Ergodic geometry for non-elementary rank one manifolds

Let $X$ be a Hadamard manifold, and $\mathrm{\Gamma }$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X\mathrm{/}\mathrm{\Gamma }$ to the behavior of the Poincar{'e} series of $\mathrm{\Gamma }$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan – well-known for manifolds of pinched negative curvature – to the framework of rank one orbifolds. Moreover, we derive some important properties for $\mathrm{\Gamma }$-invariant conformal densities supported on the geometric limit set of $\mathrm{\Gamma }$.