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Density of half-horocycles on geometrically infinite hyperbolic surfaces

Published online by Cambridge University Press:  16 May 2012

BARBARA SCHAPIRA
BARBARA SCHAPIRA*
Affiliation:
LAMFA, UMR CNRS 6140, Université Picardie Jules Verne, 33 rue St Leu, 80000 Amiens, France (email: barbara.schapira@u-picardie.fr)
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Abstract

On the unit tangent bundle of a hyperbolic surface, we study the density of positive orbits $(h^s v)_{s\ge 0}$ under the horocyclic flow. More precisely, given a full orbit $(h^sv)_{s\in {\mathbb R}}$, we prove that under a weak assumption on the vector $v$, both half-orbits $(h^sv)_{s\ge 0}$ and $(h^s v)_{s\le 0}$ are simultaneously dense or not in the non-wandering set $\mathcal {E}$of the horocyclic flow. We give also a counterexample to this result when this assumption is not satisfied.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1162 - 1177
Copyright
Copyright © 2012 Cambridge University Press 

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References

[C-D-P]Coornaert, M., Delzant, T. and Papadopoulos, A.. Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov (Lecture Notes in Mathematics, 1441). Springer, Berlin, 1990.Google Scholar
[Da]Dal’bo, F.. Topologie du feuilletage fortement stable. Ann. Inst. Fourier (Grenoble) 50(3) (2000), 981–993.CrossRefGoogle Scholar
[E1]Eberlein, P.. Geodesic flows on negatively curved manifolds I. Ann. of Math. (2) 95 (1972), 492–510.Google Scholar
[G-Ha]Ghys, É. and de la Harpe, P.. Sur les groupes hyperboliques d’après Mikhael Gromov (Berne 1988) (Progress Mathematics, 83). Birkhäuser, 1990, pp. 1–25.Google Scholar
[H]Hedlund, G. A.. Fuchsian groups and transitive horocycles. Duke Math. J. 2 (1936), 530–542.Google Scholar
[Sa]Sarig, O.. The horocycle flow and the Laplacian on hyperbolic surfaces of infinite genus. Geom. Funct. Anal. 19(6) (2010), 1757–1812.CrossRefGoogle Scholar
[Scha]Schapira, B.. Density and equidistribution of half-horocycles on a geometrically finite hyperbolic surface. J. Lond. Math. Soc. 84(3) (2011), 785–806.CrossRefGoogle Scholar