Combinatorics and topology of the Robinson tiling

Franz Gähler; Antoine Julien; Jean Savinien

doi:10.1016/j.crma.2012.06.007

Dynamical Systems

Combinatorics and topology of the Robinson tiling
[Combinatoire et topologie des pavages de Robinson]

Présenté par : the Editorial Board

Franz Gähler ¹ ; Antoine Julien ² ; Jean Savinien ³

¹ Faculty of Mathematics, University of Bielefeld, Universitätsstraße 25, 33615 Bielefeld, Germany
² Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, V8P 5C2 Victoria, BC, Canada
³ Département de Mathématiques, Université de Metz, Ile du Saulcy, 57045 Metz cedex 1, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 627-631.

Résumé
Abstract

Nous étudions lʼespace de tous les pavages qui peuvent sʼobtenir à partir des tuiles de Robinson (il sʼagit dʼun sous-décalage de type fini). Cet espace contient un unique sous-espace minimal, que nous décrivons par le biais dʼune substitution. En conséquence, il est possible de calculer les groupes de cohomologie associés, et de montrer quʼil sʼagit dʼun pavage de coupe et projection. Cet article a une annexe qui a été transmise à lʼAcadémie des Sciences.

We study the space of all tilings which can be obtained using the Robinson tiles (this is a two-dimensional subshift of finite type). We prove that it has a unique minimal subshift, and describe it by means of a substitution. This description allows to compute its cohomology groups, and prove that it is a model set. This article has an annex which was transmitted to the Académie des Sciences.

Reçu le : 2012-03-07
Accepté le : 2012-06-18
Publié le : 2012-06-01

DOI : 10.1016/j.crma.2012.06.007

Affiliations des auteurs :

Franz Gähler ¹ ; Antoine Julien ² ; Jean Savinien ³

¹ Faculty of Mathematics, University of Bielefeld, Universitätsstraße 25, 33615 Bielefeld, Germany
² Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, V8P 5C2 Victoria, BC, Canada
³ Département de Mathématiques, Université de Metz, Ile du Saulcy, 57045 Metz cedex 1, France

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     author = {Franz G\"ahler and Antoine Julien and Jean Savinien},
     title = {Combinatorics and topology of the {Robinson} tiling},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {627--631},
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     year = {2012},
     doi = {10.1016/j.crma.2012.06.007},
     language = {en},
}

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Franz Gähler; Antoine Julien; Jean Savinien. Combinatorics and topology of the Robinson tiling. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 627-631. doi : 10.1016/j.crma.2012.06.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.06.007/

Bibliographie
Cité par

[1] J.E. Anderson; I.F. Putnam Topological invariants for substitution tilings and their associated $C^{⁎}$ -algebras, Ergodic Theory Dynam. Systems, Volume 18 (1998), pp. 509-537

[2] M. Baake; M. Schlottmann; P.D. Jarvis Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability, J. Phys. A: Math. Gen., Volume 24 (1991), pp. 4637-4654

[3] A. Johnson; K. Madden Putting the pieces together: understanding Robinsonʼs nonperiodic tilings, College Math. J., Volume 28 (1997), pp. 172-181

[4] J.-Y. Lee; R.V. Moody Lattice substitution systems and model sets, Discrete Comput. Geom., Volume 25 (2001) no. 2, pp. 173-201

[5] R. Robinson Undecidability and nonperiodicity for tilings of the plane, Invent. Math., Volume 12 (1971), pp. 177-209

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