{
  "id": "1206.1190",
  "version": "v1",
  "published": "2012-06-06T11:41:03.000Z",
  "updated": "2012-06-06T11:41:03.000Z",
  "title": "Measures of the full Hausdorff dimension for a general Sierpiński carpet",
  "authors": [
    "Jung-Chao Ban",
    "Chih-Hung Chang",
    "Ting-Ju Chen"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The measure of the full dimension for a general Sierpi\\'{n}ski carpet is studied. In the first part of this study, we give a criterion for the measure of the full Hausdorff dimension of a Sierpi\\'{n}ski carpet. Meanwhile, it is the conditional equilibrium measure of zero potential with respect to some Gibbs measure $\\nu_{\\alpha}$ of matrix-valued potential $\\alpha\\mathbf{N}$ (defined later). On one hand, this investigation extends the result of [17] without condition \\textbf{(H)}. On the other hand, it provides a checkable condition to ensure the existence and uniqueness of the measure of the full Hausdorff dimension for a general Sierpi\\'{n}ski carpet. In the second part of this paper we give a criterion for the Markov projection measure and estimate its number of steps by means of the induced matrix-valued potential. The results enable us to answer some questions which arise from [1] and [4] on the projection measure and factors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-06T11:41:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37C45"
    ],
    "keywords": [
      "full hausdorff dimension",
      "general sierpiński carpet",
      "markov projection measure",
      "conditional equilibrium measure",
      "matrix-valued potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.1190B"
    }
  }
}