## Generalized Model Sets and Dynamical Systems (1999)

Venue: | CRM Monograph Series |

Citations: | 64 - 0 self |

### BibTeX

@INPROCEEDINGS{Schlottmann99generalizedmodel,

author = {Martin Schlottmann},

title = {Generalized Model Sets and Dynamical Systems},

booktitle = {CRM Monograph Series},

year = {1999},

pages = {143--159}

}

### Years of Citing Articles

### OpenURL

### Abstract

It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigma-compact Abelian groups with arbitrary locally compact Abelian groups as internal spaces. It is then shown that these regular model sets possess pure point diffraction spectra.

### Citations

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Citation Context ...rbitrary size in G which contain no points of P . We don't want to exclude this class of sets from the start, because it contains interesting examples like the set of visible points in a lattice (cf. =-=[A]-=-). In these cases, the empty set ; will be an element of XP . In the following description of C(XP ), this turns out to be rather inconvenient, so we consider the locally compact space XP nf;g and the... |

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Citation Context ...ion, and several other brands, which have been developed independently [KN, DK, KKL], it turned out to be possible to consider all these methods as special cases of a construction introduced by Meyer =-=[Me], du-=-bbed \model sets". Although the understanding of the quasicrystal diraction spectra has been the primary goal of the application of model sets in physics, it took until the late nineties (in comp... |

47 |
sets and their duals, in The Mathematics of Long-Range Aperiodic
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(Show Context)
Citation Context ...s usually restricted to the case of Euclidean space as \internal space". This restriction seems, in the spirit of Meyer's book, unnecessary and even unnatural, and, according to a suggestion by M=-=oody [M-=-], it is immediately clear that the theory should extend to general locally compact Abelian groups as internal spaces. This impression is conrmed by the observation that model sets with these general ... |

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Citation Context ...impression is conrmed by the observation that model sets with these general internal spaces can be characterized by a natural criterion which is formulated exclusively in terms of the \direct space&qu=-=ot; [Sch]-=-. The latter paper gives some preliminary results mostly concerning the geometrical properties of general model sets and the existence of relative frequencies of local It is my pleasure to express my ... |

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Citation Context ...pecic dynamical system constructed from the point sets. The present paper is devoted to a further generalization of this scheme which has already proved fruitful in the study of substitution tilings [=-=BMS]. The-=- class of model sets used in physical applications is usually restricted to the case of Euclidean space as \internal space". This restriction seems, in the spirit of Meyer's book, unnecessary and... |

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Citation Context ... a local indistinguishability (LI) class is carried out in Section 3; here, Dworkin's argument is introduced. Finally, in Section 4, we will introduce Pleasant's \torus parametrization" of LI cla=-=sses [BHP]-=- and apply the preceding results to model sets. 1. Translationally bounded measures In this section, we collect some technicalities and notations concerning translationally bounded measures on locally... |

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Citation Context ...ide for arbitrary P 2 XP0 for a given van Hove sequence i they exist and coincide only in P 0 , but for arbitrary van Hove sequences. This way, one can connect this Corollary with similar results in [=-=LP]-=-. In the uniquely ergodic case, we can prove the existence of the autocorrelation measure by an argument due to Dworkin [D]. We denote the uniquely determined G-invariant probability measure on XP0 by... |

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Citation Context ... late nineties (in comparison with the discovery of quasicrystals in 1984) before the Bragg character of the diraction of model sets was rigorously established under fairly general conditions by Hof [H]. Hof used a beautiful trick invented in a dierent context by Dworkin [D] which connects the diraction spectrum with the spectrum of a specic dynamical system constructed from the point sets. The p... |

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Citation Context ... and the dual of ^ G). If we allow for the sign in (3.7) we obtain that, for ';s2 C c (G), the inverse Fourier transform of A 7! hsf0g; t 0 ; E(A)' f0g; t 0 i is the functions ' sP . Hence (cf. [AG]), ^s^ '^sP = hsf0g; t 0 ; E()' f0g; t 0 i (3.9) Because in general the functions of the form ' f0g; t 0 do not form a total set in the Hilbert space L 2 () (cf. 2.5), we do not get a one-to-one c... |

5 |
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Citation Context ...]. 2. Uniform structure on sets of point sets In the following, we denote by D the set of all closed and discrete subsets of G. Oursrst task is to dene a topology, more specically a uniformity (cf. [K]) on D which makes it a complete uniform Hausdor space. For this, we have to dene what it means for two point sets, P 1 , P 2 , to be close; basically P 1 will be close to P 2 if, after a small tran... |

3 |
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(Show Context)
Citation Context ... before the Bragg character of the diraction of model sets was rigorously established under fairly general conditions by Hof [H]. Hof used a beautiful trick invented in a dierent context by Dworkin [D] which connects the diraction spectrum with the spectrum of a specic dynamical system constructed from the point sets. The present paper is devoted to a further generalization of this scheme which h... |

3 |
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Citation Context ... w.r.t Haar measure. (This shows that the dynamical system (XP0 ; G) is an \almost 1-1 extension" of a minimal rotation action on a \torus", a result that was also obtained, in the Euclidean=-= case, in [FHK]-=-.) This has an important consequence: Theorem 4.5. Let P 0 be a regular model set. Then 1) The dynamical system of P 0 is uniquely ergodic (informally: frequencies ofsnite patterns exist uniformly, cf... |

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