{
  "id": "math/9410221",
  "version": "v1",
  "published": "1994-10-01T00:00:00.000Z",
  "updated": "1994-10-01T00:00:00.000Z",
  "title": "Frontiers in complex dynamics",
  "authors": [
    "Curtis T. McMullen"
  ],
  "comment": "18 pages. Abstract added in migration.",
  "journal": "Bull. Amer. Math. Soc. (N.S.) 31 (1994) 155-172",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.). The rational maps of a given degree form a finite-dimensional manifold, so exploration of this {\\em parameter space} is especially tractable. Finally, some of the conjectures once proposed for {\\em smooth} dynamical systems (and now known to be false) seem to have a definite chance of holding in the arena of rational maps. In this article we survey a small constellation of such conjectures centering around the density of {\\em hyperbolic} rational maps --- those which are dynamically the best behaved. We discuss some of the evidence and logic underlying these conjectures, and sketch recent progress towards their resolution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1994-10-01T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational maps",
      "complex dynamics",
      "riemann sphere occupy",
      "conjectures",
      "potential theory"
    ],
    "tags": [
      "journal article",
      "expository article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Bull. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1994math.....10221M"
    }
  }
}