{
  "id": "2006.07238",
  "version": "v1",
  "published": "2020-06-11T15:27:28.000Z",
  "updated": "2020-06-11T15:27:28.000Z",
  "title": "Nonsingular Gaussian actions: beyond the mixing case",
  "authors": [
    "Amine Marrakchi",
    "Stefaan Vaes"
  ],
  "categories": [
    "math.DS",
    "math.GR",
    "math.OA"
  ],
  "abstract": "Every affine isometric action $\\alpha$ of a group $G$ on a real Hilbert space gives rise to a nonsingular action $\\hat{\\alpha}$ of $G$ on the associated Gaussian probability space. In the recent paper [AIM19], several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation $\\pi$ of $G$ is mixing. We develop new methods to prove ergodicity when $\\pi$ is only weakly mixing. We determine the type of $\\hat{\\alpha}$ in full generality. Using Cantor measures, we give examples of type III$_1$ ergodic Gaussian actions of $\\mathbb{Z}$ whose underlying representation is non mixing, and even has a Dirichlet measure as spectral type. We also provide very general ergodicity results for Gaussian skew product actions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-11T15:27:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonsingular gaussian actions",
      "mixing case",
      "gaussian skew product actions",
      "real hilbert space",
      "general ergodicity results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}