Annales Henri Poincaré

, Volume 11, Issue 1–2, pp 69–99 | Cite as

Tiling Groupoids and Bratteli Diagrams

  • Jean Bellissard
  • Antoine Julien
  • Jean Savinien


Let T be an aperiodic and repetitive tiling of \({{\mathbb R}^d}\) with finite local complexity. Let Ω be its tiling space with canonical transversal \({\Xi}\) . The tiling equivalence relation \({R_\Xi}\) is the set of pairs of tilings in \({\Xi}\) which are translates of each others, with a certain (étale) topology. In this paper \({R_\Xi}\) is reconstructed as a generalized “tail equivalence” on a Bratteli diagram, with its standard AF -relation as a subequivalence relation. Using a generalization of the Anderson–Putnam complex (Bellissard et al. in Commun. Math. Phys. 261:1–41, 2006) Ω is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram \({{\mathcal B}}\) is built from this sequence, and its set of infinite paths \({\partial {\mathcal B}}\) is homeomorphic to \({\Xi}\) . The diagram \({{\mathcal B}}\) is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an étale equivalence relation \({R_{\mathcal B}}\) on \({\partial {\mathcal B}}\) which is homeomorphic to \({R_\Xi}\) , and contains the AF-relation of “tail equivalence”.


Equivalence Relation Inverse Limit Horizontal Edge Maximal Path Nest Sequence 
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Authors and Affiliations

  • Jean Bellissard
    • 1
  • Antoine Julien
    • 2
  • Jean Savinien
    • 2
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 du CNRSVilleurbanne CedexFrance

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