{
  "id": "1003.5174",
  "version": "v1",
  "published": "2010-03-26T16:06:40.000Z",
  "updated": "2010-03-26T16:06:40.000Z",
  "title": "Variations of Hausdorff Dimension in the Exponential Family",
  "authors": [
    "Guillaume Havard",
    "Mariusz Urbanski",
    "Michel Zinsmeister"
  ],
  "comment": "32 pages. A para\\^itre dans Annales Academi{\\ae} Scientiarum Fennic{\\ae} Mathematica",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we deal with the following family of exponential maps $(f_\\lambda:z\\mapsto \\lambda(e^z-1))_{\\lambda\\in [1,+\\infty)}$. Denoting $d(\\lambda)$ the hyperbolic dimension of $f_\\lambda$. It is known that the function $\\lambda\\mapsto d(\\lambda)$ is real analytic in $(1,+\\infty)$, and that it is continuous in $[1,+\\infty)$. In this paper we prove that this map is C$^1$ on $[1,+\\infty)$, with $d'(1^+)=0$. Moreover, depending on the value of $d(1)$, we give estimates of the speed of convergence towards 0.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-03-26T16:06:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff dimension",
      "exponential family",
      "variations",
      "exponential maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.5174H"
    }
  }
}