{
  "id": "2506.20797",
  "version": "v1",
  "published": "2025-06-25T19:41:15.000Z",
  "updated": "2025-06-25T19:41:15.000Z",
  "title": "Automorphism groups of measures on the Cantor space. Part II: Abstract homogeneous measures",
  "authors": [
    "Piotr Niemiec"
  ],
  "comment": "54 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "The main aim of the paper is to introduce a new class of (semigroup-valued) measures that are ultrahomogeneous on the Boolean algebra of all clopen subsets of the Cantor space and to study their automorphism groups. A characterisation, in terms of the so-called trinary spectrum of a measure, of ultrahomogeneous measures such that the action of their automorphism groups is (topologically) transitive or minimal is given. Also sufficient and necessary conditions for the existence of a dense (or co-meager) conjugacy class in these groups are offered. In particular, it is shown that there are uncountably many full non-atomic probability Borel measures m on the Cantor space such that m and all its restrictions to arbitrary non-empty clopen sets have all the following properties: this measure is ultrahomogeneous and not good, the action of its automorphism group G is minimal (on a respective clopen set), and G has a dense conjugacy class. It is also shown that any minimal homeomorphism h on the Cantor space induces a homogeneous h-invariant probability measure that is universal among all h-invariant probability measures (which means that it `generates' all such measures) and this property determines this measure (in a certain sense) up to a Q-linear isomorphism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-25T19:41:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A15",
      "28D15",
      "28D05",
      "03E15"
    ],
    "keywords": [
      "automorphism group",
      "cantor space",
      "abstract homogeneous measures",
      "h-invariant probability measure",
      "full non-atomic probability borel measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}