Critical exponent and Hausdorff dimension in pseudoRiemannian hyperbolic geometry
Abstract
The aim of this article is to understand the geometry of limit sets in pseudoRiemannian hyperbolic geometry. We focus on a class of subgroups of $\mathrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud and Kassel, called $\mathbb{H}^{p,q}$convex cocompact. We define a pseudoRiemannian 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 $\mathbb{H}^{2,1}=\mathrm{ADS}^3$ which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in $3$dimensional hyperbolic geometry.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.05512
 Bibcode:
 2016arXiv160605512G
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Dynamical Systems;
 Mathematics  Geometric Topology;
 Mathematics  Metric Geometry