Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation
Résumé
In this second paper, we study the case of substitution tilings of Rd. 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 Bj, j = 0, . . . d, of all those substitutions. The set of infinite paths in Bd is identified with the canonical transversal of the tiling. Any such path has a “border”, which is a set of tails in Bj for some j ≤ d, and this corresponds to a natural notion of border for its associated tiling. We define an ´etale equivalence relation RB on B by saying that two infinite paths are equivalent if they have borders which are tail equivalent in Bj for some j
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