{
  "id": "1211.4303",
  "version": "v3",
  "published": "2012-11-19T05:07:26.000Z",
  "updated": "2014-09-26T08:37:06.000Z",
  "title": "Rational functions with identical measure of maximal entropy",
  "authors": [
    "Hexi Ye"
  ],
  "comment": "2 figures, 21 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We discuss when two rational functions $f$ and $g$ can have the same measure of maximal entropy. The polynomial case was completed by (Beardon, Levin, Baker-Eremenko,Schmidt-Steinmetz, etc., 1980s-90s), and we address the rational case following Levin-Przytycki (1997). We show: $\\mu_f = \\mu_g$ implies that $f$ and $g$ share an iterate ($f^n = g^m$ for some $n$ and $m$) for general $f$ with degree $d \\geq 3$. And for generic $f\\in \\Rat_{d\\geq 3}$, $\\mu_f = \\mu_g$ implies $g=f^n$ for some $n \\geq 1$. For generic $f\\in \\Rat_2$, $\\mu_f = \\mu_g$ implies that $g= f^n$ or $\\sigma_f\\circ f^n$ for some $n\\geq 1$, where $\\sigma_f\\in PSL_2(\\C)$ permutes two points in each fiber of $f$. Finally, we construct examples of $f$ and $g$ with $\\mu_f = \\mu_g$ such that $f^n \\neq \\sigma\\circ g^m$ for any $\\sigma \\in PSL_2(\\C)$ and $m,n\\geq 1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-12-16T03:12:09.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-26T08:37:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational functions",
      "maximal entropy",
      "identical measure",
      "polynomial case",
      "rational case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4303Y"
    }
  }
}