{
  "id": "math/0005304",
  "version": "v1",
  "published": "2000-05-01T00:00:00.000Z",
  "updated": "2000-05-01T00:00:00.000Z",
  "title": "Entropy and mixing for amenable group actions",
  "authors": [
    "Daniel J. Rudolph",
    "Benjamin Weiss"
  ],
  "comment": "32 pages, published version",
  "journal": "Ann. of Math. (2) 151 (2000), no. 3, 1119-1150",
  "categories": [
    "math.DS"
  ],
  "abstract": "For \\Gamma a countable amenable group consider those actions of \\Gamma as measure-preserving transformations of a standard probability space, written as {T_\\gamma}_{\\gamma \\in \\Gamma} acting on (X,{\\cal F}, \\mu). We say {T_\\gamma}_{\\gamma\\in\\Gamma} has completely positive entropy (or simply cpe for short) if for any finite and nontrivial partition P of X the entropy h(T,P) is not zero. Our goal is to demonstrate what is well known for actions of \\Bbb Z and even \\Bbb Z^d, that actions of completely positive entropy have very strong mixing properties. Let S_i be a list of finite subsets of \\Gamma. We say the S_i spread if any particular \\gamma \\neq id belongs to at most finitely many of the sets S_i S_i^{-1}. Theorem 0.1. For {T_\\gamma}_{\\gamma \\in \\Gamma} an action of \\Gamma of completely positive entropy and P any finite partition, for any sequence of finite sets S_i\\subseteq \\Gamma which spread we have \\frac 1{\\# S_i} h(\\spans{S_i}{P}){\\mathop{\\to}_i} h(P). The proof uses orbit equivalence theory in an essential way and represents the first significant application of these methods to classical entropy and mixing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-05-01T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "amenable group actions",
      "positive entropy",
      "standard probability space",
      "orbit equivalence theory",
      "first significant application"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......5304R"
    }
  }
}