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arXiv · 1609.08505

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

Jaeyoo Choy·Hahng-Yun Chu
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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.

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Jaeyoo Choy, Hahng-Yun Chu. 2016-09-27. Polycycle omega-limit sets of flows on the compact Riemann surfaces and Eulerian path. https://arxiv.org/abs/1609.08505

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