There are two well-known equivalence relations on the family of measure-preserving ergodic automorphisms (of a given Lebesgue space): isomorphism and Kakutani equivalence. Two such automorphisms are said to be isomorphic if there is a measure-preserving relabeling of the points in the space that changes one automorphism into the other. Kakutani equivalence is a weaker equivalence relation in which we are also allowed to replace the automorphisms by their first return maps to measurable subsets of the space before relabeling the points.

We consider the complexity of the problem of classifying ergodic automorphisms up to isomorphism or up to Kakutani equivalence. For example, are these equivalence relations, considered as subsets of the Cartesian product of the space of ergodic automorphisms with itself, Borel sets? The first breakthroughs were the anti-classification results of Foreman, Rudolph, and Weiss for isomorphism of ergodic automorphisms, and the subsequent anti-classification results of Foreman and Weiss for isomorphism of ergodic automorphisms that are also smooth diffeomorphisms preserving a given smooth measure on a manifold. 

I will describe further anti-classification results for the Kakutani equivalence relation, and for both isomorphism and Kakutani equivalence, when we restrict to smooth diffeomorphisms with additional properties, such as the mixing property or the K property.