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Equidistribution of parabolic fixed points in the limit set of Kleinian groups. (English) Zbl 0947.30033

The purpose of this paper is to prove that the Patterson-Sullivan measure on the limit set of a geometrically finite Kleininan group can be realized as a limit of measures supported on an orbit of a given parabolic cusp. In proving this the author also obtains some asymptotic results on the distribution of such orbits. These estimates correspond to various statements in classical analytic number theory concerning the Euler function or Farey series. The methods used here are a combination of geometric methods (due to Stratmann and Velani, and to Kleinbock and Margulis) combined with ergodic results of Rudolph.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
37A25 Ergodicity, mixing, rates of mixing
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
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