Beyond Poincar{\'e} recurrence: on a topological trichotomy of orbits and on the dependence of limit sets for flows on surfaces

This paper investigates the global structure of orbits for continuous flows on surfaces. One of our main results provides a topological trichotomy of orbits of flows with finitely many connected components of the singular point set on surfaces with finite genus and finite ends: each orbit is either recurrent, wandering, or corresponds to a circuit with wandering holonomy or a strictly limit circuit, two new concepts that we introduce in this paper. These concepts capture non-recurrent behaviors lying beyond the reach of classical tools such as the Poincar{'e} recurrence theorem and the Poincar{'e}-Bendixson theorem. Moreover, the non-recurrent non-wandering behaviors reveal the dependency between the existence of infinitely many non-degenerate $\omega$-limit sets and $\alpha$-limit sets, and provide a necessary and sufficient condition for a flow to be minimal on (possibly non-compact) surfaces.