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Annales Henri Poincaré

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

Tiling Groupoids and Bratteli Diagrams

  • Jean Bellissard
  • Antoine Julien
  • Jean Savinien
Article

Abstract

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”.

Keywords

Equivalence Relation Inverse Limit Horizontal Edge Maximal Path Nest Sequence 
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 2010

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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