{
  "id": "1410.0129",
  "version": "v1",
  "published": "2014-10-01T07:31:08.000Z",
  "updated": "2014-10-01T07:31:08.000Z",
  "title": "Note on the points with dense orbit under $\\times 2$ and $\\times 3$ maps",
  "authors": [
    "Wenya Wang"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "It was conjectured by Furstenberg that for any $x\\in [0,1]\\backslash Q$, $$ \\dim_H \\bar{\\{2^nx ({\\text{mod}}\\ 1): n\\ge 1\\}}+ \\dim_H \\bar{\\{3^nx ({\\text{mod}}\\ 1): n\\ge 1\\}}\\ge 1. $$ When $x$ is a normal number, the above result holds trivially. In this note, we give explicit non-normal numbers for which the above dimensional formula holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-01T07:31:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dense orbit",
      "dimensional formula holds",
      "explicit non-normal numbers",
      "furstenberg",
      "result holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.0129W"
    }
  }
}