Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Abstract

Suppose that we have a repetitive and aperiodic tiling $\mathbf{T}$ of ${\mathbb{R}}^n$, and two mass distributions $f_1$ and $f_2$ on ${\mathbb{R}}^n$, each pattern equivariant with respect to $\mathbf{T}$. 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 $\mathbf{T}$, properties that in most common examples are already well-understood.

16 pages