{
  "id": "1108.1900",
  "version": "v3",
  "published": "2011-08-09T11:21:16.000Z",
  "updated": "2014-09-02T16:44:29.000Z",
  "title": "On semiconjugate rational functions",
  "authors": [
    "F. Pakovich"
  ],
  "comment": "In this version a more strong and precise result is proved",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\\circ X=X\\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of such functions, then either $B$ can be obtained from $A$ by a certain iterative process, or $A$ and $B$ can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-06-20T15:02:30.000Z",
      "abstract": "We classify rational solutions of the functional equation A(X)=X(B) in terms of groups acting properly discontinuously on the complex plane or the Riemann sphere, generalizing classical results of Julia, Fatou, and Ritt about commuting rational functions. We also give a description of rational solutions of the more general functional equation A(C)=D(B) under the condition that the algebraic curve A(x)-D(y)=0 is irreducible.",
      "comment": "In the second version the approach is simplified and new results are added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-02T16:44:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semiconjugate rational functions",
      "general functional equation",
      "complex plane",
      "classify rational solutions",
      "riemann sphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.1900P"
    }
  }
}