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Nonsingular Poisson Suspensions

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 Aut2(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 Aut2(X, μ) is ergodic and of Krieger type III1. Let G be a locally compact Polish group and let A: G → Aut2(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.

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Correspondence to Alexandre I. Danilenko.

Additional information

The research of Z. K. was partially supported by ISF grant No. 1570/17

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Danilenko, A.I., Kosloff, Z. & Roy, E. Nonsingular Poisson Suspensions. JAMA 146, 741–790 (2022). https://doi.org/10.1007/s11854-022-0213-8

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  • DOI: https://doi.org/10.1007/s11854-022-0213-8