Institut de Mathématiques de Marseille, UMR 7373

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HILION Arnaud

Fonction : maître de conférences

Organisme : Aix Marseille Université
Maître de Conférences
(HDR)

Equipe: Groupes, Dynamique, Arithmétique, Combinatoire (GDAC)

arnaud.hilion _AT_

univ-amu.fr

0413551307

Bureau: 325, Etg: 3, Bât: CMI - Site : Chateau-Gombert

Page Web perso

Domaines de Recherche:

HDR

Arbres, laminations et groupes libres

Auteur(s): Hilion A.

(H.D.R.) , 2011

Ref HAL: tel-01218369_v1
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Résumé:

Ce document donne un aperçu des travaux récents de l'auteur. On rappelle notamment la définition de la lamination duale à un arbre réel dans le bord de l'outre espace. On explique comment utiliser les systèmes d'isométries partielles pour étudier cette lamination.

Dernieres productions scientifiques :

Topological substitutions and Rauzy fractals

Auteur(s): Bedaride N., Hilion A., Jolivet Timo

(Document sans référence bibliographique) 2016-03-09

Ref HAL: hal-01322691_v1
Ref Arxiv: 1603.02790
Ref. & Cit.: NASA ADS
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Résumé:

We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal.

Commentaires: 27 pages, 13 figures. arXiv admin note: text overlap with arXiv:1101.3905

Indecomposable $F_N$-trees and minimal laminations

Auteur(s): Coulbois T., Hilion A., Reynolds Patrick

(Article) Publié: Groups Geometry And Dynamics, vol. 9 p.567–597 (2015)

Ref HAL: hal-01218334_v1
Ref Arxiv: 1110.3506
Ref. & Cit.: NASA ADS
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Résumé:

We extend the techniques of [CH] to build an inductive procedure for studying actions in the boundary of the Culler-Vogtmann Outer Space, the main novelty being an adaptation of the classical Rauzy-Veech induction for studying actions of surface type. As an application, we prove that a tree in the boundary of Outer space is free and indecomposableif and only if its dual lamination is minimal up to diagonal leaves. Our main result generalizes [BFH97, Proposition 1.8] as well as the main result of [KL11].

Invariant measures for train track towers

Auteur(s): Bedaride N., Hilion A., Lustig M.

(Document sans référence bibliographique)

Ref HAL: hal-01218333_v1
Ref Arxiv: 1503.08000
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Résumé:

In this paper we present a combinatorial machinery, consisting of a graph tower Γ← and a weight towers ω← on Γ←, which allow us to efficiently describe invariant measures μ=μω← on rather general discrete dynamicals system over a finite alphabet. A train track map f:Γ→Γ defines canonically a stationary such graph tower Γf←. In the most important two special cases the measure μ specializes to a (typically ergodic) invariant measure on a substitution subshift, or to a projectively f∗-invariant current on the free group π1Γ. Our main result establishes a 1-1 correspondence between such measures μ and the non-negative eigenvectors of the incidence ("transition") matrix of f.

Ergodic currents dual to a real tree

Auteur(s): Coulbois T., Hilion A.

(Document sans référence bibliographique) 2014-05-22

Ref HAL: hal-01066575_v1
Ref Arxiv: 1302.3766
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Résumé:

Let $T$ be an $\R$-tree in the boundary of Outer space with dense orbits. When the free group $\FN$ acts freely on $T$, we prove that the number of projective classes of ergodic currents dual to $T$ is bounded above by $3N-5$. We combine Rips induction and splitting induction to define unfolding induction for such an $\R$-tree $T$. Given a current $\mu$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards $\mu$. We also give a unique ergodicity criterion.

Commentaires: 14 pages, minor corrections from previous version

The hyperbolicity of the sphere complex via surgery paths

Auteur(s): Hilion Arnaud, Horbez Camille

(Document sans référence bibliographique) 2012-10-23

Ref HAL: hal-00746622_v1
Ref Arxiv: 1210.6183
Ref. & Cit.: NASA ADS
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Résumé:

Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's sphere complex (another model for the free splitting complex), instead of Handel and Mosher's fold paths. As a byproduct, we get that surgery paths are unparameterized quasi-geodesics in the sphere complex.

Commentaires: 23 pages, 11 figures

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