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

Pierre Arnoux 1 Valerie Berthe 2 Arnaud Hilion 3 Anne Siegel 4
2 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
4 SYMBIOSE - Biological systems and models, bioinformatics and sequences
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
Abstract : 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., so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers ({\it 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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Pierre Arnoux, Valerie Berthe, Arnaud Hilion, Anne Siegel. Fractal representation of the attractive lamination of an automorphism of the free group. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2006, 56 (7), pp.2161-2212. ⟨inria-00178799⟩

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