Prescribing the behaviour of geodesics in negative curvature

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.