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- W1989760636 abstract "In this second paper, we study the case of substitution tilings of $${{mathbb R}^d}$$ . 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 $${{mathcal B}}$$ which is made of the Bratteli diagrams $${{mathcal B}^j, j=0, ldots d}$$ , of all those substitutions. The set of infinite paths in $${{mathcal B}^d}$$ is identified with the canonical transversal Ξ of the tiling. Any such path has a “border”, which is a set of tails in $${{mathcal B}^j}$$ for some j ≤ d, and this corresponds to a natural notion of border for its associated tiling. We define an étale equivalence relation $${{mathcal R}_{mathcal B}}$$ on $${{mathcal B}}$$ by saying that two infinite paths are equivalent if they have borders which are tail equivalent in $${{mathcal B}^j}$$ for some j ≤ d. We show that $${{mathcal R}_{mathcal B}}$$ is homeomorphic to the tiling’s equivalence relation $${{mathcal R}_Xi}$$ ." @default.
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- W1989760636 date "2011-07-01" @default.
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- W1989760636 title "Tiling Groupoids and Bratteli Diagrams II: Structure of the Orbit Equivalence Relation" @default.
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- W1989760636 doi "https://doi.org/10.1007/s00023-011-0121-4" @default.
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