{
  "id": "math/0505099",
  "version": "v2",
  "published": "2005-05-05T23:19:13.000Z",
  "updated": "2007-08-21T14:17:08.000Z",
  "title": "Hausdorff dimension, its properties, and its surprises",
  "authors": [
    "Dierk Schleicher"
  ],
  "comment": "28 pages, 4 figures. Minor revision in process of publication",
  "journal": "American Mathematical Monthly 114 (2007), 509-528",
  "categories": [
    "math.DS"
  ],
  "abstract": "We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension has some surprising properties: we construct a set $E\\subset\\C$ of positive planar measure and with dimension 2 such that each point in $E$ can be joined to $\\infty$ by one or several curves in $\\C$ such that all curves are disjoint from each other and from $E$, and so that their union has Hausdorff dimension 1. We can even arrange things so that every point in $\\C$ which is not on one of these curves is in $E$. These examples have been discovered very recently; they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as $z\\mapsto \\pi\\sin(z)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-08-21T14:17:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37-02",
      "37F10",
      "37F35",
      "30D05",
      "54G20",
      "28A78"
    ],
    "keywords": [
      "hausdorff dimension",
      "complex dynamics",
      "fundamental properties",
      "arise quite",
      "metric spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......5099S"
    }
  }
}