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Annales Henri Poincaré

, Volume 13, Issue 2, pp 297–332 | Cite as

Tiling Groupoids and Bratteli Diagrams II: Structure of the Orbit Equivalence Relation

  • Antoine Julien
  • Jean Savinien
Article

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 jd. We show that \({{\mathcal R}_{\mathcal B}}\) is homeomorphic to the tiling’s equivalence relation \({{\mathcal R}_\Xi}\) .

Keywords

Equivalence Relation Generalize Path Horizontal Edge Substitution Rule Bratteli Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Institut Camille Jordan, Université Claude Bernard Lyon 1Villeurbanne CedexFrance

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