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On the distribution of orbits of geometrically finite hyperbolic groups on the boundary

Published online by Cambridge University Press:  05 April 2011

SEONHEE LIM  and
HEE OH
SEONHEE LIM
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-747, Korea (email: slim@snu.ac.kr)
HEE OH
Affiliation:
Mathematics Department, Brown University, Providence, RI, USA (email: heeoh@math.brown.edu) Korea Institute for Advanced Study, Seoul, Korea
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Abstract

We investigate the distribution of orbits of a non-elementary discrete hyperbolic subgroup Γ acting on ℍn and its geometric boundary ∂∞(ℍn). In particular, we show that if Γ admits a finite Bowen–Margulis–Sullivan measure (for instance, if Γ is geometrically finite), then every Γ-orbit in ∂∞(ℍn) is equidistributed with respect to the Patterson–Sullivan measure supported on the limit set Λ(Γ). The appendix by Maucourant is the extension of a part of his PhD thesis where he obtains the same result as a simple application of Roblin’s theorem. Our approach is via establishing the equidistribution of solvable flows on the unit tangent bundle of Γ∖ℍn, which is of independent interest.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 173 - 189
Copyright
Copyright © Cambridge University Press 2011

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