Non-Classifiability of Kolmogorov Diffeomorphisms up to Isomorphism

Abstract

We consider the problem of classifying Kolmogorov automorphisms (or $K$-automorphisms for brevity) up to isomorphism. Within the collection of measure-preserving transformations, Bernoulli shifts have the ultimate mixing property, and $K$-automorphisms have the next-strongest mixing properties of any widely considered family of transformations. J. Feldman observed that unlike Bernoulli shifts, the family of $K$-automorphisms cannot be classified up to isomorphism by a complete numerical Borel invariant. This left open the possibility of classifying $K$-automorphisms with a more complex type of Borel invariant. We show that this is impossible, by proving that the isomorphism equivalence relation restricted to $K$-automorphisms, considered as a subset of the Cartesian product of the set of $K$-automorphisms with itself, is a complete analytic set, and hence not Borel. Moreover, we prove this remains true if we restrict consideration to $K$-automorphisms that are also $C^{\infty}$ diffeomorphisms. This shows in a concrete way that the problem of classifying $K$-automorphisms up to isomorphism is intractible.