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Documents  52C23 | enregistrements trouvés : 11

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Research schools;Dynamical Systems and Ordinary Differential Equations;Geometry

These lectures introduce the dynamical systems approach to tilings of Euclidean space, especially quasicrystalline tilings that have been constructed using a ‘supertile method’. Because tiling dynamics parallels one-dimensional symbolic dynamics, we discuss this case as well, highlighting the differences and similarities in the methods of study and the results that can be obtained.
In the first lecture we motivate the field with the discovery of quasicrystals, which led to D. Schectman’s winning the 2011 Nobel Prize in Chemistry. Then we set up the basics of tiling dynamics, describing tiling spaces, a tiling metric, and the shift or translation actions. Shift-invariant and ergodic measures are discussed, along with fundamental topological and dynamical properties.
The second lecture brings in the supertile construction methods, including symbolic substitutions, self-similar tilings, $S$-adic systems, and fusion rules. Numerous examples are given, most of which are not the “standard” examples, and we identify many commonalities and differences between these interrelated methods of construction. Then we compare and contrast dynamical results for supertile systems, highlighting those key insights that can be adapted to all cases.
In the third lecture we investigate one of the many current tiling research areas: spectral theory. Schectman made his Nobel-prize-winning discovery using diffraction analysis, and studying the mathematical version has been quite fruitful. Spectral theory of tiling dynamical systems is also of broad interest. We describe how these types of spectral analysis are carried out, give examples, and discuss what is known and unknown about the relationship between dynamical and diffraction analysis. Special attention is paid to the “point spectrum”, which is related to eigenfunctions and also to the bright spots that appear on diffraction images.
These lectures introduce the dynamical systems approach to tilings of Euclidean space, especially quasicrystalline tilings that have been constructed using a ‘supertile method’. Because tiling dynamics parallels one-dimensional symbolic dynamics, we discuss this case as well, highlighting the differences and similarities in the methods of study and the results that can be obtained.
In the first lecture we motivate the field with the discovery of ...

37B50 ; 37B10 ; 52C23

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- xii; 327 p.
ISBN 978-1-4704-2919-5

IAS/Park City mathematics series , 0023

Localisation : Collection 1er étage

mécanique statistique # science des matériaux # géométrie discrète # élasticité non linéaire # matériaux granulaires # cristaux liquides # auto-assemblage synthétique

82B05 ; 35Q70 ; 82B26 ; 74N05 ; 51P05 ; 52C17 ; 52C23

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Research schools;Dynamical Systems and Ordinary Differential Equations;Geometry;Number Theory

Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity.
It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization.
Nevertheless, recently Berthé, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982).
Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincaré section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well.
It is the aim of my series of lectures to review the above results.
Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword ...

11B83 ; 11K50 ; 37B10 ; 52C23 ; 53D25

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Research schools;Dynamical Systems and Ordinary Differential Equations;Geometry

In this lecture we focus on selected topics around the themes: Delone sets as models for quasicrystals, inflation symmetries and expansion constants, substitution Delone sets and tilings, and associated dynamical systems.

52C23 ; 37B50

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- xx; 386 p.
ISBN 978-0-521-86992-8

Encyclopedia of mathematics and its applications , 0166

Localisation : Collection 1er étage

ordre apériodique # cristallographie # analyse de Fourier # quasi-cristal

52-02 ; 52C23 ; 82D25

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- 273 p.
ISBN 978-3-540-43241-8

Springer tracts in modern physics , 0180

Localisation : Ouvrage RdC (Cove)

quasi-cristal # pavage quasi-périodique # recouvrement # pavage # système discret # système quasi-périodique

52C23 ; 05B40

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- 379 p.
ISBN 978-0-8218-2629-4

CRM monograph series , 0013

Localisation : Collection 1er étage

géométrie discrète # cristallographie # quasi-cristal # transformée de Fourier # fractale # k-théorie # pavage # c*-algèbre # pavage auto-affine # symétrie # diffraction

52C23 ; 43A25 ; 28A80 ; 46L80 ; 47B25 ; 11R52 ; 52C17 ; 82B20

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- xi; 428 p.
ISBN 978-3-0348-0902-3

Progress in mathematics , 0309

Localisation : Collection 1er étage

système apériodique # quasi-cristal # théorie de la diffraction # pavage # ensemble de Delone # conjecture de Pisot # opérateur de Schrödinger # théorie des nombres

52C23 ; 37B50 ; 47A35 ; 11K70 ; 58B34 ; 52-06 ; 52C22

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- 320 p.
ISBN 978-0-8218-3722-1

Localisation : Ouvrage RdC (Coxe)

groupe de Coxeter # théorie des groupes # géométrie algébrique # polytope

01A99 ; 14M25 ; 20E42 ; 20F55 ; 20E46 ; 51A20 ; 51M20 ; 52B15 ; 52B70 ; 52C23

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- 120 p.
ISBN 978-0-8218-2965-3

Memoirs of the american mathematical society , 0758

Localisation : Collection 1er étage

pavage a-périodique # cohomologie # pavage invariant # équation fonctionnelle # K-théorie # système dynamique # dynamique topologique

52C23 ; 19E20 ; 37BXX ; 46Lxx ; 55Txx ; 82D25

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- x; 118 p.
ISBN 978-0-8218-4727-5

University lecture series , 0046

Localisation : Collection 1er étage

pavage de dimensions n # pavage apériodique # quasi-cristal

52C22 ; 55-02 ; 52C23 ; 55N99 ; 55N05 ; 54-02 ; 37BXX

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