{
  "id": "0912.0505",
  "version": "v1",
  "published": "2009-12-02T20:42:03.000Z",
  "updated": "2009-12-02T20:42:03.000Z",
  "title": "Critical heights on the moduli space of polynomials",
  "authors": [
    "Laura DeMarco",
    "Kevin Pilgrim"
  ],
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "Let $M_d$ be the moduli space of one-dimensional complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights map $G: M_d \\to \\mathbb{R}^{d-1}$. For generic values of $G$, each connected component of a fiber of $G$ is the deformation space for twist deformations on the basin of infinity. We analyze the quotient space $\\mathcal{T}_d^*$ obtained by collapsing each connected component of a fiber of $G$ to a point. The space $\\mathcal{T}_d^*$ is a parameter-space analog of the polynomial tree $T(f)$ associated to a polynomial $f:\\mathbb{C}\\to\\mathbb{C}$, studied by DeMarco and McMullen, and there is a natural projection from $\\mathcal{T}_d^*$ to the space of trees $\\mathcal{T}_d$. We show that the projectivization $\\mathbb{P}\\mathcal{T}_d^*$ is compact and contractible; further, the shift locus in $\\mathbb{P}\\mathcal{T}_d^*$ has a canonical locally finite simplicial structure. The top-dimensional simplices are in one-to-one corespondence with topological conjugacy classes of structurally stable polynomials in the shift locus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-02T20:42:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45"
    ],
    "keywords": [
      "moduli space",
      "critical heights",
      "one-dimensional complex polynomial dynamical systems",
      "shift locus",
      "canonical locally finite simplicial structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.0505D"
    }
  }
}