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Pré-Publication, Document De Travail Année : 2010

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

Antoine Julien
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Jean Savinien
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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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Dates et versions

hal-00529784 , version 1 (26-10-2010)

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  • HAL Id : hal-00529784 , version 1

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Antoine Julien, Jean Savinien. Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation. 2010. ⟨hal-00529784⟩
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