### References & Citations

# Mathematics > Dynamical Systems

# Title: Nonsingular Poisson Suspensions

(Submitted on 6 Feb 2020 (v1), last revised 28 Dec 2020 (this version, v2))

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 $t$.

## Submission history

From: Alexandre Danilenko [view email]**[v1]**Thu, 6 Feb 2020 11:50:26 GMT (41kb)

**[v2]**Mon, 28 Dec 2020 13:44:59 GMT (38kb)

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