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50
When Shape Matters: Deformations of Tiling Spaces
, 2003
"... We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map I from the parameter space of possible shapes of tiles to H 1 ..."
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Cited by 40 (12 self)
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We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map I from the parameter space of possible shapes of tiles to H 1 of a model tiling space, with values in R d. Two tiling spaces that have the same image under I are mutually locally derivable (MLD). When the difference of the images is “asymptotically negligible”, then the tiling dynamics are topologically conjugate, but generally not MLD. For substitution tilings, we give a simple test for a cohomology class to be asymptotically negligible, and show that infinitesimal deformations of shape result in topologically conjugate dynamics only when the change in the image of I is asymptotically negligible. Finally, we give criteria for a (deformed) substitution tiling space to be topologically weakly mixing.
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.
Crossed products of the Cantor set by free, minimal actions of Z d
 Comm. Math. Phys
"... Abstract. Let d be a positive integer, let X be the Cantor set, and let Z d act freely and minimally on X. We prove that the crossed product C ∗ (Z d, X) has stable rank one, real rank zero, and cancellation of projections, and that the order on K0(C ∗ (Z d, X)) is determined by traces. We obtain th ..."
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Cited by 20 (2 self)
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Abstract. Let d be a positive integer, let X be the Cantor set, and let Z d act freely and minimally on X. We prove that the crossed product C ∗ (Z d, X) has stable rank one, real rank zero, and cancellation of projections, and that the order on K0(C ∗ (Z d, X)) is determined by traces. We obtain the same conclusion for the C*algebras of various kinds of aperiodic tilings. In [35], Putnam considered the C*algebra A associated with a substitution tiling system satisfying certain additional conditions, and proved that the order on K0(A) is determined by the unique tracial state τ on A. That is, if η ∈ K0(A) satisfies τ∗(η)> 0, then there is a projection p ∈ M∞(A) = ⋃∞ n=1 Mn(A) such that η = [p]. In this paper, we strengthen Putnam’s theorem, obtaining Blackadar’s Second Fundamental Comparability Question ([7], 1.3.1) for A, namely that if p, q ∈ M∞(A) are projections such that τ(p) < τ(q) for every tracial state τ on A, then p � q, that is, that p is Murrayvon Neumann equivalent to a subprojection of q. We further prove that the C*algebra A has real rank zero [10] and stable rank one [37].
A PROOF OF THE GAP LABELING CONJECTURE
, 2002
"... We will give a proof of the Gap Labeling Conjecture formulated by Bellissard, [3]. It makes use of a version of Connes ’ index theorem for foliations which is appropriate for foliated spaces, [11]. These arise naturally in dynamics and are likely to have further applications. ..."
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Cited by 17 (0 self)
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We will give a proof of the Gap Labeling Conjecture formulated by Bellissard, [3]. It makes use of a version of Connes ’ index theorem for foliations which is appropriate for foliated spaces, [11]. These arise naturally in dynamics and are likely to have further applications.
PatternEquivariant Cohomology with Integer Coefficients”, at arXiv.com math.DS/0602066
, 2006
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Transverse Laplacians for Substitution Tilings
, 2009
"... Pearson and Bellissard recently built a spectral triple — the data of Riemanian noncommutative geometry — for ultrametric Cantor sets. They derived a family of Laplace–Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversal ..."
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Cited by 9 (5 self)
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Pearson and Bellissard recently built a spectral triple — the data of Riemanian noncommutative geometry — for ultrametric Cantor sets. They derived a family of Laplace–Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversals of tiling spaces, which are particular selfsimilar Cantor sets. We use Bratteli diagrams to encode the selfsimilarity, and Cuntz–Krieger algebras to implement it. We show that the abscissa of convergence of the ζfunction of the spectral triple gives indications on the exponent of complexity of the tiling. We determine completely the spectrum of the Laplace–Beltrami operators, give an explicit method of calculation for their eigenvalues, compute their Weyl asymptotics, and a Seeley equivalent for their heat kernels.
Lectures on Foliation Dynamics: Barcelona 2010
 Proceedings of Conference on Geometry and Topology of Foliations (C.R.M
, 2013
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Groupoids, von Neumann algebras and the integrated density of states
 MATH. PHYS. ANAL. GEOM
, 2007
"... We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. The general ..."
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Cited by 8 (2 self)
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We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. The general
Exact regularity and the cohomology of tiling spaces, Ergodic Theory Dynam
 Systems
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Topological quantization of boundary forces and the integrated density of states
 J. Phys. A
"... For quantum systems described by Schrödinger operators on the halfspace R d−1 ×R ≤0 the boundary force per unit area and unit energy is topologically quantised provided the Fermi energy lies in a gap of the bulk spectrum. Under this condition it is also equal to the integrated density of states at ..."
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Cited by 6 (3 self)
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For quantum systems described by Schrödinger operators on the halfspace R d−1 ×R ≤0 the boundary force per unit area and unit energy is topologically quantised provided the Fermi energy lies in a gap of the bulk spectrum. Under this condition it is also equal to the integrated density of states at the Fermi energy. 1