Given a family of (almost) disjoint strictly convex subsets of a completenegatively curved Riemannian manifold M, such as balls, horoballs, tubularneighborhoods of totally geodesic submanifolds, etc, the aim of this paper isto construct geodesic rays or lines in M which have exactly once an exactlyprescribed (big enough) penetration in one of them, and otherwise avoid (or donot enter too much in) them. Several applications are given, including adefinite improvement of the unclouding problem of [PP1], the prescription ofheights of geodesic lines in a finite volume such M, or of spiraling timesaround a closed geodesic in a closed such M. We also prove that the Hall rayphenomenon described by Hall in special arithmetic situations and bySchmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvatureproperty.
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