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Closed geodesics and points in variable curvature. (Géodésiques fermées et pointes en courbure variable.) (French. Abridged English version) Zbl 0847.53027

Summary: One considers a family of free, discrete and geometrically finite groups of isometries of the hyperbolic ball \(\mathbb B^n\) which contains parabolic transformations. Let \(\Gamma\) be a group in this family, fix a metric \(\overline g\) on \(M = \mathbb B^n/ \Gamma\) which is a variation of the hyperbolic metric and let \(\delta\) be the critical exponent of the Poincaré series associated with \(\Gamma\) and \(\overline g\). We prove that the number of closed geodesics on \((M, \overline g)\) whose length is not larger than \(a\) is equivalent to \({e^{a \delta} \over a \delta}\) as \(a\) goes to \(+ \infty\).

MSC:

53C22 Geodesics in global differential geometry
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