Ergodic homoclinic groups, Sidon constructions and Poisson suspensions

We give some new examples of mixing transformations on a space with infinite measure: the so-called Sidon constructions of rank 1. We obtain rapid decay of correlations for a class of infinite transformations; this was recently discovered by Prikhod'ko for dynamical systems with simple spectrum acting on a probability space. We obtain an affirmative answer to Gordin's question about the existence of transformations with zero entropy and an ergodic homoclinic flow. We consider modifications of Sidon constructions inducing Poisson suspensions with simple singular spectrum and a homoclinic Bernoulli flow. We give a new proof of Roy's theorem on multiple mixing of Poisson suspensions.