{
  "id": "1207.4844",
  "version": "v2",
  "published": "2012-07-20T02:39:45.000Z",
  "updated": "2013-08-19T06:53:08.000Z",
  "title": "Exact Hausdorff and packing measures of linear Cantor sets with overlaps",
  "authors": [
    "Hua Qiu"
  ],
  "comment": "41 pages, 5 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $K$ be the attractor of a linear iterated function system (IFS) $S_j(x)=\\rho_jx+b_j,$ $j=1,\\cdots,m$, on the real line satisfying the generalized finite type condition (whose invariant open set $\\mathcal{O}$ is an interval) with an irreducible weighted incidence matrix. This condition was introduced by Lau \\& Ngai recently as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let $\\alpha$ be the dimension of $K$. In this paper, we state that \\begin{equation*} \\mathcal{H}^\\alpha(K\\cap J)\\leq |J|^\\alpha \\end{equation*} for all intervals $J\\subset\\overline{\\mathcal{O}}$, and \\begin{equation*} \\mathcal{P}^\\alpha(K\\cap J)\\geq |J|^\\alpha \\end{equation*} for all intervals $J\\subset\\overline{\\mathcal{O}}$ centered in $K$, where $\\mathcal{H}^\\alpha$ denotes the $\\alpha$-dimensional Hausdorff measure and $\\mathcal{P}^\\alpha$ denotes the $\\alpha$-dimensional packing measure. This result extends a recent work of Olsen where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of $K$. Moreover, using these densities theorems, we describe a scheme for computing $\\mathcal{H}^\\alpha(K)$ exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing $\\mathcal{P}^\\alpha(K)$ as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer \\& Strichartz and Feng, respectively, and apply to some new classes allowing us to include linear Cantor sets with overlaps.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-19T06:53:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80"
    ],
    "keywords": [
      "linear cantor sets",
      "packing measure",
      "exact hausdorff",
      "open set condition",
      "weighted incidence matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.4844Q"
    }
  }
}