## 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