## Prescribing the behaviour of geodesics in negative curvature.(English)Zbl 1191.53026

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.

### MSC:

 53C22 Geodesics in global differential geometry 52A55 Spherical and hyperbolic convexity 53D25 Geodesic flows in symplectic geometry and contact geometry 11J06 Markov and Lagrange spectra and generalizations

Zbl 1082.53035

OEIS
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