# Measure Preserving Diffeomorphisms of the Torus are Unclassifiable

## TL;DR

This paper proves that classifying measure-preserving diffeomorphisms of the torus up to isomorphism is impossible due to the complexity of their classification problem, which is shown to be a complete analytic set.

## Contribution

It demonstrates that the isomorphism problem for ergodic diffeomorphisms of the torus is unclassifiable, extending the understanding of complexity in ergodic theory.

## Key findings

- The set of isomorphic pairs of ergodic diffeomorphisms is a complete analytic set.
- The classification problem is not Borel, indicating high complexity.
- The result applies to smooth transformations on the 2-torus.

## Abstract

The isomorphism problem in ergodic theory was formulated by von Neumann in 1932 in his pioneering paper Zur Operatorenmethode in der klassischen Mechanik (Ann. of Math. (2), 33(3):587--642, 1932). The problem has been solved for some classes of transformations that have special properties, such as the collection of transformations with discrete spectrum or Bernoulli shifts. This paper shows that a general classification is impossible (even in concrete settings) by showing that the collection $E$ of pairs of ergodic, Lebesgue measure preserving diffeomorphisms $(S,T)$ of the 2-torus that are isomorphic is a complete analytic set in the $C^\infty$- topology (and hence not Borel).