Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism

Abstract

We show that the problem of classifying, up to isomorphism, the collection of zero-entropy mixing automorphisms of a standard non-atomic probability space, is intractible. More precisely, the collection of isomorphic pairs of automorphisms in this class is not Borel, when considered as a subset of the Cartesian product of the collection of measure-preserving automorphisms with itself. This remains true if we restrict to zero-entropy mixing automorphisms that are also $C^{\infty}$ diffeomorphisms of the five-dimensional torus. In addition, both of these results still hold if ``isomorphism'' is replaced by ``Kakutani equivalence.'' In our argument we show that for a uniquely and totally ergodic automorphism $U$ and a particular family of automorphisms $\mathcal{S}$, if $T\times U$ is isomorphic to $T^{-1}\times U$ with $T\in\mathcal{S}$ then $T$ is isomorphic to ${T^{-1}}$. However, this type of ``cancellation'' of factors from isomorphic Cartesian products is not true in general. We present an example due to M. Lema\'nczyk of two weakly mixing automorphisms $T$ and $S$ and an irrational rotation $R$ such that $T\times R$ is isomorphic to $S\times R$, but $T$ and $S$ are not isomorphic.