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

Pierre Arnoux[1]; Valérie Berthé[2]; Arnaud Hilion[3]; Anne Siegel[4]

  • [1] IML-UMR 6206 163 avenue de Luminy Case 907 13288 Marseille cedex 9 (France)
  • [2] Université de Montpellier II LIRMM-CNRS UMR 5506 161 rue Ada 34392 Montpellier cedex 5 (France)
  • [3] Université Aix-Marseille III LATP Avenue de l’escadrille Normandie-Niémen Case A 13397 Marseille cedex 20 (France)
  • [4] IRISA-CNRS Campus de Beaulieu 35042 Rennes cedex (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 7, page 2161-2212
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Arnoux, Pierre, et al. "Fractal representation of the attractive lamination of an automorphism of the free group." Annales de l’institut Fourier 56.7 (2006): 2161-2212. <http://eudml.org/doc/10201>.

@article{Arnoux2006,
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., 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.},
affiliation = {IML-UMR 6206 163 avenue de Luminy Case 907 13288 Marseille cedex 9 (France); Université de Montpellier II LIRMM-CNRS UMR 5506 161 rue Ada 34392 Montpellier cedex 5 (France); Université Aix-Marseille III LATP Avenue de l’escadrille Normandie-Niémen Case A 13397 Marseille cedex 20 (France); IRISA-CNRS Campus de Beaulieu 35042 Rennes cedex (France)},
author = {Arnoux, Pierre, Berthé, Valérie, Hilion, Arnaud, Siegel, Anne},
journal = {Annales de l’institut Fourier},
keywords = {Free group automorphism; attractive lamination; substitution; symbolic $\quad $ dynamics; self-similarity; Pisot number; free group automorphisms; attractive laminations; substitutions; symbolic dynamics; Pisot numbers; free monoids; automorphisms with irreducible powers automorphisms; shift maps},
language = {eng},
number = {7},
pages = {2161-2212},
publisher = {Association des Annales de l’institut Fourier},
title = {Fractal representation of the attractive lamination of an automorphism of the free group},
url = {http://eudml.org/doc/10201},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Arnoux, Pierre
AU - Berthé, Valérie
AU - Hilion, Arnaud
AU - Siegel, Anne
TI - Fractal representation of the attractive lamination of an automorphism of the free group
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2161
EP - 2212
AB - 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.
LA - eng
KW - Free group automorphism; attractive lamination; substitution; symbolic $\quad $ dynamics; self-similarity; Pisot number; free group automorphisms; attractive laminations; substitutions; symbolic dynamics; Pisot numbers; free monoids; automorphisms with irreducible powers automorphisms; shift maps
UR - http://eudml.org/doc/10201
ER -

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