Topological transitivity, representability, and a generalization of Poincare-Bendixson theorem of surfaces flows and their applications

In this talk, we discuss some topics of flows on compact surfaces. First we present a necessary and sufficient condition for the existence of dense orbits of continuous flows on compact connected surfaces, which is a generalization of a necessary and sufficient condition on area-preserving flows obtained by H. Marzougui and G. Soler L'opez. Second, we generalize the Poincare-Bendixson theorem for a flow with arbitrarily many singular points on a compact surface. In fact, the omega-limit set of any non-closed orbit is one of the following exclusively: i) a nowhere dense subset of singular points; ii) a limit cycle; iii) a limit ”quasi-circuit”; iv) a locally dense Q-set; v) a ”quasi-Q-set” which is not locally dense. Third, we consider what class of flows on compact surfaces can be characterized by finite labeled graphs. In particular, a class of surface flows, up to topological conjugacy, which contains both the set of Morse Smale flows and the set of area-preserving flows with finite singular points is classified. In fact, although the set of topological equivalent classes of minimal flows on a torus is uncountable, we enumerate the set of topological equivalent classes of flows with non-degenerate singular points and with at most finitely many limit cycles but without non-closed recurrent orbits on a compact surface using finite labeled graphs. Finally, we introduce an implementation of the representation theorem for Hamiltonian flows on punctured disks and apply to analysis on fluid phenomena.