{
  "id": "1711.08703",
  "version": "v1",
  "published": "2017-11-23T14:22:29.000Z",
  "updated": "2017-11-23T14:22:29.000Z",
  "title": "Generic Behavior of a Measure Preserving Transformation",
  "authors": [
    "Mahmood Etedadialiabadi"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Del Junco--Lema\\'nczyk showed that a generic measure preserving transformation satisfies a certain orthogonality conditions. More precisely, there is a dense $G_\\delta$ subset of measure preserving transformations such that for every $T\\in G$ and $k(1), k(2), \\dots, k(l)\\in \\mathbb{Z}^+$, $k'(1), k'(2), \\dots, k'(l')\\in \\mathbb{Z}^+$, the convolutions \\[ \\sigma_{T^{k(1)}} \\ast\\cdots\\ast \\sigma_{T^{k(l)}} \\ \\text{and} \\ \\sigma_{T^{k'(1)}} \\ast\\cdots \\ast\\sigma_{T^{k'(l')}} \\] are mutually singular, provided that $(k(1), k(2), \\dots, k(l))$ is not a rearrangement of $(k'(1), k'(2), \\dots, k'(l'))$. We will introduce an analogous orthogonality conditions for continuous unitary representations of $L^0(\\mu,\\mathbb{T})$ which we denote by DL--condition. We connect the DL--condition with a result of Solecki which states that every continuous unitary representations of $L^0(\\mu,\\mathbb{T})$ is a direct sum of action by pointwise multiplication on measure spaces $(X^{|\\kappa|},\\lambda_\\kappa)$ where $\\kappa$ is an increasing finite sequence of non-zero integers. In particular, we show that the \"probabilistic\" DL-condition translates to \"deterministic\" orthogonality conditions on the measures $\\lambda_\\kappa$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-23T14:22:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generic behavior",
      "orthogonality conditions",
      "continuous unitary representations",
      "generic measure preserving transformation satisfies",
      "dl-condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}