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On the orbit function of discrete groups in negative curvature. (Sur la fonction orbitale des groupes discrets en courbure négative.) (French) Zbl 1008.20040

Let \(\Gamma\) be a non-elementary discrete group of isometries of a metric space \(X\) and \(O\) be a fixed point in \(X\). Denote by \(N_\Gamma(R)\) the number of points of the orbit point \(O\) with respect to \(\Gamma\), which belong to the ball with center \(O\) and radius \(R\). Set \(\delta(\Gamma)=\overline\lim_{R\to\infty}(\log N_\Gamma(R)/R)\). Theorem. \(\log N_\Gamma(R)/R\to\delta(\Gamma)\) as \(R\to\infty\).

MSC:

20F69 Asymptotic properties of groups
37C35 Orbit growth in dynamical systems
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
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