Recherche
Publications
- Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry IMRN (Avec Daniel Monclair)
The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of PO(p,q+1) introduced by Danciger, Guéritaud and Kassel, called ℍp,q-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in ℍ2,1=ADS3 which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in 3-dimensional hyperbolic geometry. - Entropy of embedded surfaces in quasi-fuchsian manifolds Pacific J. of Math. We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, ℍ3, and on invariant disks embedded in ℍ3. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuchsian.
- Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds, Geometria Dedicata We propose a definition for the length of closed geodesics in a globally hyperbolic maximal compact (GHMC) Anti-De Sitter manifold. We then prove that the number of closed geodesics of length less than R grows exponentially fast with R and the exponential growth rate is related to the critical exponent associated to the two hyperbolic surfaces coming from Mess parametrization. We get an equivalent of three results for quasi-Fuchsian manifolds in the GHMC setting: Bowen’s rigidity theorem of critical exponent, Sanders’ isolation theorem and McMullen’s examples lightening the behaviour of this exponent when the surfaces range over Teichmüller space.
Pré-publications
- The embedding of the Teichmüller space in geodesic currents We prove that the embedding of the Teichmüller space in the space of geodesic currents is totally linearly independent. As a corollary we get a rigidity result for the marked length spectrum of anti-de Sitter quasi-Fuchsian manifolds.
- Hausdorff dimension of limit sets for projective Anosov representations avec Daniel Monclair et Nicolas Tholozan We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in P(ℝn)×P(ℝn∗) is bounded between two critical exponents associated respectively to a highest weight and a simple root.
- Regularity of limit sets of AdS quasi-Fuchsian groups, avec Daniel Monclair Limit sets of AdS-quasi-Fuchsian groups of PO(n,2) are always Lipschitz submanifolds. The aim of this article is to show that they are never 1, except for the case of Fuchsian groups. As a byproduct we show that AdS-quasi-Fuchsian groups that are not Fuchsian are Zariski dense in PO(n,2).
- Topological mixing of the Weyl chamber flow avec Thi Dang In this paper, we study topological properties of the right action by translation of the Weyl Chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.
- Critical exponent for geodesic currents For any geodesic current we associated a quasi-metric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.
Conferences
- Perspectives on convex projective geometry 24-28 Jun 2019 Sète
- Higher Teichmüller theory and related topics Pavia, 3-7 June 2019
- Dynamiques d'actions de groupes - Une conférence en l'honneur d'Yves Benoist 27 au 31 mai 2019
- Conference on Geometric Structures in Nice January 14-18, 2019.
- Dynamique in Aussois - 3/12/18-7/12/18
- Hyperbolic groups and their representations Ecole d'été CIMPA, Piriapolis, Uruguay. 31/03/2016-08/04/2016
- Geometry of Groups in Montevideo Montévidéo, Uruguay, 11/04/2016-15/04/2016
- Conférence en l'honneur de Francois Ledrappier Paris, France, 01/06/2016-03/06/2016
- Dynamics, Geometry and Number Theory IHP, Paris, france. 13/06/2016-17/06/2016
- Workshop for young researchers: groups acting on manifolds Teresopolis, Brazil. 20/06/2016-24/06/2016
- Geometric structures and representation variety University of Singapour, Singapour. 03/05/2017-05/05/2017
- Infinite measure dynamics Brest, France. 06/06/2017-09/06/2017