Non-Classifiability of Kolmogorov Diffeomorphisms up to Isomorphism

TL;DR

This paper proves that classifying Kolmogorov automorphisms up to isomorphism is impossible using Borel invariants, demonstrating the problem's intractability even among smooth diffeomorphisms.

Contribution

It establishes that the isomorphism relation for K-automorphisms is a complete analytic set, making classification by Borel invariants impossible.

Findings

The isomorphism relation on K-automorphisms is a complete analytic set.

This intractability persists even for K-automorphisms that are smooth diffeomorphisms.

Classifying K-automorphisms up to isomorphism is fundamentally intractable.

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.