{
  "id": "0806.0159",
  "version": "v3",
  "published": "2008-06-01T17:16:37.000Z",
  "updated": "2015-12-24T13:22:46.000Z",
  "title": "Connected components of partition preserving diffeomorphisms",
  "authors": [
    "Sergiy Maksymenko"
  ],
  "comment": "22 pages, 6 figures. In the previous version only polynomials without multiple factors were considered. Now the result is proved for all homogeneous polynomials. Moreover some proofs are rewritten with more details",
  "journal": "Methods of Functional Analysis and Topology, vol. 15, no. 3 (2009) 264-279",
  "categories": [
    "math.DS",
    "math.GT"
  ],
  "abstract": "Let $f:\\mathbb{R}^2 \\to \\mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\\mathbb{R}^2 \\to \\mathbb{R}^2$ preserving $f$, i.e. $f \\circ h = f$. Denote by $S(f,r)$, $(0\\leq r \\leq \\infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $S(f,\\infty) = \\cdots = S(f,1)$ for all such $f$ and that $S(f,1) \\not= S(f,0)$ if and only if $f$ is a product of at least two distinct irreducible over $\\mathbb{R}$ quadratic forms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-11-30T12:08:20.000Z",
      "abstract": "Let f:R^2 --> R be a real homogeneous polynomial and S(f) be the group of diffeomorphisms h:R^2 --> R^2 preserving f, i.e. f \\circ h = f. Denote by S(f,r), (0\\leq r \\leq \\infty), the identity path component of S(f) with respect to the weak Whitney C^{r}_{W}-topology. We prove that S(f,\\infty) = ... = S(f,1) for all such f and that S(f,1) \\not= S(f,0) if and only if f is a product of at least two distinct irreducible over R quadratic forms.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-12-24T13:22:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S05"
    ],
    "keywords": [
      "partition preserving diffeomorphisms",
      "connected components",
      "identity path component",
      "real homogeneous polynomial",
      "weak whitney"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.0159M"
    }
  }
}