Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism
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| Format: | Preprint |
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2025
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| author | Gerber, Marlies Kunde, Philipp |
| author_facet | Gerber, Marlies Kunde, Philipp |
| contents | We show that the problem of classifying, up to isomorphism, the collection of zero-entropy mixing automorphisms of a standard non-atomic probability space, is intractible. More precisely, the collection of isomorphic pairs of automorphisms in this class is not Borel, when considered as a subset of the Cartesian product of the collection of measure-preserving automorphisms with itself. This remains true if we restrict to zero-entropy mixing automorphisms that are also $C^{\infty}$ diffeomorphisms of the five-dimensional torus. In addition, both of these results still hold if ``isomorphism'' is replaced by ``Kakutani equivalence.''
In our argument we show that for a uniquely and totally ergodic automorphism $U$ and a particular family of automorphisms $\mathcal{S}$, if $T\times U$ is isomorphic to $T^{-1}\times U$ with $T\in\mathcal{S}$ then $T$ is isomorphic to ${T^{-1}}$. However, this type of ``cancellation'' of factors from isomorphic Cartesian products is not true in general. We present an example due to M. Lemańczyk of two weakly mixing automorphisms $T$ and $S$ and an irrational rotation $R$ such that $T\times R$ is isomorphic to $S\times R$, but $T$ and $S$ are not isomorphic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_09003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism Gerber, Marlies Kunde, Philipp Dynamical Systems Primary: 37A35, Secondary: 37A20, 37A05, 37A25, 37C40, 03E15 We show that the problem of classifying, up to isomorphism, the collection of zero-entropy mixing automorphisms of a standard non-atomic probability space, is intractible. More precisely, the collection of isomorphic pairs of automorphisms in this class is not Borel, when considered as a subset of the Cartesian product of the collection of measure-preserving automorphisms with itself. This remains true if we restrict to zero-entropy mixing automorphisms that are also $C^{\infty}$ diffeomorphisms of the five-dimensional torus. In addition, both of these results still hold if ``isomorphism'' is replaced by ``Kakutani equivalence.'' In our argument we show that for a uniquely and totally ergodic automorphism $U$ and a particular family of automorphisms $\mathcal{S}$, if $T\times U$ is isomorphic to $T^{-1}\times U$ with $T\in\mathcal{S}$ then $T$ is isomorphic to ${T^{-1}}$. However, this type of ``cancellation'' of factors from isomorphic Cartesian products is not true in general. We present an example due to M. Lemańczyk of two weakly mixing automorphisms $T$ and $S$ and an irrational rotation $R$ such that $T\times R$ is isomorphic to $S\times R$, but $T$ and $S$ are not isomorphic. |
| title | Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism |
| topic | Dynamical Systems Primary: 37A35, Secondary: 37A20, 37A05, 37A25, 37C40, 03E15 |
| url | https://arxiv.org/abs/2509.09003 |