Approximation of commuting transformations

Abstract:

Let $\sigma$ and $\tau$ be two (measure-preserving) transformations. The main purpose of the paper is to show that if $\tau$ admits approximation by partitions and that if $\sigma$ commutes with a power ${\tau ^s}$ of $\tau$, then $\sigma$ can be approximated by a finite number of powers of $\tau$. As an application of the result we solve a problem posed earlier, showing that there exist strongly mixing transformations with only a finite number of prescribed roots.