Tiling Groupoids and Bratteli Diagrams

Abstract

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".

Simple groups of dynamical origin

“…For every π ∈ A d , restrictions of F (1) π to the domains of P and D (1) are equal to P π and D (1) π , respectively, while its restriction to the domain of D (2) is trivial. Similarly, restrictions of F (2) π to the domains of P and D (2) are equal to P π and D (2) π , respectively, while its restriction to the domain of D (1) is trivial. It follows from Lemma 3.3 that A(P) and A(D (i) ), i = 1, 2, belong to A(F (1) ) ∪ A(F (2) ) .…”

“…The elements of the form γγ −1 are called units. We denote the set of units of a groupoid G by G (0) , the set of composable pairs by G (2) , the source and range maps by s and r, so that s(γ) = γ −1 γ, r(γ) = γγ −1 for all γ ∈ G. Units x, y ∈ G (0) belong to one G-orbit if there exists γ ∈ G such that s(γ) = x and r(γ) = y. A groupoid G is said to be minimal if all its orbits are dense in G (0) .…”

Embeddings of self-similar ultrametric Cantor sets

“…Bratteli diagrams were first introduced in [3] to classify AF C * -algebras. They provide also a handy formalism for substitutions [6,7,9], and tilings in general [2].…”

Lectures on Foliation Dynamics

“…The tiling space Ω ∆ was introduced by Bellisard in his study of mathematical models of electron transport [21]. This construction is the subject of many papers, as for example in [12,22,23,84,102,123,187]. The results have been generalized to quasi-periodic tilings of G-spaces in [24].…”