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.