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

TL;DR

This paper classifies polycycle omega-limit sets of real analytic flows on compact surfaces, showing they are topologically equivalent to boundaries of cacti and describing the surface's decomposition, extending previous genus-zero results.

Contribution

It provides a topological classification of omega-limit sets for flows on surfaces, generalizing known results to higher genus surfaces and broader classes of flows.

Findings

Omega-limit sets are diffeomorphic to boundaries of cacti in S^2.

The surface decomposes into a connected sum involving the sphere with the cactus boundary.

Results apply to flows with finitely many singularities and certain corner cases.

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.