Fractal representation of the attractive lamination of an automorphism of the free group

Arnoux, Pierre; Berthé, Valérie; Hilion, Arnaud; Siegel, Anne

Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 2161-2212.

Nous étendons aux automorphismes de groupes libres certains résultats et constructions associés aux morphismes de monoïdes libres, autrement appelés substitutions. Nous construisons une représentation géométrique de la lamination attractive d’une classe d’automorphismes du groupe libre (plus précisément, les automorphismes irréductibles et dont les puissances sont irréductibles) dans le cas où le coefficient de dilatation de l’automorphisme est un nombre de Pisot unitaire. On montre que, dans ce cas, l’application de décalage sur la lamination symbolique attractive est isomorphe en mesure à un échange de domaines sur un ensemble autosimilaire compact. Cet ensemble est appelé tuile centrale de l’automorphisme ; sa construction s’inspire des fractals de Rauzy associés à une substitution primitive Pisot. La tuile centrale admet des symétries liées à l’inversion dans le groupe libre. On conjecture dans le cas général que la tuile centrale est un domaine fondamental pour une translation sur un groupe compact.

In this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, i.e., the so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers (iwip) automorphisms) is given in the case where the dilation coefficient of the automorphism is a unit Pisot number. The shift map associated with the attractive symbolic lamination is, in this case, proved to be measure-theoretically isomorphic to a domain exchange on a self-similar Euclidean compact set. This set is called the central tile of the automorphism, and is inspired by Rauzy fractals associated with Pisot primitive substitutions. The central tile admits some specific symmetries, and is conjectured under the Pisot hypothesis to be a fundamental domain for a toral translation.

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