# Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation - Mathematics > Dynamical Systems

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Abstract: In this second paper, we study the case of substitution tilings of R^d. Thesubstitution on tiles induces substitutions on the faces of the tiles of alldimensions j=0,

., d-1. We reconstruct the tiling-s equivalence relation in apurely combinatorial way using the AF-relations given by the lower dimensionalsubstitutions. We define a Bratteli multi-diagram B which is made of theBratteli diagrams B^j, j=0,

., d, of all those substitutions. The set ofinfinite paths in B^d is identified with the canonical transversal Xi of thetiling. Any such path has a -border-, which is a set of tails in B^j for some jless than or equal to d, and this corresponds to a natural notion of border forits associated tiling. We define an etale equivalence relation R B on B bysaying that two infinite paths are equivalent if they have borders which aretail equivalent in B^j for some j less than or equal to d. We show that R B ishomeomorphic to the tiling-s equivalence relation R Xi.

Author: ** Antoine Julien, Jean Savinien**

Source: https://arxiv.org/