{
  "id": "2006.08154",
  "version": "v1",
  "published": "2020-06-15T06:16:59.000Z",
  "updated": "2020-06-15T06:16:59.000Z",
  "title": "On symmetries of iterates of rational functions",
  "authors": [
    "Fedor Pakovich"
  ],
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "Let $A$ be a rational function of degree $n\\geq 2$. We denote by $ G(A)$ the group of M\\\"obius transformations $\\sigma$ such that $ A\\circ \\sigma=\\nu \\circ A$ for some M\\\"obius transformations $\\nu$, and by $\\Sigma(A)$ and ${\\rm Aut}(A)$ subgroups of $ G(A)$, consisting of M\\\"obius transformations $\\sigma$ such that $ A\\circ \\sigma= A$ and $ A\\circ \\sigma= \\sigma \\circ A$, correspondingly. We show that, unless $A$ has a very special form, the orders of the groups $ G(A^{\\circ k})$, $k\\geq 1,$ are finite and uniformly bounded in terms of $n$ only. We also prove a number of results allowing us in some cases to calculate explicitly the groups $\\Sigma_{\\infty}(A)=\\cup_{k=1}^{\\infty} \\Sigma(A^{\\circ k})$ and ${\\rm Aut}_{\\infty}(A)=\\cup_{k=1}^{\\infty} {\\rm Aut}(A^{\\circ k})$, especially interesting from the dynamical perspective. In addition, we prove that the number of rational functions $B$ of degree $d$ sharing an iterate with $A$ is finite and bounded in terms of $n$ and $d$ only.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-15T06:16:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational function",
      "symmetries",
      "transformations",
      "special form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}