Finite ergodic index and asymmetry for infinite measure preserving actions

TL;DR

This paper constructs specific infinite measure preserving actions of Abelian groups with prescribed ergodic indices and explores their properties, including asymmetry and ergodic behavior of their products.

Contribution

It introduces new constructions of rank-one infinite measure preserving actions with specified ergodic indices and analyzes their ergodic and conservative properties.

Findings

Constructed rank-one actions with ergodic index k

Demonstrated existence of actions where T×T^{-1} is not ergodic

Showed T×T^{-1} is conservative for all such actions

Abstract

Given $k>0$ and an Abelian countable discrete group $G$ with elements of infinite order, we construct $(i)$ rigid funny rank-one infinite measure preserving (i.m.p.) $G$-actions of ergodic index $k$, $(ii)$ 0-type funny rank-one i.m.p. $G$-actions of ergodic index $k$, $(iii)$ funny rank-one i.m.p. $G$-actions $T$ of ergodic index 2 such that the product $T\times T^{-1}$ is not ergodic. It is shown that $T\times T^{-1}$ is conservative for each funny rank-one $G$-action $T$.