Substitution rules and topological properties of the Robinson tilings

TL;DR

This paper introduces a simple overlapping substitution rule for Robinson tilings, demonstrating their pure point diffraction and computing their topological and dynamical properties using cohomology and zeta functions.

Contribution

It presents a new, minimal substitution rule for Robinson tilings and analyzes their topological and dynamical structure, establishing their status as a model set.

Findings

Robinson tilings form a model set with pure point diffraction.

The paper computes the Cech cohomology of the tiling hull.

It describes the structure of the hull via the dynamical zeta function.

Abstract

A relatively simple substitution for the Robinson tilings is presented, which requires only 56 tiles up to translation. In this substitution, due to Joan M. Taylor, neighboring tiles are substituted by partially overlapping patches of tiles. We show that this overlapping substitution gives rise to a normal primitive substitution as well, implying that the Robinson tilings form a model set and thus have pure point diffraction. This substitution is used to compute the Cech cohomology of the hull of the Robinson tilings via the Anderson-Putnam method, and also the dynamical zeta function of the substitution action on the hull. The dynamical zeta function is then used to obtain a detailed description of the structure of the hull, relating it to features of the cohomology groups.