Polycycle omega-limit sets of flows on the compact Riemann surfaces and Eulerian path

Abstract

Let $(S,Φ)$ be a pair of a closed oriented surface and $Φ$ be a real analytic flow with finitely many singularities. Let $x$ be a point of $S$ with the polycycle $ω$-limit set $ω(x)$. In this paper we give topological classification of $ω(x)$. Our main theorem says that $ω(x)$ is diffeomorphic to the boundary of a cactus in the $2$-sphere $S^{2}$. Moreover $S$ is a connected sum of the above $S^{2}$ and a closed oriented surface along finitely many embedded circles which are disjoint from $ω(x)$. This gives a natural generalization to the higher genus of the main result of \cite{JL} for the genus $0$ case. Our result is further applicable to a larger class of surface flows, a compact oriented surface with corner and a $C^{1}$-flow with finitely many singularities locally diffeomorphic to an analytic flow.