Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T03:53:32.725Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
HEE OH
Affiliation:
Mathematics Department, Brown University, Providence, RI, USA (email: [email protected]) Korea Institute for Advanced Study, Seoul, Korea
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ballmann, W.. Lectures on spaces of nonpositive curvature. DMV Seminar, Band 25. Birkhäuser, Basel, 1995.Google Scholar
[2]Bartels, H.-J.. Nichteuklidische Gitterpunktprobleme und Gleichverteilung in linearen algebraischen Gruppen. Comment. Math. Helv. 57(1) (1982), 158–172.CrossRefGoogle Scholar
[3]Bowen, R.. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
[4]Burger, M.. Horocycle flow on geometrically finite surfaces. Duke Math. J. 61(3) (1990), 779–803.CrossRefGoogle Scholar
[5]Duke, W., Rudnick, Z. and Sarnak, P.. Density of integer points on affine homogeneous varieties. Duke Math. J. 71(1) (1993), 143–179.CrossRefGoogle Scholar
[6]Eskin, A. and McMullen, C. T.. Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71(1) (1993), 181–209.CrossRefGoogle Scholar
[7]Gorodnik, A. and Oh, H.. Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary. Duke Math. J. 139(3) (2007), 483–525.Google Scholar
[8]Gorodnik, A., Shah, N. and Oh, H.. Strong wavefront lemma and counting lattice points in sectors. Israel J. Math. 176 (2010), 419–444.CrossRefGoogle Scholar
[9]Gorodnik, A. and Maucourant, F.. Proximality and equidistribution on the Furstenberg boundary. Geom. Dedicata 113 (2005), 197–213.Google Scholar
[10]Lax, P. D. and Phillips, R. S.. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal. 46(3) (1982), 280–350.CrossRefGoogle Scholar
[11]Margulis, G.. On Some Aspects of the Theory of Anosov Systems (Springer Monographs in Mathematics). Springer, Berlin, 2004, with a survey by Richard Sharp: Periodic orbits of hyperbolic flows, translated from the Russian by Valentina Vladimirovna Szulikowska.Google Scholar
[12]Maucourant, F.. Approximation diophantienne, dynamique des chambres de Weyl et répartition d’orbites de réseaux. PhD Thesis, Université Lille I, http://tel.archives-ouvertes.fr/tel-00158036/fr.Google Scholar
[13]Oh, H. and Shah, N.. Equidistribution and counting for orbits of geometrically finite hyperbolic groups. Preprint, arXiv:1001.2096.Google Scholar
[14]Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.CrossRefGoogle Scholar
[15]Peigné, M.. On the Patterson–Sullivan measure of some discrete groups of isometries. Israel J. Math. 133 (2003), 77–88.CrossRefGoogle Scholar
[16]Roblin, T.. Sur l’ergodicité rationnelle et les propriétés ergodiques du flot géodésique dans les variétés hyperboliques. Ergod. Th. & Dynam. Sys. 20 (2000), 1785–1819.CrossRefGoogle Scholar
[17]Roblin, T.. Ergodicité et équidistribution en courbure négative. Mém. Soc. Math. Fr. (N.S.) 95 (2003), vi+96.Google Scholar
[18]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171–202.CrossRefGoogle Scholar
[19]Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153(3–4) (1984), 259–277.CrossRefGoogle Scholar