Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation

TL;DR

This paper explores the structure of orbit equivalence relations in substitution tilings of R^d, using Bratteli multi-diagrams and AF-relations to provide a combinatorial reconstruction of the tiling's equivalence relation.

Contribution

It introduces a new combinatorial framework using Bratteli multi-diagrams to analyze the orbit structure of substitution tilings, extending previous methods to higher dimensions.

Findings

The tiling's equivalence relation is homeomorphic to an etale relation on the Bratteli multi-diagram.

A natural notion of border for tilings is characterized via tails in the Bratteli diagrams.

The approach unifies the analysis of tilings across all dimensions j=0,...,d.

Abstract

In this second paper, we study the case of substitution tilings of R^d. The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j=0, ..., d-1. We reconstruct the tiling's equivalence relation in a purely combinatorial way using the AF-relations given by the lower dimensional substitutions. We define a Bratteli multi-diagram B which is made of the Bratteli diagrams B^j, j=0, ..., d, of all those substitutions. The set of infinite paths in B^d is identified with the canonical transversal Xi of the tiling. Any such path has a "border", which is a set of tails in B^j for some j less than or equal to d, and this corresponds to a natural notion of border for its associated tiling. We define an etale equivalence relation R_B on B by saying that two infinite paths are equivalent if they have borders which are tail equivalent in B^j for some j less than or equal to d. We show that R_B is homeomorphic to the tiling's equivalence relation R_Xi.