On the Fundamental Group of self-affine plane Tiles

Soient $A\in {\mathbb{Z}}^2\times {\mathbb{Z}}^2$ une matrice expansive, $𝒟\subset {\mathbb{Z}}^2$ un ensemble à $|det(A\left)\right|$ éléments et $𝒯$ l’ensemble défini par l’équation $A𝒯=𝒯+𝒟$. 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 ${\pi }_1\left(𝒯\right)$ 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 $\mathrm{int}\left(𝒯\right)$ 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 ${\pi }_1\left(𝒯\right)=0$, nous montrons même que l’adhérence de chaque composante de $\mathrm{int}\left(𝒯\right)$ est homéomorphe au disque unité.

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

Let $A\in {\mathbb{Z}}^2\times {\mathbb{Z}}^2$ be an expanding matrix, $𝒟\subset {\mathbb{Z}}^2$ a set with $|det(A\left)\right|$ 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 ${\pi }_1\left(𝒯\right)$ of $𝒯$ is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of ${\pi }_1\left(𝒯\right)$. Furthermore, we give a short proof of the fact that the closure of each component of $\mathrm{int}\left(𝒯\right)$ 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 ${\pi }_1\left(𝒯\right)=0$ we even show that the closure of each component of $\mathrm{int}\left(𝒯\right)$ is homeomorphic to a closed disk.

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

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