Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds

@article{Glorieux2015CountingCG,
title={Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds},
author={Olivier Glorieux},
journal={Geometriae Dedicata},
year={2015},
volume={188},
pages={63-101}
}
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…
8 Citations
Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry
• Mathematics
International Mathematics Research Notices
• 2019
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 $\textrm{PO}(p,q+1)$ introduced by Danciger,
Correlation of the renormalized Hilbert length for convex projective surfaces
• Mathematics
• 2020
In this paper we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same
Degeneration of globally hyperbolic maximal anti-de Sitter structures along rays.
Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle of the Teichm\"uller space of $S$, we study how some geometric
High energy harmonic maps and degeneration of minimal surfaces
Let $S$ be a closed surface of genus $g \geq 2$ and let $\rho$ be a maximal $\mathrm{PSL}(2, \mathbb{R}) \times \mathrm{PSL}(2, \mathbb{R})$ surface group representation. By a result of Schoen, there
The embedding of the space of negatively curved surfaces in geodesic currents.
We prove by an algebraic method that the embedding of the Teichmuller space in the space of geodesic currents is totally linearly independent. We prove a similar result for all negatively curved
The embedding of the Teichm\"uller space in geodesic currents
We prove that the embedding of the Teichm\"uller 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
Entropy degeneration of globally hyperbolic maximal compact anti-de Sitter structures
Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle of the Teichm\"uller space of $S$, we study how some geometric
Critical exponent and Hausdorff dimension for quasi-Fuchsian AdS manifolds
• Mathematics
• 2016
The aim of this article is to understand the geometry of limit sets in Anti-de Sitter space. We focus on a particular type of subgroups of $\mathrm{SO}(2,n)$ called quasi-Fuchsian groups (which are

References

SHOWING 1-10 OF 38 REFERENCES
Métriques prescrites sur le bord du coeur convexe d'une variété anti-de Sitter globalement hyperbolique maximale compacte de dimension trois
Le but de cette these est d'apporter une reponse partielle positive a l'une des conjectures de Geoffrey Mess, datant des annees 90, sur la geometrie du bord du coeur convexe d'une variete anti-de
Critical exponent and Hausdorff dimension for quasi-Fuchsian AdS manifolds
• Mathematics
• 2016
The aim of this article is to understand the geometry of limit sets in Anti-de Sitter space. We focus on a particular type of subgroups of $\mathrm{SO}(2,n)$ called quasi-Fuchsian groups (which are
Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups
AbstractLet X be a globally symmetric space of noncompact type, $$G = \textrm{Isom}^o (X)$$ and $$\Gamma \subset G$$ a discrete subgroup. Introducing an appropriate notion of Hausdorff
A VARIATIONAL PRINCIPLE FOR THE PRESSURE OF CONTINUOUS TRANSFORMATIONS.
Introduction. Ruelle ([17]) has defined the concept of pressure for a continuous Zn action on a compact metric space and proved a variational principle when the action is expansive and satisfies the