Ergodic homoclinic groups, Sidon constructions and Poisson suspensions

Abstract

. 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 mod-ifications 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.