{
  "id": "2105.13110",
  "version": "v1",
  "published": "2021-05-27T13:14:19.000Z",
  "updated": "2021-05-27T13:14:19.000Z",
  "title": "Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics",
  "authors": [
    "Olga Pochinka",
    "Danila Shubin"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of $n$-manifolds with two limit cycles is given. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincar\\'{e}-Hopf theorem Euler characteristic of closed manifold $M^n$ which admits the considered flows is equal to zero. Only torus and Klein bottle can be ambient manifolds for such flows in case of $n=2$. Authors established that there exist exactly two classes of topological equivalence of such flows of torus and three of the Klein bottle. There are no constraints for odd-dimensional manifolds which follow from the fact that Euler characteristic is zero. However, it is known that orientable $3$-manifold admits a flow of considered class if and only if it is a lens space. In this paper, it is proved that up to topological equivalence each of $\\mathbb S^3$ and $\\mathbb RP^3$ admit one such flow and other lens spaces two flows each. Also, it is shown that the only non-orientable $n$-manifold (for $n>2$), which admits considered flows is the twisted I-bundle over $(n-1)$-sphere. Moreover, there are exactly two classes of topological equivalence of such flows. Among orientable $n$-manifolds only the product of $(n-1)$-sphere and the circle can be ambient manifold of a considered flow and the flows are split into two classes of topological equivalence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-27T13:14:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonsingular morse-smale flows",
      "attractor-repeller dynamics",
      "topological equivalence",
      "klein bottle",
      "ambient manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}