Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

TL;DR

This paper investigates conditions under which mass distributions on aperiodic tilings can be transported in a pattern-equivariant manner, linking the problem to properties of the tiling's ch cohomology.

Contribution

It connects the problem of bounded and equivariant mass transport to the ch cohomology of tiling hulls, providing a cohomological framework for understanding transport possibilities.

Findings

Transport possibilities are characterized by ch cohomology properties.

Most common tilings have well-understood cohomology, simplifying analysis.

Conditions for strongly and weakly pattern-equivariant transport are established.

Abstract

Suppose that we have a repetitive and aperiodic tiling $\bf T$ of $\mathbb{R}^n$, and two mass distributions $f_1$ and $f_2$ on $\mathbb{R}^n$, each pattern equivariant with respect to $\bf T$. Under what circumstances is it possible to do a bounded transport from $f_1$ to $f_2$? When is it possible to do this transport in a strongly or weakly pattern-equivariant way? We reduce these questions to properties of the \v Cech cohomology of the hull of $\bf T$, properties that in most common examples are already well-understood.