Coboundaries for commuting transformations

Abstract

Let $\tau$ and $\sigma$ be two commuting ergodic measure preserving transformations of a probablity space, and $\mathrm{Cob}(\tau)$, $\mathrm{Cob}(\sigma)$ be the sets of their coboundaries. We show that the inclusion $\mathrm{Cob}(\sigma) \subseteq \mathrm{Cob}(\tau)$ holds if and only if $\sigma = \tau^{n}$ for some $n \in \mathbb{Z}$. The transformations $\tau$ and $\sigma$ have exactly the same coboundaries if and only if $\sigma = \tau^{\pm1}$. Some related results and open questions are discussed.