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55
Twisted Ktheory of differentiable stacks
 ANN. SCI. ÉCOLE NORM. SUP
, 2004
"... In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framew ..."
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Cited by 75 (13 self)
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In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted Ktheories including the usual twisted Ktheory of topological spaces, twisted equivariant Ktheory, and the twisted Ktheory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted Kgroups can be expressed by socalled “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of Ktheory (KKtheory) of C ∗algebras.
Tiling Semigroups
 11TH ICALP, LECTURE NOTES IN COMPUTER SCIENCE 199
, 1999
"... It has recently been shown how to construct an inverse semigroup from any tiling: a construction having applications in Ktheoretical gaplabelling. In this paper, we provide the categorical basis for this construction in terms of an appropriate group acting partially and without fixed points on ..."
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Cited by 52 (13 self)
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It has recently been shown how to construct an inverse semigroup from any tiling: a construction having applications in Ktheoretical gaplabelling. In this paper, we provide the categorical basis for this construction in terms of an appropriate group acting partially and without fixed points on an inverse category associated with the tiling.
The local structure of tilings and their integer group of invariants
 Comm. Math. Phys
, 1997
"... The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the K0group of the groupoid C ∗algebra for ..."
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Cited by 51 (13 self)
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The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the K0group of the groupoid C ∗algebra for tilings which reduce to decorations of Z d. The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and primitive. This yields in particular the set of possible gap labels predicted by Ktheory for Schrödinger operators describing the particle motion in such a tiling. KCLTH956 1
Tiling spaces are inverse limits
 J. Math. Phys
"... Abstract. Let M be an arbitrary Riemannian homogeneous space, and let Ω be a space of tilings of M, with finite local complexity (relative to some symmetry group Γ) and closed in the natural topology. Then Ω is the inverse limit of a sequence of compact finitedimensional branched manifolds. The bra ..."
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Cited by 39 (8 self)
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Abstract. Let M be an arbitrary Riemannian homogeneous space, and let Ω be a space of tilings of M, with finite local complexity (relative to some symmetry group Γ) and closed in the natural topology. Then Ω is the inverse limit of a sequence of compact finitedimensional branched manifolds. The branched manifolds are (finite) unions of cells, constructed from the tiles themselves and the group Γ. This result extends previous results of Anderson and Putnam [AP], of Ormes, Radin and Sadun [ORS], of Bellissard, Benedetti and Gambaudo [BBG], and of Gähler [Gäh]. In particular, the construction in this paper is a natural generalization of Gähler’s. 1. Background In the last few years, it has become clear that many spaces of tilings of R d can be viewed as inverse limit spaces. Anderson and Putnam [AP] began this program for substitution tilings. Given a substitution, they showed that the corresponding space of tilings of R d is the inverse limit of a branched dmanifold K under an expansive map from K to itself. If the substitution has a property called “forcing the border ” [Kel], then the manifold K is constructed by stitching all the tile types together along possible common boundaries.
An ergodic theorem for Delone dynamical systems and existence of the density of states
 J. Anal. Math
"... Abstract. We study strictly ergodic Delone dynamical systems and prove an ergodic theorem for Banach space valued functions on the associated set of pattern classes. As an application, we prove existence of the integrated density of states in the sense of uniform convergence in distribution for the ..."
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Cited by 37 (14 self)
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Abstract. We study strictly ergodic Delone dynamical systems and prove an ergodic theorem for Banach space valued functions on the associated set of pattern classes. As an application, we prove existence of the integrated density of states in the sense of uniform convergence in distribution for the associated random operators.
Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets
 J. Noncommut. Geom
"... Abstract. An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree [28]. This tree allows to define a family of spectral triples (CLip(C), H, D) using the ℓ 2 ..."
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Cited by 31 (3 self)
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Abstract. An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree [28]. This tree allows to define a family of spectral triples (CLip(C), H, D) using the ℓ 2space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here CLip(C) denotes the space of Lipschitz continuous functions on (C, d). The family of spectral triples is indexed by the space of choice functions which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the Dirac operator D then allows to recover the metric on C. The corresponding ζfunction is shown to have abscissa of convergence, s0, equal to the upper box dimension of (C, d). Taking the residue at this singularity leads to the definition of a canonical probability measure on C which in certain cases coincides with the Hausdorff measure at dimension s0. This measure in turns induces a measure on the space of choices. Given a choice, the commutator of D with
A homeomorphism invariant for substitution tiling spaces
, 2000
"... We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed dire ..."
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Cited by 26 (14 self)
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We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheellike tiling spaces. We also introduce a module structure on cohomology which is very convenient as well as of intuitive value.
Patternequivariant functions and cohomology
 J. Phys. A
"... Abstract. The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier accessible. It is based on generalizing the notion ..."
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Cited by 26 (8 self)
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Abstract. The cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern. Here we present a construction which is more direct and therefore easier accessible. It is based on generalizing the notion of equivariance from lattices to point patterns of finite local complexity. 1.
Topological Equivalence of Tilings
 J.MATH.PHYS., VOL
, 1997
"... We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomor ..."
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Cited by 24 (7 self)
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We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic.