Ergodic Properties of a Class of Discrete Abelian Group Extensions of Rank-One Transformations

TL;DR

This paper investigates the ergodic properties of a specific class of group extensions of rank-one transformations, establishing conditions for weak mixing, recurrence, and conservativity in infinite measure settings.

Contribution

It introduces a new class of abelian group extensions of rank-one transformations and characterizes their ergodic and recurrence properties.

Findings

Conditions for power weakly mixing are established.

All transformations in the class are multiply recurrent.

Examples of conservative transformations are provided.

Abstract

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.