{
  "id": "1006.4498",
  "version": "v1",
  "published": "2010-06-23T12:22:19.000Z",
  "updated": "2010-06-23T12:22:19.000Z",
  "title": "On Hausdorff dimension of the set of closed orbits for a cylindrical transformation",
  "authors": [
    "Krzysztof Fraczek",
    "Mariusz Lemanczyk"
  ],
  "comment": "32 pages, 1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "We deal with Besicovitch's problem of existence of discrete orbits for transitive cylindrical transformations $T_\\varphi:(x,t)\\mapsto(x+\\alpha,t+\\varphi(x))$ where $Tx=x+\\alpha$ is an irrational rotation on the circle $\\T$ and $\\varphi:\\T\\to\\R$ is continuous, i.e.\\ we try to estimate how big can be the set $D(\\alpha,\\varphi):=\\{x\\in\\T:|\\varphi^{(n)}(x)|\\to+\\infty\\text{as}|n|\\to+\\infty\\}$. We show that for almost every $\\alpha$ there exists $\\varphi$ such that the Hausdorff dimension of $D(\\alpha,\\varphi)$ is at least $1/2$. We also provide a Diophantine condition on $\\alpha$ that guarantees the existence of $\\varphi$ such that the dimension of $D(\\alpha,\\varphi)$ is positive. Finally, for some multidimensional rotations $T$ on $\\T^d$, $d\\geq3$, we construct smooth $\\varphi$ so that the Hausdorff dimension of $D(\\alpha,\\varphi)$ is positive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-06-23T12:22:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "37C45",
      "37C29"
    ],
    "keywords": [
      "hausdorff dimension",
      "cylindrical transformation",
      "closed orbits",
      "discrete orbits",
      "construct smooth"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/0951-7715/23/10/003",
      "journal": "Nonlinearity",
      "year": 2010,
      "month": "Oct",
      "volume": 23,
      "number": 10,
      "pages": 2393
    },
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010Nonli..23.2393F"
    }
  }
}