On the Fundamental Group of self-affine plane Tiles
[Groupe fondamental de motifs auto-affines]
Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2493-2524.

Soient A 2 × 2 une matrice expansive, 𝒟 2 un ensemble à |det(A)| éléments et 𝒯 l’ensemble défini par l’équation A𝒯=𝒯+𝒟. Si 𝒯 a une mesure de Lebesgue sur 2 strictement supérieure à zéro, alors 𝒯 est appelé motif plan auto-affine. Cet article établit certaines propriétés topologiques de 𝒯. Nous montrons que le groupe fondamental π 1 (𝒯) de 𝒯 est soit trivial, soit infini non dénombrable, et nous donnons des critères associés à chacun des deux cas. De plus, nous incluons une courte preuve de la propriété que l’adhérence de chaque composante connexe de int (𝒯) est un continuum localement connexe (nous démontrons même ce résultat dans le cas plus général d’attracteurs plans d’IFS satisfaisant la condition de l’ensemble ouvert). Si π 1 (𝒯)=0, nous montrons même que l’adhérence de chaque composante de int(𝒯) est homéomorphe au disque unité.

Nous appliquons nos résultats à plusieurs examples de motifs étudiés dans la littérature.

Let A 2 × 2 be an expanding matrix, 𝒟 2 a set with |det(A)| elements and define 𝒯 via the set equation A𝒯=𝒯+𝒟. If the two-dimensional Lebesgue measure of 𝒯 is positive we call 𝒯 a self-affine plane tile. In the present paper we are concerned with topological properties of 𝒯. We show that the fundamental group π 1 (𝒯) of 𝒯 is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of π 1 (𝒯). Furthermore, we give a short proof of the fact that the closure of each component of int (𝒯) is a locally connected continuum (we prove this result even in the more general case of plane IFS attractors fulfilling the open set condition). If π 1 (𝒯)=0 we even show that the closure of each component of int (𝒯) is homeomorphic to a closed disk.

We apply our results to several examples of tiles which are studied in the literature.

DOI : 10.5802/aif.2247
Classification : 52C20, 14F35, 11A63, 05B45
Keywords: Tile, tiling, fundamental group, number system
Mot clés : Motif, pavage, groupe fondamental, Systme de numŽration
Luo, Jun 1 ; Thuswaldner, Jörg M. 2

1 Sun Yat-Sen University School of Mathematics and Computational Science Guangzhou 510275 (China)
2 Institut für Mathematik und Angewandte Geometrie Abteilung für Mathematik und Statistik Montanuniversität Leoben Franz-Josef-Strasse 18 8700 Leoben (Austria)
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Luo, Jun; Thuswaldner, Jörg M. On the Fundamental Group of self-affine plane Tiles. Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2493-2524. doi : 10.5802/aif.2247. http://www.numdam.org/articles/10.5802/aif.2247/

[1] Ahlfors, L. V. Complex Analysis, McGraw-Hill Book Comp, 1966 | MR | Zbl

[2] Akiyama, S.; M. Thuswaldner, J. A survey on topological properties of tiles related to number systems, Geom. Dedicata, Volume 109 (2004), pp. 89-105 | DOI | MR | Zbl

[3] Akiyama, S.; M. Thuswaldner, J. Topological structure of fractal tilings generated by quadratic number systems, Comput. Math. Appl., Volume 49 (2005), pp. 1439-1485 | DOI | MR | Zbl

[4] Bandt, C.; Wang, Y. Disk-like self-affine tiles in 2 , Discrete Comput. Geom., Volume 26 (2001), pp. 591-601 | MR | Zbl

[5] Conner, G. R.; W. Lamoreaux, J. On the existence of universal covering spaces for metric spaces and subsets of the euclidean plane (preprint) | MR | Zbl

[6] Croft, H. T.; J. Falconer, K.; K. Guy, R. Unsolved problems in geometry. Problem Books in Mathematics, Unsolved Problems in Intuitive Mathematics, II, Springer-Verlag, New York, 1994 (Corrected reprint of the 1991 original) | MR | Zbl

[7] Eilenberg, S. Languages and Machines, A, Academic Press, New York, 1974

[8] Falconer, K. Fractal Geometry, John Wiley and Sons, 1990 | MR | Zbl

[9] Gröchenig, K.; Haas, A. Self-similar lattice tilings, J. Fourier Anal. Appl., Volume 1 (1994), pp. 131-170 | DOI | MR | Zbl

[10] Hata, M. On the structure of self-similar sets, Japan J. Appl. Math., Volume 2 (1985), pp. 381-414 | DOI | MR | Zbl

[11] Hatcher, A. Algebraic Topology, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[12] Hutchinson, J. E. Fractals and self-similarity, Indiana Univ. Math. J., Volume 30 (1981), pp. 713-747 | DOI | MR | Zbl

[13] Kátai, I. Number Systems and Fractal Geometry, Janus Pannonius University Pecs, 1995 | Zbl

[14] Kovács, A. On the computation of attractors for invertible expanding linear operators in k , Publ. Math. Debrecen, Volume 56 (2000), pp. 97-120 | MR | Zbl

[15] Kuratowski, K. Topology, I, Academic Press, New York and London, 1966 | MR | Zbl

[16] Kuratowski, K. Topology, II, Academic Press, New York and London, 1968 | Zbl

[17] Luo, J.; Akiyama, S.; M. Thuswaldner, J. On the boundary connectedness of connected tiles, Math. Proc. Cambridge Philos. Soc., Volume 137 (2004), pp. 397-410 | DOI | MR | Zbl

[18] Luo, J.; Rao, H.; Tan, B. Topological structure of self-similar sets, Fractals, Volume 2 (2002), pp. 223-227 | DOI | MR | Zbl

[19] Moise, E. E. Geometric Topology in Dimensions 2 and 3 , Graduate Texts in Mathematics, 47, Springer-Verlag, New York-Heidelberg, 1977 | MR | Zbl

[20] Ngai, S.-M.; Tang, T.-M. Vertices of self-similar tiles (Illinois J. Math, to appear) | MR

[21] Ngai, S.-M.; Tang, T.-M. A technique in the topology of connected self-similar tiles, Fractals, Volume 12 (2004), pp. 389-403 | DOI | MR

[22] Ngai, S.-M.; Tang, T.-M. Toplogy of self-similar tiles in the plane with disconnected interiors, Topology Appl., Volume 150 (2005), pp. 139-155 | DOI | MR | Zbl

[23] Pommerenke, C. Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975 | MR | Zbl

[24] Scheicher, K.; M. Thuswaldner, J. Canonical number systems, counting automata and fractals, Math. Proc. Cambridge Philos. Soc., Volume 133 (2002), pp. 163-182 | DOI | MR | Zbl

[25] Scheicher, K.; M. Thuswaldner, J.; Grabner, P. Neighbours of self-affine tiles in lattice tilings, Fractals in Graz 2001. Analysis, dynamics, geometry, stochastics (Proceedings of the conference), Birkhäuser, Graz, Austria, 2003, pp. 241-262 | MR | Zbl

[26] Strichartz, R.; Wang, Y. Geometry of self-affine tiles I, Indiana Univ. Math. J., Volume 23 (1999), pp. 1-23 | MR | Zbl

[27] Thuswaldner, J. M. Attractors of invertible expanding linear operators and number systems in 2 , Publ. Math. Debrecen, Volume 58 (2001), pp. 423-440 | MR | Zbl

[28] Vince, A. Digit tiling of euclidean space, Directions in Mathematical Quasicrystals, American Mathematical Society Providence, RI, 2000, pp. 329-370 | MR | Zbl

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