Topological equivalence of tilings

TL;DR

This paper introduces a new notion of equivalence for tilings based on local structure and demonstrates its relation to existing concepts, showing that finite type tilings are topologically equivalent if their groupoids are isomorphic.

Contribution

It defines a novel local-structure-based equivalence for tilings and links it to groupoid isomorphisms, advancing the understanding of tiling topological classification.

Findings

Two tilings of finite type are topologically equivalent if their groupoids are isomorphic.

The introduced equivalence aligns with existing local derivation concepts.

Groupoid isomorphism characterizes topological equivalence of tilings.

Abstract

We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic.