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

Abstract

Let $(S,\Phi )$ be a pair of a closed oriented surface and $\Phi$ be a real analytic flow with finitely many singularities. Let $x$ be a point of $S$ with the polycycle $\omega$-limit set $\omega (x)$. In this paper we give topological classification of $\omega (x)$. Our main theorem says that $\omega (x)$ 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 $\omega (x)$. 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.