{
  "id": "0907.0354",
  "version": "v2",
  "published": "2009-07-02T12:07:08.000Z",
  "updated": "2015-12-24T14:26:14.000Z",
  "title": "Reparametrizations of vector fields and their shift maps",
  "authors": [
    "Sergiy Maksymenko"
  ],
  "comment": "7 pages, no figures",
  "journal": "Topological problems and related questions. Proceedings of Intsitute of Mathematics of Ukrainian NAS, vol. 3, no. 3 (2006) 269-308",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $M$ be a smooth manifold, $F$ be a smooth vector field on $M$, and $F_t$ be the local flow of $F$. Denote by $Sh(F)$ the space of smooth maps $h:M\\to M$ of the following form: $h(x) = F_{f(x)}(x)$, where $f:M\\to\\mathbb{R}$ runs over all smooth functions on $M$ which can be substituted into the flow $F_t$ instead of time. This space often coincides with the identity component of the group of diffeomorphisms preserving orbits of $F$. In this note it is shown that $Sh(F)$ is not changed under reparametrizations and pushforwards of $F$. As an application it is proved that a vector field $F$ without non-closed orbits can be reparametrized to induce a circle action on $M$ if and only if there exists a smooth function $f:M\\to (0,+\\infty)$ such that for each non-singular point $x$ of $M$, the value $f(x)$ is an integer multiple of the period of $x$ with respect to $F$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-02T12:07:08.000Z",
      "abstract": "Let M be a smooth manifold, F be a smooth vector field on M, and F_t be the local flow of F. Denote by Sh(F) the space of smooth maps h:M-->M of the following form: h(x) = F_{f(x)}(x), where f:M-->R runs over all smooth functions on M which can be substituted into the flow F_t instead of time. This space often coincides with the identity component of the group of diffeomorphisms preserving orbits of F. In this note it is shown that Sh(F) is not changed under reparametrizations and pushforwards of F. As an application it is proved that a vector field F without non-closed orbits can be reparametrized to induce a circle action on M if and only if there exists a smooth function f:M-->(0,+\\infty) such that for each non-singular point x of M, the value f(x) is an integer multiple of the period of x with respect to F.",
      "comment": "To appear in the Proceeding of Institute of Mathematics of NAS of Ukraine. 7 pages, no figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-24T14:26:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C10",
      "37C27",
      "37C55"
    ],
    "keywords": [
      "shift maps",
      "reparametrizations",
      "smooth function",
      "smooth vector field",
      "smooth manifold"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0354M"
    }
  }
}