#### Volume 14, issue 1 (2010)

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Prescribing the behaviour of geodesics in negative curvature

### Jouni Parkkonen and Frédéric Paulin

Geometry & Topology 14 (2010) 277–392
##### Abstract

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 [Geom. Func. Anal. 15 (2005) 491–533], 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.

##### Keywords
geodesics, negative curvature, horoballs, Lagrange spectrum, Hall ray
##### Mathematical Subject Classification 2000
Primary: 53C22, 11J06, 52A55
Secondary: 53D25
##### Publication
Received: 1 June 2007
Revised: 21 July 2009
Accepted: 15 April 2009
Preview posted: 27 October 2009
Published: 2 January 2010
Proposed: Martin Bridson
Seconded: Walter Neumann, Jean-Pierre Otal
##### Authors
 Jouni Parkkonen Department of Mathematics and Statistics PO Box 35 40014 University of Jyväskylä Finland Frédéric Paulin Département de Mathématique et Applications UMR 8553 CNRS Ecole Normale Supérieure 45 rue d’Ulm 75230 PARIS Cedex 05 France