On conservative sequences and their application to ergodic multiplier problems

TL;DR

This paper investigates conservative sequences in ergodic theory to construct transformations with specific ergodic properties, providing new insights into product transformations and their ergodic or conservative nature.

Contribution

It introduces methods to construct rank-one transformations affecting ergodicity and conservativity of product systems, extending results to actions and flows.

Findings

Existence of transformations $S$ making $T_i imes S$ non-ergodic for given collections.

Conditions under which $T imes S$ is ergodic or conservative but not ergodic.

The infinite Chacón transformation satisfies both ergodic and conservative conditions.

Abstract

The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n \in \mathbb{Z}$ such that $T^n A \cap A \not = \varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure $\{T_i\}$, there exists a rank-one transformation $S$ such that $T_i \times S$ is not ergodic for all $i$. Moreover, $S$ can be chosen to be rigid or have infinite ergodic index. We establish similar results for $\mathbb{Z}^d$ actions and flows. Then, we find sufficient conditions on rank-one transformations $T$ that guarantee the existence of a rank-one transformation $S$ such that $T \times S$ is ergodic, or, alternatively, conditions that guarantee that $T \times S$ is conservative but not ergodic. In particular, the infinite Chac\'on transformation satisfies both conditions. Finally, for a given ergodic transformation $T$, we study the Baire categories of the sets $E(T)$, $\bar{E}C(T)$ and $\bar{C}(T)$ of transformations $S$ such that $T \times S$ is ergodic, ergodic but not conservative, and conservative, respectively.