Non-classifiability of Ergodic Flows up to Time Change
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929620050247680 |
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| author | Gerber, Marlies Kunde, Philipp |
| author_facet | Gerber, Marlies Kunde, Philipp |
| contents | A time change of a flow $\{T_t\}$, ${t\in\mathbb{R}}$, is a reparametrization of the orbits of the flow such that each orbit is mapped to itself by an orientation-preserving homeomorphism of the parameter space. If a flow $\{S_t\}$ is isomorphic to a flow obtained by a reparametrization of a flow $\{T_t\}$, then we say that $\{S_t\}$ and $\{T_t\}$ are isomorphic up to a time change. For ergodic flows $\{S_t\}$ and $\{T_t\}$, Kakutani showed that this happens if and only if the two flows have Kakutani equivalent transformations as cross-sections.
We prove that the Kakutani equivalence relation on ergodic invertible measure-preserving transformations of a standard non-atomic probability space is not a Borel set. This shows in a precise way that classification of ergodic transformations up to Kakutani equivalence is impossible. In particular, our results imply the non-classifiability of ergodic flows up to isomorphism after a time change. Moreover, we obtain anti-classification results under isomorphism for ergodic invertible transformations of a sigma-finite measure space.
We also obtain anti-classification results under Kakutani equivalence for ergodic area-preserving smooth diffeomorphisms of the disk, annulus, and 2-torus, as well as real-analytic diffeomorphisms of the $2$-torus. Our work generalizes the anti-classification results under isomorphism for ergodic transformations obtained by Foreman, Rudolph, and Weiss. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_06086 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Non-classifiability of Ergodic Flows up to Time Change Gerber, Marlies Kunde, Philipp Dynamical Systems Primary: 37A20, Secondary: 37A35, 37A05, 37C40, 03E15 A time change of a flow $\{T_t\}$, ${t\in\mathbb{R}}$, is a reparametrization of the orbits of the flow such that each orbit is mapped to itself by an orientation-preserving homeomorphism of the parameter space. If a flow $\{S_t\}$ is isomorphic to a flow obtained by a reparametrization of a flow $\{T_t\}$, then we say that $\{S_t\}$ and $\{T_t\}$ are isomorphic up to a time change. For ergodic flows $\{S_t\}$ and $\{T_t\}$, Kakutani showed that this happens if and only if the two flows have Kakutani equivalent transformations as cross-sections. We prove that the Kakutani equivalence relation on ergodic invertible measure-preserving transformations of a standard non-atomic probability space is not a Borel set. This shows in a precise way that classification of ergodic transformations up to Kakutani equivalence is impossible. In particular, our results imply the non-classifiability of ergodic flows up to isomorphism after a time change. Moreover, we obtain anti-classification results under isomorphism for ergodic invertible transformations of a sigma-finite measure space. We also obtain anti-classification results under Kakutani equivalence for ergodic area-preserving smooth diffeomorphisms of the disk, annulus, and 2-torus, as well as real-analytic diffeomorphisms of the $2$-torus. Our work generalizes the anti-classification results under isomorphism for ergodic transformations obtained by Foreman, Rudolph, and Weiss. |
| title | Non-classifiability of Ergodic Flows up to Time Change |
| topic | Dynamical Systems Primary: 37A20, Secondary: 37A35, 37A05, 37C40, 03E15 |
| url | https://arxiv.org/abs/2109.06086 |