Comptes Rendus
no. 15-16
Dynamical Systems
Affability of Euclidean tilings
[Affabilité des pavages euclidiens]

Présenté par : Étienne Ghys

Fernando Alcalde Cuesta1 ; Pablo González Sequeiros1 ; Álvaro Lozano Rojo2
1 Dpto. Xeometría e Topoloxía, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2 Dpto. Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea, 48940 Leioa, Spain
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 947-952.

Nous prouvons que toute relation d'équivalence définie sur l'ensemble de Cantor par l'enveloppe d'un pavage euclidien apériodique et répétitif est affable.

We prove that every minimal equivalence relation on a Cantor set arising from the continuous hull of an aperiodic and repetitive Euclidean tiling is affable.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.06.011
Fernando Alcalde Cuesta 1 ; Pablo González Sequeiros 1 ; Álvaro Lozano Rojo 2

1 Dpto. Xeometría e Topoloxía, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2 Dpto. Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea, 48940 Leioa, Spain 
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Fernando Alcalde Cuesta; Pablo González Sequeiros; Álvaro Lozano Rojo. Affability of Euclidean tilings. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 947-952. doi : 10.1016/j.crma.2009.06.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.011/

[1] F. Alcalde Cuesta; Á. Lozano Rojo; M. Macho Stadler Dynamique transverse de la lamination de Ghys–Kenyon, Astérisque, Volume 323 (2009)

[2] J. Bellissard; R. Benedetti; J.M. Gambaudo Spaces of tilings, finite telescopic approximations and gap-labelling, Comm. Math. Phys., Volume 261 (2006), pp. 1-41

[3] E. Ghys Laminations par surfaces de Riemann, Panor. Syntheses, Volume 8 (1999), pp. 49-95

[4] T. Giordano; I. Putnam; C. Skau Affable equivalence relations and orbit structure of Cantor minimal systems, Ergodic Theory Dynam. Systems, Volume 24 (2004), pp. 441-475

[5] T. Giordano; H. Matui; I. Putnam; C. Skau Orbit equivalence for Cantor minimal Z2-systems, J. Amer. Math. Soc., Volume 21 (2008), pp. 863-892

[6] T. Giordano; H. Matui; I. Putnam; C. Skau The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, Volume 28 (2008), pp. 1509-1531

[7] T. Giordano; H. Matui; I. Putnam; C. Skau Orbit equivalence for Cantor minimal Zd-systems | arXiv

[8] B. Grünbaum; G.C. Shephard Tilings and Patterns, W.H. Freeman & Co., New York, 1987

[9] Á. Lozano Rojo, Dinámica transversa de laminaciones definidas por grafos repetitivos, UPV-EHU Ph.D. thesis, 2008

[10] H. Matui Affability of equivalence relations arising from two-dimensional substitution tilings, Ergodic Theory Dynam. Systems, Volume 26 (2006), pp. 467-480

[11] J.C. Oxtoby Ergodics sets, Bull. Amer. Math. Soc., Volume 58 (1952), pp. 116-136

[12] S. Petite On invariant measures of finite affine type tilings, Ergodic Theory Dynam. Systems, Volume 26 (2006), pp. 1159-1176

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

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Cité par Sources :

☆ This work was supported by the Spanish Ministry of Education and Science (Research Projects MTM2004-08214 and MTM2007-66262), the University of the Basque Country (R. Project EHU 06/05), and the Spanish Network of Topology.

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