## Abstract

The classical Poisson functor associates to every infinite measure preserving dynamical system (X, μ, T) a probability preserving dynamical system (X*, μ*, T_{*}) called the Poisson suspension of T. In this paper we generalize this construction: a subgroup Aut_{2}(X, μ) of μ-nonsingular transformations T of X is specified as the largest subgroup for which T_{*} is μ*-nonsingular. The topological structure of this subgroup is studied. We show that a generic element in Aut_{2}(X, μ) is ergodic and of Krieger type III_{1}. Let G be a locally compact Polish group and let A: G → Aut_{2}(X, μ) 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 κ on G and t > 0, a nonsingular Poisson G-action is constructed whose Furstenberg κ-entropy is t.

Original language | American English |
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Pages (from-to) | 741-790 |

Number of pages | 50 |

Journal | Journal d'Analyse Mathematique |

Volume | 146 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2022 |

### Bibliographical note

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