Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Abstract

Abstract Suppose that we have a repetitive and aperiodic tiling ${\textbf{T}}$ of ${\mathbb{R}}^n$ and two mass distributions $f_1$ and $f_2$ on ${\mathbb{R}}^n$, each pattern equivariant (PE) with respect to ${\textbf{T}}$. 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 PE way? We reduce these questions to properties of the Čech cohomology of the hull of ${\textbf{T}}$, properties that in most common examples are already well understood.