Noncommutative Geometry of Tilings and Gap Labelling

Kellendonk, Johannes

doi:10.1142/s0129055x95000426

Cited by 116 publications

(164 citation statements)

References 31 publications

(74 reference statements)

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“…This result is considerably stronger than the corresponding earlier results of Kellendonk [20], and Hof [15], which only gave weak convergence. It fits well within the general point of view that quasicrystals should behave very uniformly due to their proximity to crystals.…”

Section: Introductioncontrasting

confidence: 69%

An ergodic theorem for Delone dynamical systems and existence of the integrated density of states

Lenz

Stollmann

2005

J. Anal. Math.

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Section: Introductioncontrasting

confidence: 69%

An ergodic theorem for Delone dynamical systems and existence of the integrated density of states

Lenz

Stollmann

2005

J. Anal. Math.

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“…Such patch spaces are said to force their borders. This condition was introduced by Kellendonk in [41] for substitution tilings and was required in order to be able to recover the hull as the inverse limit of such spaces. It was generalized in [13] for branched manifolds of repetitive tilings with FLC and coincides here with the above definition.…”

Section: Definition 10mentioning

confidence: 99%

A spectral sequence for theK-theory of tiling spaces

Savinien

Bellissard

2009

Ergod. Th. Dynam. Sys.

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Let T be an aperiodic and repetitive tiling of R d with finite local complexity. We present a spectral sequence that converges to the K-theory of T with page-2 given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of T generalizes the cohomology of the base space of a fibration with local coefficients in the K-theory of its fiber. We prove that it is isomorphic to thě Cech cohomology of the hull of T (a compactification of the family of its translates). Main ResultsLet T be an aperiodic and repetitive tiling of R d with finite local complexity (definition 3). The hull Ω is a compactification, with respect to an appropriate topology, of the family of translates of T by vectors of R d (definition 4). The tiles of T are given compatible ∆-complex decompositions (section 4.2), with each simplex punctured, and the ∆-transversal Ξ ∆ is the subset of Ω corresponding to translates of T having the puncture of one of those simplices at the origin 0 R d . The prototile space B 0 (definition 9) is built out of the prototiles of T (translational equivalence classes of tiles) by gluing them together according to the local configurations of their representatives in the tiling. The hull is given a dynamical system structure via the natural action of the group R d on itself by translation [13]. The C * -algebra of the hull is isomorphic to the crossed-productThere is a map p o from the hull onto the prototile space (proposition 1)which, thanks to a lamination structure on Ω (remark 2), resembles (although is not) a fibration with base space B 0 and fiber Ξ ∆ (remark 4). The Pimsner-Voiculescu (PV) *

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“…Therefore the C * -algebra C * (G) can be thought as the non-commutative analogue of the subshift space [5], corresponding to the matrix A G . It is worth noticing that beyond directed lattices, the Cuntz-Krieger algebras are ultimately connected with various other topics, like wavelets [4], tilings [22,2], generalised sub-shifts [5], noncommutative geometry [10], etc.…”

Section: A C * -Algebraic Description Of Oriented Latticesmentioning

confidence: 99%

On the physical relevance of random walks: an example of random walks on a randomly oriented lattice

Kaimanovich¹

2004

Random Walks and Geometry

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Abstract. Random walks on general graphs play an important role in the understanding of the general theory of stochastic processes. Beyond their fundamental interest in probability theory, they arise also as simple models of physical systems. A brief survey of the physical relevance of the notion of random walk on both undirected and directed graphs is given followed by the exposition of some recent results on random walks on randomly oriented lattices. It is worth noticing that general undirected graphs are associated with (not necessarily Abelian) groups while directed graphs are associated with (not necessarily Abelian) C * -algebras. Since quantum mechanics is naturally formulated in terms of C * -algebras, the study of random walks on directed lattices has been motivated lately by the development of the new field of quantum information and communication.

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Noncommutative Geometry of Tilings and Gap Labelling

Cited by 116 publications

References 31 publications

An ergodic theorem for Delone dynamical systems and existence of the integrated density of states

An ergodic theorem for Delone dynamical systems and existence of the integrated density of states

A spectral sequence for theK-theory of tiling spaces

On the physical relevance of random walks: an example of random walks on a randomly oriented lattice

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