{
  "id": "math/0605729",
  "version": "v2",
  "published": "2006-05-29T17:25:05.000Z",
  "updated": "2006-10-18T16:18:52.000Z",
  "title": "A generic $C^1$ map has no absolutely continuous invariant probability measure",
  "authors": [
    "Artur Avila",
    "Jairo Bochi"
  ],
  "comment": "12 pages",
  "journal": "Nonlinearity 19 (2006) 2717-2725",
  "doi": "10.1088/0951-7715/19/11/001",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \\ge 1$. We consider the set of $C^1$ maps $f:M\\to M$ which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual (dense $G_\\delta) set in the $C^1$ topology. In the course of the proof, we need a generalization of the usual Rokhlin tower lemma to non-invariant measures. That result may be of independent interest.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-10-18T16:18:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C40"
    ],
    "keywords": [
      "absolutely continuous invariant probability measure",
      "usual rokhlin tower lemma",
      "smooth compact manifold"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}