An Algebraic Invariant for Substitution Tiling Systems

TL;DR

This paper introduces an algebraic invariant for substitution tiling systems, like Penrose tilings, and extends the analysis to broader dynamical systems, providing a new tool for understanding their structure.

Contribution

It defines a novel algebraic invariant associated with substitution tilings and computes it for various examples, extending the approach to general dynamical systems.

Findings

Computed the invariant for several substitution tilings

Extended the invariant to general dynamical systems

Provided insights into the structure of tiling systems

Abstract

We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.