Abstract
Let F be a closed surface. Let V be a smooth vector field on F inducing a flow t. A theorem due to Poincaré says that if t is area preserving, then almost every point of F is recurrent under t. In this paper we examine the limit sets of orbits, and prove in particular that if V has hyperbolic singularities, and t has no closed orbits and no saddle connections (or more generally, the graph of saddle connections does not separate the surface), then t has an orbit which is dense in the entire surface.
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