{
  "id": "math/0311480",
  "version": "v6",
  "published": "2003-11-26T16:45:17.000Z",
  "updated": "2007-10-28T19:44:15.000Z",
  "title": "Bifurcations in the Space of Exponential Maps",
  "authors": [
    "Lasse Rempe",
    "Dierk Schleicher"
  ],
  "comment": "29 pages, 3 figures. The main change in the new version is the introduction of Theorem 1.1 on the connectivity of the bifurcation locus, which follows from the results of the original version but was not explicitly stated. Also, some small revisions have been made and references updated",
  "journal": "Invent. Math. 175 (2009), No. 1, 103 - 135",
  "doi": "10.1007/s00222-008-0147-5",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "This article investigates the parameter space of the exponential family $z\\mapsto \\exp(z)+\\kappa$. We prove that the boundary (in $\\C$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon. In fact, we prove the stronger statement that the exponential bifurcation locus is connected in $\\C$, which is an analog of Douady and Hubbard's celebrated theorem that the Mandelbrot set is connected. We show furthermore that $\\infty$ is not accessible through any nonhyperbolic (\"queer\") stable component. The main part of the argument consists of demonstrating a general \"Squeezing Lemma\", which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2007-10-28T19:44:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "30D05"
    ],
    "keywords": [
      "exponential maps",
      "parameter space",
      "hyperbolic component",
      "lyubich concerning bifurcation trees",
      "exponential bifurcation locus"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Inventiones Mathematicae",
      "year": 2008,
      "month": "Sep",
      "volume": 175,
      "number": 1,
      "pages": 103
    },
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008InMat.175..103R"
    }
  }
}