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Masse des pointes, temps de retour et enroulements en courbure négative. (Mass of cusps, time of return and windings in negative curvature). (French) Zbl 1035.37026
The Liouville measure on hyperbolic finite volume constant curvature manifolds is replaced by the Bowen-Margulis-Sullivan-Patterson measure on infinite volume or on pinched variable curvature geometrically finite manifolds. This measure \(m\) is invariant under the geodesic flow and is singular. Let \(\Gamma\) be a geometrically finite discrete group of isometries of a Hadamard manifold \(X\) and let \({\mathcal P}\) be a cusp of the orbifold associated to \({\mathcal M}:= \Gamma \backslash X\).
The goal of the article is to estimate the asymptotic mass of a small horocyclic neighborhood of the cusp \({\mathcal P}\) and to deduce the asymptotic, under \(m\), of the return times of the geodesic flow on a small horocycle of the cusp \({\mathcal P}\), via the mass of the Palm measure induced on \({\mathcal P}\) by \(m\). The computations are achieved in the context of geometrically finite orbifolds of pinched negative curvature, whith an additional hypothesis on the growth of the holonomy group of \({\mathcal P}\). The computations are specialized to the case of constant curvature, namely to hyperbolic real or complex orbifolds. The article under review is in continuation with a previous work of the authors jointly with Y. Le Jan, in the context of constant real curvature [Probab. Theory Relat. Fields {119}, 213–255 (2001; Zbl 1172.37311)].

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37A50 Dynamical systems and their relations with probability theory and stochastic processes
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