When Periodicity Fails to Guarantee the Existence of Rotation: A Counterexample on $\mathbb{T}^3$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908981369241600 |
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| author | Oukil, Walid |
| author_facet | Oukil, Walid |
| contents | In this manuscript, we construct an explicit counterexample of a smooth \(C^{\infty}\), periodic dynamical system on the torus \(\mathbb{T}^3\) for which the rotation vector exists in a weak sense, but fails to exist in the strong sense of bounded deviation (also referred to as {\it {frequencies}} in parts of the physics and biology literature). The construction exploits Liouville-type arithmetic properties and demonstrates that smoothness and periodicity alone do not ensure bounded deviation, even within the class of integrable systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21006 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | When Periodicity Fails to Guarantee the Existence of Rotation: A Counterexample on $\mathbb{T}^3$ Oukil, Walid Dynamical Systems 34D05, 37B65, 34C15 In this manuscript, we construct an explicit counterexample of a smooth \(C^{\infty}\), periodic dynamical system on the torus \(\mathbb{T}^3\) for which the rotation vector exists in a weak sense, but fails to exist in the strong sense of bounded deviation (also referred to as {\it {frequencies}} in parts of the physics and biology literature). The construction exploits Liouville-type arithmetic properties and demonstrates that smoothness and periodicity alone do not ensure bounded deviation, even within the class of integrable systems. |
| title | When Periodicity Fails to Guarantee the Existence of Rotation: A Counterexample on $\mathbb{T}^3$ |
| topic | Dynamical Systems 34D05, 37B65, 34C15 |
| url | https://arxiv.org/abs/2504.21006 |