We define a class of discrete Abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that Cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations.

Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in $L^p$, 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^p$. For p = 1 we find that the maximal ergodic...

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T × T is ergodic, but the product $T\times T^{-1}$ is not. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

We construct ${}^{\infty }$ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ${\left(n^{-1}{\sum }_{k=0}^{n-1}\nu \circ T^{-k}\right)}_{n\ge 1}$ of arithmetical averages of image measures does not converge weakly.