{
  "id": "2002.02207",
  "version": "v1",
  "published": "2020-02-06T11:50:26.000Z",
  "updated": "2020-02-06T11:50:26.000Z",
  "title": "Nonsingular Poisson Suspensions",
  "authors": [
    "Alexandre I. Danilenko",
    "Zemer Kosloff",
    "Emmanuel Roy"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The classical Poisson functor associates to every infinite measure preserving dynamical system $(X,\\mu,T)$ a probability preserving dynamical system $(X^*,\\mu^*,T_*)$ called the Poisson suspension of $T$. In this paper we generalize this construction: a subgroup Aut$_2(X,\\mu)$ of $\\mu$-nonsingular transformations $T$ of $X$ is specified as the largest subgroup for which $T_*$ is $\\mu^*$-nonsingular. Topological structure of this subgroup is studied. We show that a generic element in Aut$_2(X,\\mu)$ is ergodic and of Krieger type III$_1$. Let $G$ be a locally compact Polish group and let $A:G\\to\\text{Aut}_2(X,\\mu)$ be a $G$-action. We investigate dynamical properties of the Poisson suspension $A_*$ of $A$ in terms of an affine representation of $G$ associated naturally with $A$. It is shown that $G$ has property (T) if and only if each nonsingular Poisson $G$-action admits an absolutely continuous invariant probability. If $G$ does not have property $(T)$ then for each generating probability $\\kappa$ on $G$ and $t>0$, a nonsingular Poisson $G$-action is constructed whose Furstenberg $\\kappa$-entropy is $t$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-06T11:50:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A40",
      "37A50"
    ],
    "keywords": [
      "nonsingular poisson suspensions",
      "infinite measure preserving dynamical system",
      "classical poisson functor associates",
      "absolutely continuous invariant probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}