Tiling groupoids and Bratteli diagrams

TL;DR

This paper links tiling spaces with Bratteli diagrams by reconstructing the tiling equivalence relation as a tail equivalence on a diagram, enhancing understanding of tiling dynamics through combinatorial structures.

Contribution

It introduces a novel method to represent tiling spaces and their equivalence relations using Bratteli diagrams with horizontal structures, generalizing previous approaches.

Findings

Tiling space X is homeomorphic to the infinite path space of a constructed Bratteli diagram.

The tiling equivalence relation R_X is modeled as a tail equivalence on the diagram.

The approach generalizes the Anderson-Putnam complex to include adjacency information.

Abstract

Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. Let O be its tiling space with canonical transversal X. The tiling equivalence relation R_X is the set of pairs of tilings in X which are translates of each others, with a certain (etale) topology. In this paper R_X is reconstructed as a generalized "tail equivalence" on a Bratteli diagram, with its standard AF-relation as a subequivalence relation. Using a generalization of the Anderson-Putnam complex, O is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is built from this sequence, and its set of infinite paths dB is homeomorphic to X. The diagram B is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an etale equivalence relation R_B on dB which is homeomorphic to R_X, and contains the AF-relation of "tail equivalence".