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65
Weighted Dirac combs with pure point diffraction
, 2002
"... A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a ..."
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Cited by 36 (24 self)
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A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the theory of the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.
Dynamical Systems on Translation Bounded Measures: PURE POINT DYNAMICAL AND DIFFRACTION SPECTRA
, 2003
"... Certain topological dynamical systems are considered that arise from actions of σcompact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure ..."
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Cited by 31 (17 self)
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Certain topological dynamical systems are considered that arise from actions of σcompact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
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 24 (15 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.
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
, 2002
"... ... links pure point diffractivity to the asymptotic density of the almostperiods under repeated iteration of Q. We apply these results to Dekking's wellknown criterion for pure point diffractivity, generalizing it completely from its original setting in equallength alphabetic substitutions ..."
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Cited by 19 (9 self)
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... links pure point diffractivity to the asymptotic density of the almostperiods under repeated iteration of Q. We apply these results to Dekking's wellknown criterion for pure point diffractivity, generalizing it completely from its original setting in equallength alphabetic substitutions to the case of lattice substitution systems in arbitrary dimensions.
Delone dynamical systems and associated random operators
 Proc. OAMP
, 2003
"... ABSTRACT. We carry out a careful study of basic topological and ergodic features of Delone dynamical systems. We then investigate the associated topological groupoids and in particular their representations on certain direct integrals with non constant fibres. Via noncommutative integration theory t ..."
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Cited by 18 (14 self)
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ABSTRACT. We carry out a careful study of basic topological and ergodic features of Delone dynamical systems. We then investigate the associated topological groupoids and in particular their representations on certain direct integrals with non constant fibres. Via noncommutative integration theory these representations give rise to von Neumann algebras of random operators. Features of these algebras and operators are discussed. Restricting our attention to a certain subalgebra of tight binding operators, we then discuss a Shubin trace formula.
Deformation of Delone dynamical systems and pure point spectrum
 J. Fourier Anal. Appl
, 2005
"... Abstract. This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss ..."
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Cited by 13 (10 self)
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Abstract. This paper deals with certain dynamical systems built from point sets and, more generally, measures on locally compact Abelian groups. These systems arise in the study of quasicrystals and aperiodic order, and important subclasses of them exhibit pure point diffraction spectra. We discuss the relevant framework and recall fundamental results and examples. In particular, we show that pure point diffraction is stable under “equivariant” local perturbations and discuss various examples, including deformed model sets. A key step in the proof of stability consists in transforming the problem into a question on factors of dynamical systems. 1.
A note on shelling
 Discr. Comput. Geom
, 2003
"... Abstract. The radial distribution function is a characteristic geometric quantity of a point set in Euclidean space that reflects itself in the corresponding diffraction spectrum and related objects of physical interest. The underlying combinatorial and algebraic structure is well understood for cry ..."
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Cited by 11 (6 self)
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Abstract. The radial distribution function is a characteristic geometric quantity of a point set in Euclidean space that reflects itself in the corresponding diffraction spectrum and related objects of physical interest. The underlying combinatorial and algebraic structure is well understood for crystals, but less so for nonperiodic arrangements such as mathematical quasicrystals or model sets. In this note, we summarise several aspects of central versus averaged shelling, illustrate the difference with explicit examples, and discuss the obstacles that emerge with aperiodic order. 1.
Point sets and dynamical systems in the autocorrelation topology
 Can. Math. Bulletin
, 2004
"... This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups G. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of G is complete in this to ..."
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Cited by 9 (3 self)
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This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups G. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of G is complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced. The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to a Gdynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as a G dynamical system, is a factor of the dynamical local hull. ∗RVM is grateful for the continuing support of the Natural Sciences and Engineering Research Council of Canada.