{
  "id": "1604.04771",
  "version": "v1",
  "published": "2016-04-16T16:15:28.000Z",
  "updated": "2016-04-16T16:15:28.000Z",
  "title": "Finiteness theorems for commuting and semiconjugate rational functions",
  "authors": [
    "F. Pakovich"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we study the functional equation $A\\circ X=X\\circ B,$ where $A$, $B$, and $X$ are rational functions of degree at least two. More precisely, we assume that $B$ is fixed and study solutions of the above equation in $A$ and $X$. Roughly speaking, our main result states that, up to some natural transformations, the set of such solutions is finite, unless $B$ is a Latt\\`es function or is conjugated to $z^{\\pm d}$ or $\\pm T_d$. In more details, we show that there exist rational functions $A_1, A_2,\\dots, A_r$ and $X_1, X_2,\\dots, X_r$ such that the equality $A\\circ X=X\\circ B$ holds if and only if $A=\\mu \\circ A_j\\circ \\mu^{-1}$ for some $j,$ $1\\leq j \\leq r,$ and M\\\"obius transformation $\\mu$, and $X=\\mu \\circ X_j\\circ B^{\\circ k}$ for some $k\\geq 1$. We also show that the degrees of $X_j,$ $1\\leq j \\leq r,$ and $r$ can be bounded from above by constants depending on degree of $B$ only. As an application we obtain \"effective\" analogues of classical results about commuting rational functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-16T16:15:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semiconjugate rational functions",
      "finiteness theorems",
      "main result states",
      "functional equation",
      "natural transformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160404771P"
    }
  }
}