On the Fundamental Group of self-affine plane Tiles

Soient A ∈ Z 2 x Z 2 une matrice expansive, D c Z 2 un ensemble a |det(A)| elements et T l'ensemble defini par l'equation AT = T + D. Si T a une mesure de Lebesgue sur R 2 strictement superieure a zero, alors T est appele motif plan auto-affine. Cet article etablit certaines proprietes topologiques de T. Nous montrons que le groupe fondamental π 1 (T) de T est soit trivial, soit infini non denombrable, et nous donnons des criteres associes a chacun des deux cas. De plus, nous incluons une courte preuve de la propriete que l'adherence de chaque composante connexe de int(T) est un continuum localement connexe (nous demontrons meme ce resultat dans le cas plus general d'attracteurs plans d'IFS satisfaisant la condition de l'ensemble ouvert). Si π 1 (T) = 0, nous montrons meme que l'adherence de chaque composante de int(T) est homeomorphe au disque unite. Nous appliquons nos resultats a plusieurs examples de motifs etudies dans la litterature.

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