{
  "id": "math/0608759",
  "version": "v2",
  "published": "2006-08-30T17:13:33.000Z",
  "updated": "2008-01-04T21:31:47.000Z",
  "title": "Trees and the dynamics of polynomials",
  "authors": [
    "Laura G. DeMarco",
    "Curtis T. McMullen"
  ],
  "comment": "60 pages",
  "categories": [
    "math.DS",
    "math.GT"
  ],
  "abstract": "The basin of infinity of a polynomial map $f : {\\bf C} \\arrow {\\bf C}$ carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial tree $(T,\\eta)$, with limiting dynamics $F : T \\arrow T$. In this paper we characterize the trees that arise as limits, and show they provide a natural boundary $\\PT_d$ compactifying the moduli space of polynomials of degree $d$. We show that $(T,\\eta,F)$ records the limiting behavior of multipliers at periodic points, and that any divergent meromorphic family of polynomials $\\{f_t(z) : t \\mem \\Delta^* \\}$ can be completed by a unique tree at its central fiber. Finally we show that in the cubic case, the boundary of moduli space $\\PT_3$ is itself a tree. The metrized trees $(T,\\eta,F)$ provide a counterpart, in the setting of iterated rational maps, to the ${\\bf R}$-trees that arise as limits of hyperbolic manifolds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-01-04T21:31:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "moduli space",
      "natural foliation",
      "iterated rational maps",
      "flat metric",
      "cubic case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8759D"
    }
  }
}