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arXiv · 2303.12900

Anti-classification results for weakly mixing diffeomorphisms

Philipp Kunde
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Abstract

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of ``computability''. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the $2$-torus.

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Philipp Kunde. 2023-03-22. Anti-classification results for weakly mixing diffeomorphisms. https://arxiv.org/abs/2303.12900

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