{
  "id": "math/0411612",
  "version": "v4",
  "published": "2004-11-27T14:46:50.000Z",
  "updated": "2015-12-24T14:07:32.000Z",
  "title": "Stabilizers and orbits of smooth functions",
  "authors": [
    "Sergey Maksymenko"
  ],
  "comment": "This is the second version of the paper. Now the cases $P=\\mathbb{R}$ and $P=S^1$ considered from unique point of view. In particular, this covers the results of the preprint http://xxx.lanl.gov/math.FA/0503734. Moreover, the role of the condition for a function to belong to its Jacobi ideal is explained",
  "journal": "Bulletin des Sciences Mathematiques, 130 (2006) 279-311",
  "doi": "10.1016/j.bulsci.2005.11.001",
  "categories": [
    "math.FA",
    "math.AT",
    "math.DS"
  ],
  "abstract": "Let $f:R^m \\to R$ be a smooth function such that $f(0)=0$. We give a condition on $f$ when for arbitrary preserving orientation diffeomorphism $\\phi:\\mathbb{R} \\to \\mathbb{R}$ such that $\\phi(0)=0$ the function $\\phi\\circ f$ is right equivalent to $f$, i.e. there exists a diffeomorphism $h:\\mathbb{R}^m \\to \\mathbb{R}^m$ such that $\\phi \\circ f = f \\circ h$ at $0\\in \\mathbb{R}^m$. The requirement is that $f$ belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated critical points, simple singularities and many others. We also globalize this result as follows. Let $M$ be a smooth compact manifold, $f:M \\to [0,1]$ a surjective smooth function, $\\mathrm{Diff}(M)$ the group of diffeomorphisms of $M$, and $\\mathrm{Diff}^{[0,1]}(\\mathbb{R})$ the group of diffeomorphisms of $\\mathbb{R}$ that have compact support and leave $[0,1]$ invariant. There are two natural right and left-right actions of $\\mathrm{Diff}(M)$ and $\\mathrm{Diff}(M) \\times \\mathrm{Diff}^{[0,1]}(\\mathbb{R})$ on $C^{\\infty}(M,R)$. Let $S_M(f)$, $S_{MR}(f)$, $O_{M}(f)$, and $O_{MR}(f)$ be the corresponding stabilizers and orbits of $f$ with respect to these actions. Under mild assumptions on $f$ we get the following homotopy equivalences $S_M(f) \\approx S_{MR}(f)$ and $O_M \\approx O_{MR}$. Similar results are obtained for smooth mappings $M \\to S^1$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-08-14T14:27:57.000Z",
      "abstract": "Let $f:R^m --> R$ be a smooth function such that $f(0)=0$. We give a condition on $f$ when for arbitrary preserving orientation diffeomorphism $\\phi:R --> R$ such that $\\phi(0)=0$ the function $\\phi\\circ f$ is right equivalent to $f$, i.e. there exists a diffeomorphism $h:R^m --> \\R^m$ such that $\\phi \\circ f = f \\circ h$ at $0\\in R^m$. The requirement is that $\\mrsfunc$ belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated critical points, simple singularities and many others. We also globalize this result as follows. Let $M$ be a smooth compact manifold, $f:M --> [0,1]$ a surjective smooth function, $Diff(M)$ the group of diffeomorphisms of $M$, and $Diff^{[0,1]}(R)$ the group of diffeomorphisms of $R^1$ that have compact support and leave $[0,1]$ invariant. There are two natural right and left-right actions of $Diff(M)$ and $Diff(M) \\times Diff^{[0,1]}(R)$ on $C^{\\infty}(M,R)$. Let $S_M(f)$, $S_{MR}(f)$, $O_{M}(f)$, and $O_{MR}(f)$ be the corresponding stabilizers and orbits of $f$ with respect to these actions. Under mild assumptions on $f$ we get the following homotopy equivalences $S_M(f) \\approx S_{MR}(f)$ and $O_M \\approx O_{MR}$. Similar results are obtained for smooth mappings $M-->S^1$.",
      "comment": "This is the second version of the paper. Now the cases P=R and P=S^1 considered from unique point of view. In particular, this covers the results of the preprint http://xxx.lanl.gov/math.FA/0503734 Moreover, the role of the condition for a function to belong to its Jacobi ideal is explained"
    },
    {
      "version": "v4",
      "updated": "2015-12-24T14:07:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C05",
      "57S05",
      "57R45"
    ],
    "keywords": [
      "stabilizers",
      "arbitrary preserving orientation diffeomorphism",
      "smooth compact manifold",
      "smooth mappings",
      "right equivalent"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11612M"
    }
  }
}