Nonsingular Poisson Suspensions

Abstract

The classical Poisson functor associates to every infinite measure preserving dynamical system $(X,\mu ,T)$ a probability preserving dynamical system $(X^{\ast },{\mu }^{\ast },T_{\ast })$ 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 $T_{\ast }$ is ${\mu }^{\ast }$-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 $A_{\ast }$ 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 $t$.

Little corrections. Subsection 8.1 is completely rewritten