Ergodicity and Conservativity of products of infinite transformations and their inverses

TL;DR

This paper constructs specific infinite measure-preserving transformations demonstrating that the product with itself can be ergodic while the product with its inverse is not, revealing nuanced properties of ergodicity and conservativity.

Contribution

It introduces a class of rank-one transformations with distinct ergodic properties for their self-product and inverse-product, advancing understanding of infinite measure dynamics.

Findings

Product of the transformation with itself is ergodic.

Product of the transformation with its inverse is not ergodic.

Product of any rank-one transformation with its inverse is conservative.

Abstract

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation $T$ in the class, the cartesian product $T\times T$ of the transformation with itself is ergodic, but the product $T\times T^{-1}$ of the transformation with its inverse is not ergodic. 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.