Abstract:

A classical problem in ergodic theory, posed by J. von Neumann in 1932, is the isomorphism problem: classify measure-preserving transformations up to conjugacy. Two great successes are the Halmos-von Neumann classification of ergodic transformations with pure-point spectrum in 1942 and Ornstein’s classification of Bernoulli shifts by their metric entropy in 1970. However, in 2001, Hjorth proved in a precise way that the isomorphism problem for general measure-preserving transformations is intractable, and in 2011, Foreman-Rudolph-Weiss showed that this is still true when the isomorphism problem is restricted to ergodic measure-preserving transformations. Nonetheless, it seemed feasible that the isomorphism problem might be solvable when restricted to families of measure-preserving transformations more general than Bernoulli shifts that have sufficiently strong mixing properties.

We consider the isomorphism problem restricted to K-automorphisms. 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. In particular, K-automorphisms have positive entropy and are mixing of all orders. It is known 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 is complete analytic, and hence not Borel.

Our work is primarily in the context of measurable dynamics, but I will also mention anti-classification results due to Foreman-Weiss, Foreman-Gorodetski, and Kunde in topological and differentiable dynamics.