and the information they encode in aperiodic order.

Contribution

It generalizes local derivability concepts to a wide class of objects, establishing their interrelations and showing how key maps preserve mutual local derivability.

Findings

Canonical maps preserve MLD relations

Existence of MLD objects within classes under mild conditions

Pattern-equivariant functions encode original objects up to MLD

Abstract

We define a general framework that includes objects such as tilings, Delone sets, functions and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in the literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object P to one that is MLD with P. Moreover we show that, for an object P and a class S of objects, a mild condition on them assures that there exists some Q in S that is MLD with P. As an application, we study pattern equivariant functions. In particular, we show that the space of all pattern-equivariant functions contains all the information of the original object up to MLD in a quite general setting.