{
  "id": "1102.1072",
  "version": "v1",
  "published": "2011-02-05T14:14:59.000Z",
  "updated": "2011-02-05T14:14:59.000Z",
  "title": "Infinite measures on Cantor spaces",
  "authors": [
    "Olena Karpel"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the set $M_\\infty(X)$ of all infinite full non-atomic Borel measures on a Cantor space X. For a measure $\\mu$ from $M_\\infty(X)$ we define a defective set $M_\\mu = \\{x \\in X : for any clopen set U which contains x we have \\mu(U) = \\infty \\}$. We call a measure $\\mu$ from $M_\\infty(X)$ non-defective ($\\mu \\in M_\\infty^0(X)$) if $\\mu(M_\\mu) = 0$. The paper is devoted to the classification of measures $\\mu$ from $M_\\infty^0(X)$ with respect to a homeomorphism. The notions of goodness and clopen values set $S(\\mu)$ are defined for a non-defective measure $\\mu$. We give a criterion when two good non-defective measures are homeomorphic and prove that there exist continuum classes of weakly homeomorphic good non-defective measures on a Cantor space. For any group-like subset $D \\subset [0,\\infty)$ we find a good non-defective measure $\\mu$ on a Cantor space X with $S(\\mu) = D$ and an aperiodic homeomorphism of X which preserves $\\mu$. The set $S$ of infinite ergodic R-invariant measures on non-simple stationary Bratteli diagrams consists of non-defective measures. For $\\mu \\in S$ the set $S(\\mu)$ is group-like, a criterion of goodness is proved for such measures. We show that a homeomorphism class of a good measure from $S$ contains countably many distinct good measures from $S$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-05T14:14:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37B05",
      "28D05",
      "28C15"
    ],
    "keywords": [
      "cantor space",
      "non-defective measure",
      "infinite measures",
      "non-simple stationary bratteli diagrams consists",
      "infinite full non-atomic borel measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.1072K"
    }
  }
}