On the Fundamental Group of self-affine plane Tiles
Annales de l'Institut Fourier, Volume 56 (2006) no. 7, p. 2493-2524

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.

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.

DOI : https://doi.org/10.5802/aif.2247
Classification:  52C20,  14F35,  11A63,  05B45
Keywords: Tile, tiling, fundamental group, number system
@article{AIF_2006__56_7_2493_0,
     author = {Luo, Jun and Thuswaldner, J\"org M.},
     title = {On the Fundamental Group of self-affine plane Tiles},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {7},
     year = {2006},
     pages = {2493-2524},
     doi = {10.5802/aif.2247},
     mrnumber = {2290788},
     zbl = {1119.52012},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_7_2493_0}
}
Luo, Jun; Thuswaldner, Jörg M. On the Fundamental Group of self-affine plane Tiles. Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2493-2524. doi : 10.5802/aif.2247. http://www.numdam.org/item/AIF_2006__56_7_2493_0/

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