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Pure point diffraction and cut and project schemes for measures: The smooth case
 Math. Z. 256 (2007) 347–378; math.DS/0603453. DANIEL LENZ AND NICOLAE STRUNGARU
"... Abstract. We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic images of the underlying torus. In particular, they ..."
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Abstract. We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic images of the underlying torus. In particular, they are strictly ergodic with pure point spectrum and continuous eigenfunctions. Their diffraction can be calculated explicitly. Our results cover and extend corresponding earlier results on dense Dirac combs and continuous weight functions with compact support. They also mark a clear difference in terms of factor maps between the case of continuous and noncontinuous weight functions. 1.
An Ultrametric State Space with a Dense Discrete Overlap Distribution: Paperfolding Sequences
 J STAT PHYS
, 2011
"... We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support (as a closed set) is the full interval [−1,+1]. The space of paperfolding sequences has an ultrametric structure. O ..."
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Cited by 2 (1 self)
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We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support (as a closed set) is the full interval [−1,+1]. The space of paperfolding sequences has an ultrametric structure. Our example provides an illustration of some properties which were suggested to occur for pure states in spin glass models.
MODULATED QUASICRYSTALS
"... Abstract. There is some confusion in the literature what “modulated quasicrystals ” are: Some people treat “modulated quasicrystals ” and “deformed model sets ” as exchangeable termini (compare [6, 9, 5]), others claim that “[...] the projection method becomes powerless against incommensurate modula ..."
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Cited by 1 (0 self)
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Abstract. There is some confusion in the literature what “modulated quasicrystals ” are: Some people treat “modulated quasicrystals ” and “deformed model sets ” as exchangeable termini (compare [6, 9, 5]), others claim that “[...] the projection method becomes powerless against incommensurate modulated structures ” (e.g., [12, p. 148]). We use a mathematical approach and propose the following classification: While deformed model sets are characterised by a deformation that does not change the location of the Bragg peaks, a modulated phase yields “satellites ” in the diffraction pattern and is achieved “by enlarging the internal space ” with a torus. 1.
Quasicrystals, C*algebras and Ktheory
, 2004
"... Quasicrystals are a phase between crystals and amorphous materials, exhibiting longrange order without periodicity. We review attempts to provide a theory of electronic transport in quasicrystals that may generalize Bloch theory. To this end, we introduce groupoid C*algebras, and use these to deve ..."
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Quasicrystals are a phase between crystals and amorphous materials, exhibiting longrange order without periodicity. We review attempts to provide a theory of electronic transport in quasicrystals that may generalize Bloch theory. To this end, we introduce groupoid C*algebras, and use these to develop Noncommutative Topology. This can be used to obtain a noncommutative C*algebra of observables in the aperiodic case that is a generalization of its periodic counterpart. The Ktheory of this C*algebra is used to obtain a labelling of the gaps in the spectrum of oneelectron Hamiltonians, and this labelling can be linked to the value of the integrated density of states on the gaps. Finally, we study a