Affability of Euclidean tilings

Let T be an aperiodic and repetitive tiling of R d with finite local complexity. Let Ω be its tiling space with canonical transversal Ξ. The tiling equivalence relation RΞ is the set of pairs of tilings in Ξ which are translates of each others, with a certain (étale) topology. In this paper RΞ 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 B is built from this sequence, and its set of infinite paths ∂B is homeomorphic to Ξ. The diagram B is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T . This allows to define ań etale equivalence relation RB on ∂B which is homeomorphic to RΞ, and contains the AF -relation of "tail equivalence".

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Tiling Groupoids and Bratteli Diagrams

Bellissard¹,

Julien²,

Savinien³

2010

Ann. Henri Poincaré

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Transversely Cantor laminations as inverse limits

Cuesta

Rojo

Stadler

2010

Proc. Amer. Math. Soc.

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Abstract. We demonstrate that any minimal transversely Cantor compact lamination of dimension p and class C 1 without holonomy is an inverse limit of compact branched manifolds of dimension p. To prove this result, we extend the triangulation theorem for C 1 manifolds to transversely Cantor C 1 laminations. In fact, we give a simple proof of this classical theorem based on the existence of C 1 -compatible differentiable structures of class C ∞ .

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Affability of Euclidean tilings

Cited by 2 publications

References 11 publications

Tiling Groupoids and Bratteli Diagrams

Tiling Groupoids and Bratteli Diagrams

Transversely Cantor laminations as inverse limits

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