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26
Spaces of tilings, finite telescopic approximations and gaplabelling
"... For a large class of tilings of R d, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull ΩT of such a tiling T inherits a minimal R dlamination structure with flat leaves and a transversal ΓT which is a Cantor set. In this case, we show ..."
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Cited by 51 (3 self)
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For a large class of tilings of R d, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull ΩT of such a tiling T inherits a minimal R dlamination structure with flat leaves and a transversal ΓT which is a Cantor set. In this case, we show that the continuous hull can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact dmanifolds. Truncated sequences furnish better and better finite approximations of the asymptotic dynamical system and the algebraic topological features related to this sequence reflect the dynamical properties of the R daction on the continuous hull. In particular the set of positive invariant measures of this action turns to be a convex cone, canonically associated with the orientation, in the projective limit of the d th homology groups of the branched manifolds. As an application of this construction we prove a gaplabelling theorem: Consider the C ∗algebra AT of ΩT, and the group K0(AT), then for every finite R dinvariant measure µ on ΩT, the image of the group K0(AT) by the µtrace satisfies: Tµ(K0(AT)) = ΓT dµ t C(ΓT,Z), where µ t is the transverse invariant measure on ΓT induced by µ and C(ΓT,Z) the set of continuous functions on ΓT with integer values. 1 1
Diffraction of Random Tilings: Some Rigorous Results
 J. STAT. PHYS
, 1999
"... The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar rando ..."
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Cited by 30 (17 self)
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The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous and absolutely continuous parts.
Strictly ergodic subshifts and associated operators
, 2005
"... We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have in ..."
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Cited by 30 (17 self)
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We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.
Singular spectrum of Lebesgue measure zero for quasicrystals
 Commun. Math. Phys
, 2002
"... exponent, linear repetitivity, primitive substitution Abstract. The spectrum of onedimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. T ..."
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Cited by 21 (5 self)
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exponent, linear repetitivity, primitive substitution Abstract. The spectrum of onedimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples as e.g. the RudinShapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. 1.
A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem
 Duke Math. J
, 2006
"... Abstract. This paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan, every locally constant SL(2,R)valued cocycle is uniform. As a consequence, the correspond ..."
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Cited by 18 (10 self)
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Abstract. This paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan, every locally constant SL(2,R)valued cocycle is uniform. As a consequence, the corresponding Schrödinger operators exhibit Cantor spectrum of Lebesgue measure zero. An investigation of Boshernitzan’s condition then shows that these results cover all earlier results of this type and, moreover, provide various new ones. In particular, Boshernitzan’s condition is shown to hold for almost all circle maps and almost all ArnouxRauzy subshifts. 1.
Hierarchical Structures in Sturmian Dynamical Systems
"... The paper is concerned with hierarchical structures in subshifts over a nite alphabet. In particular, we present a hierarchy based approach to Sturmian systems. This approach is then used to characterize the linearly repetitive Sturmian systems (among the Sturmian systems) by uniform positivity ..."
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Cited by 14 (13 self)
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The paper is concerned with hierarchical structures in subshifts over a nite alphabet. In particular, we present a hierarchy based approach to Sturmian systems. This approach is then used to characterize the linearly repetitive Sturmian systems (among the Sturmian systems) by uniform positivity of certain weights. More generally, we discuss various bounds on weights and their relationship.
Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
"... Abstract. Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity ..."
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Cited by 13 (8 self)
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Abstract. Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.
Linearly Recurrent Circle Map Subshifts And An Application To Schrödinger Operators
"... We discuss circle map sequences and subshifts generated by them. ..."
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Cited by 12 (5 self)
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We discuss circle map sequences and subshifts generated by them.
Substitution Delone sets with pure point spectrum are intermodel sets
 J. Geom. Phys
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Substitution Dynamical Systems: Characterization Of Linear Repetitivity And Applications
"... We consider dynamical systems arising from substitutions over a nite alphabet. We prove that such a system is linearly repetitive if and only if it is minimal. Based on this characterization we extend various results from primitive substitutions to minimal substitutions. This includes application ..."
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Cited by 10 (4 self)
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We consider dynamical systems arising from substitutions over a nite alphabet. We prove that such a system is linearly repetitive if and only if it is minimal. Based on this characterization we extend various results from primitive substitutions to minimal substitutions. This includes applications to random Schrodinger operators and to number theory.