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


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 \({{\mathcal R}_\Xi}\) .


Equivalence Relation Generalize Path Horizontal Edge Substitution Rule Bratteli Diagram 
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© 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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