Ergodicity of Poisson products and applications

In this paper we study the Poisson process over 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 .-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.T. under the assumption that T is ergodic and conservative. We then show, assuming ergodicity of T.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 Rd. 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.