On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations

TL;DR

This paper investigates weak rational ergodicity and rational weak mixing in infinite measure-preserving rank-one transformations, establishing which families exhibit these properties and exploring their relation to other mixing notions.

Contribution

It characterizes the presence of weak rational ergodicity and rational weak mixing in different families of rank-one transformations and relates these to existing mixing concepts.

Findings

Certain families of rank-one transformations are weakly rationally ergodic.

Some families do not exhibit rational weak mixing.

The paper clarifies the relationship between these properties and other mixing notions.

Abstract

We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure.