{
  "id": "1609.08505",
  "version": "v1",
  "published": "2016-09-27T15:49:35.000Z",
  "updated": "2016-09-27T15:49:35.000Z",
  "title": "Polycycle omega-limit sets of flows on the compact Riemann surfaces and Eulerian path",
  "authors": [
    "Jaeyoo Choy",
    "Hahng-Yun Chu"
  ],
  "categories": [
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-27T15:49:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C10",
      "37C70",
      "05C10"
    ],
    "keywords": [
      "compact riemann surfaces",
      "polycycle omega-limit sets",
      "eulerian path",
      "closed oriented surface",
      "real analytic flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}