Let $D^-$ and $D^+$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ from $D^-$ to $D^+$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^-$ and $D^+$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic formula for the number of connected components of the domain of discontinuity of Kleinian groups as their diameter goes to $0$.Comment: 38 pages. Major revision (20 pages shorter). Equidistribution of equidistant surfaces and the case with potential and Gibbs measures moved to book in preparation [BPP
Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M , such as balls, horoballs, tubular neighbourhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much into) them. Several applications are given, including a definite improvement of the unclouding problem of our paper [47], the prescription of heights of geodesic lines in a finite volume such M , or of spiraling times around a closed geodesic in a closed such M . We also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvature property.53C22, 11J06, 52A55; 53D25
Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects. We prove analogs of the theorems of Dirichlet, Hall and Cusick in this context. As a consequence, we obtain Diophantine approximation results of elements of R, C or the Heisenberg group by quadratic irrational ones.
Let C be a locally convex subset of a negatively curved Riemannian manifold M . We define the skinning measure σ C on the outer unit normal bundle to C in M by pulling back the Patterson-Sullivan measures at infinity, and give a finiteness result of σ C , generalising the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to C equidistribute to the Bowen-Margulis measure m BM on T 1 M , assuming only m BM is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution. 1
Soit M une variété hyperbolique de volume fini, nous montrons que les hypersurfaces équidistantes à une sous-variété C de volume fini totalement géodésique s'équidistribuent dans M . Nous donnons un asymptotique précis du nombre de segments géodésiques de longueur au plus t, perpendiculaires communs à C et au bord d'un voisinage cuspidal de M . Nous en déduisons des résultats sur le comptage d'irrationnels quadratiques sur Q ou sur une extension quadratique imaginaire de Q, dans des orbites données des sous-groupes de congruence des groupes modulaires. AbstractLet M be a finite volume hyperbolic manifold, we show the equidistribution in M of the equidistant hypersurfaces to a finite volume totally geodesic submanifold C.We prove a precise asymptotic on the number of geodesic arcs of lengths at most t, that are perpendicular to C and to the boundary of a cuspidal neighbourhood of M . We deduce from it counting results of quadratic irrationals over Q or over imaginary quadratic extensions of Q, in given orbits of congruence subgroups of the modular groups. 1 IntroductionSoit M une variété hyperbolique (c'est-à-dire riemannienne lisse, complète et à courbure sectionnelle constante −1) connexe, de volume fini, de dimension n au moins 2. Soit C une sous-variété immergée de volume fini de M , connexe, de dimension k < n, totalement géodésique (par exemple une géodésique fermée). Soit Σ une composante connexe du fibré normal unitaire de C. Pour tout t > 0, notons Σ(t) = {exp tv : v ∈ Σ} l'hypersurface strictement convexe immergée de M , poussée au temps t le long des rayons géodésiques orthogonaux à C dans la direction Σ.Notre premier résultat dit que la moyenne riemannienne de Σ(t) s'équidistribue quand t tend vers +∞ vers la moyenne riemannienne de M . Théorème 1.1 Pour toute fonction ϕ continue à support compact dans M , nous avons lim t→+∞ M ϕ(x) d vol Σ(t) (x) = M ϕ(x) d vol M (x) . 1. Keywords : Equidistribution, counting, quadratic irrational, hyperbolic manifold, binary quadratic form, perpendicular geodesic. AMS codes : 37A45, 11R11, 53A35, 22F30, 20H10, 11H06, 53C40, 11E16 √ D 2 l'unité fondamentale de O, et R 0 = log ǫ 0 le régulateur de O. Notons n 0 = 2 si O contient une unité de norme −1, et 1 sinon.Corollaire 1.4 Quand t tend vers +∞, nous avonsNous renvoyons aux parties 5.2 et 5.4 pour des résultats plus généraux (en particulier concernant le comptage d'irrationnels quadratiques dans les orbites de sous-groupes de congruence de PSL 2 (Z)), et des résultats de comptage d'irrationnels quadratiques sur une extension quadratique imaginaire de Q dans des orbites par les groupes de Bianchi PSL 2 (O Q(i √ d) ), dont un cas très particulier est le suivant.3Soient X un espace métrique géodésique CAT(−1) propre, de bord à l'infini ∂ ∞ X, et Γ un groupe discret d'isométries de X, d'exposant critique noté δ = δ Γ . Rappelons que, pour tous x, y fixés dans X, si f Γ,x,y (t) = Card{γ ∈ Γ : d(x, γy) ≤ t}, alors δ Γ = lim sup t→+∞ 6 Le résultat en découle.Montrons maintenant qu'il existe c − > 0 tel quesont des o(e δ ′ ...
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