Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds
Metric Geometry
2017-01-12 v3
Abstract
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 grows exponentially fast with 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 : R. Bowen's rigidity theorem of critical exponent, A. Sanders' isolation theorem and C. McMullen's examples lightening the behaviour of this exponent when the surfaces range over Teichm\"uller space.
Cite
@article{arxiv.1503.09067,
title = {Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds},
author = {Olivier Glorieux},
journal= {arXiv preprint arXiv:1503.09067},
year = {2017}
}