Prescribing the behaviour of geodesics in negative curvature (Q1035309)

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 an earlier paper of the authors [Geom. Funct. Anal. 15, No.~2, 491--533 (2005; Zbl 1082.53035)], the prescription of heights of geodesic lines in such an \(M\) of finite volume, or of spiraling times around a closed geodesic in a closed such \(M\). They 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. The work has deep connections with Diophantine approximation problems.