# Infinite symmetric ergodic index and related examples in infinite measure

## TL;DR

This paper introduces the concept of infinite symmetric ergodic index for measure-preserving transformations, explores its properties, and distinguishes it from power weak mixing, providing new conditions and classifications for such transformations.

## Contribution

It defines infinite symmetric ergodic index, establishes its properties, and characterizes classes of transformations with this property, clarifying their relationship with power weak mixing.

## Key findings

- Infinite symmetric ergodic index does not imply power weak mixing.
- Provided a sufficient condition for k-fold and infinite symmetric ergodic index.
- Characterized classes of transformations with infinite symmetric ergodic index.

## Abstract

We define an infinite measure-preserving transformation to have infinite symmetric ergodic index if all finite Cartesian products of the transformation and its inverse are ergodic, and show that infinite symmetric ergodic index does not imply that all products of powers are conservative, so does not imply power weak mixing. We provide a sufficient condition for $k$-fold and infinite symmetric ergodic index and use it to answer a question on the relationship between product conservativity and product ergodicity. We also show that a class of rank-one transformations that have infinite symmetric ergodic index are not power weakly mixing, and precisely characterize a class of power weak transformations that generalizes existing examples.