{
  "id": "1402.0742",
  "version": "v2",
  "published": "2014-02-04T14:22:44.000Z",
  "updated": "2014-02-10T20:37:50.000Z",
  "title": "Asymmetries in Asymptotic 3-fold Properties of Ergodic Actions",
  "authors": [
    "V. V. Ryzhikov"
  ],
  "comment": "Ergodic theory, in Russian",
  "categories": [
    "math.DS"
  ],
  "abstract": "We present: 1) a mixing $Z ^ 2$-action with the following asymmetry of multiple mixing property: for some commuting measure-preserving transformations $S$, $T$ and a sequence $n_j$ $$ \\lim_{j\\to \\infty}\\mu(A\\bigcap S^{-n_j}A\\bigcap T^{-n_j}A)=\\mu(A)^3$$ for all measurable sets $A$, but there is $A_0$, $\\mu(A_0)=\\frac 1 2$, such that $$ \\lim_{j\\to \\infty}\\mu(A_0\\bigcap S^{n_j}A_0\\bigcap T^{n_j}A_0)=0;$$ 2) $Z $-actions with the asymmetry of the partial multiple mixing and the partial multiple rigidity: $$ \\lim_{j\\to \\infty}\\mu(A\\bigcap T^{k_j}A\\bigcap T^{m_j}A)= \\frac23 \\mu(A)^3+\\frac13\\mu(A),$$ $$ \\lim_{j\\to \\infty}\\mu(A\\bigcap T^{-k_j}A\\bigcap T^{-m_j}A)= \\mu(A)^2;$$ 3) infinite transformations $T$ such that for all $A$, $\\mu(A)<\\infty$, $$\\lim_{j\\to \\infty}\\mu(A\\bigcap T^{k_j}A\\bigcap T^{m_j}A)= \\frac13\\mu(A)$$ and $$\\lim_{j\\to \\infty}\\mu(A\\bigcap T^{-k_j}A\\bigcap T^{-m_j}A)=0.$$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-10T20:37:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ergodic actions",
      "asymptotic",
      "partial multiple rigidity",
      "infinite transformations",
      "multiple mixing property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "ru",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.0742R"
    }
  }
}