{
  "id": "math/0507555",
  "version": "v2",
  "published": "2005-07-27T10:03:12.000Z",
  "updated": "2005-09-21T12:55:09.000Z",
  "title": "Bifurcation currents in holomorphic dynamics on ${\\bf P}^k$",
  "authors": [
    "Giovanni Bassanelli",
    "François Berteloot"
  ],
  "comment": "32 pages",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "We establish a formula for the sum of the Lyapounov exponents of an holomorphic endomorphism of ${\\bf P}^k$. For an holomorphic family of such endomorphisms we define the {\\em bifurcation current} as $dd^cL$ and show that it vanishes when the repulsive cycles move holomorphically. We then prove a formula which relates this current with the interaction between the Green current and the current of integration on the critical set. In the 1-dimensional case (i.e. for ${\\bf P}^1$) we find a geometrical description of the support of this current and its powers. Finally we introduce the {\\em bifurcation measure} giving some applications. This last part may be interpreted as a generalization of Mane-Sad-Sullivan theory based on pluri-potentialist methods.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-09-21T12:55:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45",
      "37F10"
    ],
    "keywords": [
      "bifurcation current",
      "holomorphic dynamics",
      "lyapounov exponents",
      "holomorphic endomorphism",
      "green current"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7555B"
    }
  }
}