On Infinite Transformations with Maximal Control of Ergodic Two-fold Product Powers

TL;DR

This paper constructs infinite measure-preserving transformations with highly controlled ergodic and conservative behaviors in their Cartesian products, demonstrating diverse and maximal control over these properties.

Contribution

It introduces a class of transformations with prescribed ergodic behavior of product powers based on rational parameters, and constructs rank-one transformations with specified ergodic and conservative indices.

Findings

Existence of transformations with ergodic product powers precisely for rational ratios

Contrast with finite measure case where all product powers are ergodic

Construction of rank-one transformations with finite ergodic index but infinite conservative index

Abstract

We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure preserving transformations. A class of transformations is constructed such that for any subset $R\subset \mathbb Q\cap (0,1)$ there exists $T$ in this class such that $T^p\times T^q$ is ergodic if and only if $\frac{p}{q} \in R$. This contrasts with the finite measure preserving case where $T^p\times T^q$ is ergodic for all nonzero $p$ and $q$ if and only if $T\times T$ is ergodic. We also show that our class is rich in the behavior of conservative products. For each positive integer $k$, a family of rank-one infinite measure preserving transformations is constructed which have ergodic index $k$, but infinite conservative index.