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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: [email protected])
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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.

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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