ERGODICITY OF POISSON PRODUCTS AND APPLICATIONS

In this paper we study the Poisson process over a a-finite measure-space equipped with a measure preserving transformation or a group of measure preserving transformations. For a measure-preserving transformation T acting on a a-finite measure-space X, the Poisson suspension of T is the associated probability preserving transformation T-* which acts on realization of the Poisson process over X. We prove ergodicity of the Poisson-product T x T-* under the assumption that T is ergodic and conservative. We then show, assuming ergodicity of T x T-*, that it is impossible to deterministically perform natural equivariant operations: thinning, allocation or matching. In contrast, there are well-known results in the literature demonstrating the existence of isometry equivariant thinning, matching and allocation of homogenous Poisson processes on R-d. We also prove ergodicity of the "first return of left-most transformation" associated with a measure preserving transformation on R+, and discuss ergodicity of the Poisson-product of measure preserving group actions, and related spectral properties.