On the half-trajectories of horocyclic flow on geometrically infinite hyperbolic surfaces

Abstract

We study the density of half-horocycles or half-orbits of the horocyclic flow on the unit tangent bundle of geometrically infinite hyperbolic surfaces. In [10] Schapira proved that under some assumptions, both half-horocycles (h s v) s≥0 and (h s v) s≤0 are simultaneously dense or not in the nonwandering set of the horocyclic flow. We construct a counterexample, when the assumptions are not satisfied, on a surface of first kind, answering a question of Schapira [10].