Convergence and Counting in Infinite Measure

Abstract

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.

Transition de phase de fonctions orbitales pour des groupes de Schottky en courbure négative

“…L'existence de telles fonctions T nécessite quelques précautions sur les recollements à effectuer ; nous renvoyons le lecteur à [6,14] pour les détails.…”

“…Dans ce paragraphe, nous expliquons comment apparaissent les différentes situations énumérées dans l'énoncé du Théorème 1.1 ainsi que la façon dont chacune d'entre elles est traitée ; nous renvoyons pour les détails des démonstrations aux articles [6,15,17].…”

“…Combinant plusieurs travaux récents autour des groupes de Schottky dit exotiques [6,15,17], nous pouvons énoncer le théorème suivant qui précise l'influence du paramètre a sur le comportement asymptotique de la fonction orbitale N a,Γ . (1) Cas sur-critique a > a * : Le groupe Γ est non exotique (i.e.…”

Entropy rigidity of negatively curved manifolds of finite volume

“…Let us also point out that Knieper's approach in [22] does not allow to deduce the above characterization in the finite volume case. Although G. Knieper's horospherical measure µ H can be perfectly defined in this context (following §3 of [22]), it can easily be infinite, as well as the Bowen-Margulis measure µ BM : given a finite volume surfacē X with convergent fundamental group Γ and with a cusp whose metric, in horospherical coordinates, writes as A 2 (t)dx 2 + dt 2 , it is not difficult to show that µ H is infinite as soon as [14]). Therefore, all formulas in [22] relating Ent top (X) to the trace of the second fundamental form of unstable horospheres need to be justified in some other way 5 .…”

Asymptotic geometry of negatively curved manifolds of finite volume

“…Convergent, exotic lattices are constructed by the authors in [16]; also, one can find in [16] some original counting results for the orbital function of in infinite Bowen-Margulis measure, more precise than (b).…”