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- W4300842722 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." @default.
- W4300842722 created "2022-10-04" @default.
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- W4300842722 date "2010-05-17" @default.
- W4300842722 modified "2023-10-17" @default.
- W4300842722 title "Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation" @default.
- W4300842722 hasPublicationYear "2010" @default.
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