Let Γ be a Kleinian group. i.e. a discrete, torsionless group of isometries of a Hadamard space X of negative, pinched curvature −B 2 ≤ K X ≤ −A 2 < 0, with quotientX = Γ\X. This paper is concerned with two mutually related problems :1) The description of the distribution of the orbits of Γ on X, namely of fine asymptotic properties of the orbital function :This has been the subject of many investigations since Margulis' [27] (see Roblin's book [33] and Babillot's report on [1] for a clear overview). The motivations to understand the behavior of the orbital function are numerous : for instance, a simple but important invariant is its exponential growth ratewhich has a major dynamical significance, since it coincides with the topological entropy of the geodesic flow whenX is compact, and is related to many interesting rigidity results and characterization of locally symmetric spaces, cp.[23], [9], [6].2) The pointwise behavior of the Poincaré series associated with Γ :x, y ∈ X for and s = δ Γ , which coincides with its exponent of convergence. The group Γ is said to be convergent if P Γ (x, y, δ Γ ) < ∞, and divergent otherwise. Divergence can also be understood in terms of dynamics as, by Hopf-Tsuju-Sullivan theorem, it is equivalent to ergodicity and total conservativity of the geodesic flow with respect to the Bowen-Margulis measure on the unit tangent bundle UX (see again [33] for a complete account).The regularity of the asymptotic behavior of v Γ , in full generality, is well expressed in Roblin's results, which trace back to Margulis' work in the compact case : [33]). Let X be a Hadamard manifold with pinched negative curvature and Γ a non elementary, discrete subgroup of isometries of X with non-arithmetic length spectrum 1 : (i) the exponential growth rate δ Γ is a true limit ;where (µ x ) x∈X denotes the family of Patterson conformal densities of Γ, and m Γ the Bowen-Margulis measure on UX.Here, f ∼ g means that f (t)/g(t) → 1 when t → ∞ ; for c ≥ 1, we will write f c ≍ g when 1 c ≤ f (t)/g(t) ≤ c for t ≫ 0 (or simply f ≍g when the constant c is not specified). The best asymptotic regularity to be expected is the existence of an equivalent, as in (ii) ; an explicit computation of the second term in the asymptotic development of v Γ is a difficult question for locally symmetric spaces (and almost a hopeless question in the general Riemannian setting).1. This means that the set L(X) = {ℓ(γ) ; γ ∈ Γ} of lengths of all closed geodesics ofX = Γ\X is not contained in a discrete subgroup of R.

International audienceWe study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices. These objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold

Let X be a Hadamard manifold, and Γ ⊂ Is(X) a non-elementary discrete subgroup 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/Γ to the behavior of the Poincaré series of Γ. 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 Γ-invariant conformal densities supported on the geometric limit set of Γ.

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We study the asymptotic behaviour of simply connected, Riemannian manifolds X of strictly negative curvature admitting a non-uniform lattice. If the quotient manifoldX = \X is asymptotically 1/4-pinched, we prove that is divergent and UX has finite Bowen-Margulis measure (which is then ergodic and totally conservative with respect to the geodesic flow); moreover, we show that, in this case, the volume growth of balls B(x, R) in X is asymptotically equivalent to a purely exponential function c(x)e R , where is the topological entropy of the geodesic flow ofX. This generalizes Margulis' celebrated theorem to negatively curved spaces of finite volume. In contrast, we exhibit examples of lattices in negatively curved spaces X (not asymptotically 1/4-pinched) where, depending on the critical exponent of the parabolic subgroups and on the finiteness of the Bowen-Margulis measure, the growth function is exponential, lower-exponential or even upper-exponential.