Diophantine approximation for negatively curved manifolds |
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Authors: | Sa'ar Hersonsky Frédéric Paulin |
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Institution: | (1) Caltech, Department of Mathematics, Pasadena CA 91125, USA (e-mail: saar@cco.caltech.edu) , US;(2) Laboratoire de Mathématiques, UMR 8628 CNRS (Bat. 425), Equipe de Topologie et Dynamique, Université Paris-Sud, 91405 Orsay Cedex, France (e-mail: Frederic.Paulin@math.u-psud.fr) , FR |
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Abstract: | Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of
Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic
lines starting from a given cusp by ones returning to it. We define a new invariant for M, theHurwitz constant of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of
constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp
neighborhood. Using the cut locus of the cusp, we define an explicit approximation sequence for a geodesic line starting from
the cusp and explore its properties. We prove that the modular once-punctured hyperbolic torus has the minimum Hurwitz constant
in its moduli space.
Received: 24 October 2000; in final form: 10 November 2001 / Published online: 17 June 2002 |
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Keywords: | Mathematics Subject Classification (2002): 53 C 22 11 J 06 30 F 40 11 J 70 |
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