On the Fundamental Group of self-affine plane Tiles

Luo, Jun; Thuswaldner, Jörg M.

doi:10.5802/aif.2247

On the Fundamental Group of self-affine plane Tiles
[Groupe fondamental de motifs auto-affines]

Luo, Jun ; Thuswaldner, Jörg M.

Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2493-2524.

Résumé
Abstract

Soient $A \in ℤ^{2} \times ℤ^{2}$ une matrice expansive, $𝒟 \subset ℤ^{2}$ un ensemble à $| det (A) |$ éléments et $𝒯$ l’ensemble défini par l’équation $A 𝒯 = 𝒯 + 𝒟$ . Si $𝒯$ a une mesure de Lebesgue sur $ℝ^{2}$ 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 $A \in ℤ^{2} \times ℤ^{2}$ be an expanding matrix, $𝒟 \subset ℤ^{2}$ a set with $| det (A) |$ elements and define $𝒯$ via the set equation $A 𝒯 = 𝒯 + 𝒟$ . 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.

MR 2290788 | Zbl 1119.52012 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2247
Classification : 52C20, 14F35, 11A63, 05B45
Mots clés : Motif, pavage, groupe fondamental, Systme de numration

@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},
     pages = {2493--2524},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {7},
     year = {2006},
     doi = {10.5802/aif.2247},
     mrnumber = {2290788},
     zbl = {1119.52012},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2247/}
}

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://archive.numdam.org/articles/10.5802/aif.2247/

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