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The aim of this article is to understand the geometry of limit sets inpseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of$\mathrm{PO}(p,q+1)$ introduced by Danciger, Gu\'eritaud and Kassel, called$\mathbb{H}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue ofcritical exponent and Hausdorff dimension of the limit set. We show that theyare equal and bounded from above by the usual Hausdorff dimension of the limitset. We also prove a rigidity result in $\mathbb{H}^{2,1}=\mathrm{ADS}^3$ whichcan be understood as a Lorentzian version of a famous Theorem of R. Bowen in$3$-dimensional hyperbolic geometry.

Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry