A counterexample to the singular Weinstein conjecture
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914011707080704 |
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| author | Fontana-McNally, Josep Miranda, Eva Oms, Cédric Peralta-Salas, Daniel |
| author_facet | Fontana-McNally, Josep Miranda, Eva Oms, Cédric Peralta-Salas, Daniel |
| contents | In this article, we study the dynamical properties of Reeb vector fields on b-contact manifolds. We show that in dimension 3, the number of so-called singular periodic orbits can be prescribed. These constructions illuminate some key properties of escape orbits and singular periodic orbits, which play a central role in formulating singular counterparts to the Weinstein conjecture and the Hamiltonian Seifert conjecture. In fact, we prove that the above-mentioned constructions lead to counterexamples of these conjectures as stated in [23]. Our construction shows that there are b-contact manifolds with no singular periodic orbit and no regular periodic orbit away from Z. We do not know whether there are constructions with no generalized escape orbits whose $α$ and $ω$-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19918 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A counterexample to the singular Weinstein conjecture Fontana-McNally, Josep Miranda, Eva Oms, Cédric Peralta-Salas, Daniel Symplectic Geometry Differential Geometry Dynamical Systems In this article, we study the dynamical properties of Reeb vector fields on b-contact manifolds. We show that in dimension 3, the number of so-called singular periodic orbits can be prescribed. These constructions illuminate some key properties of escape orbits and singular periodic orbits, which play a central role in formulating singular counterparts to the Weinstein conjecture and the Hamiltonian Seifert conjecture. In fact, we prove that the above-mentioned constructions lead to counterexamples of these conjectures as stated in [23]. Our construction shows that there are b-contact manifolds with no singular periodic orbit and no regular periodic orbit away from Z. We do not know whether there are constructions with no generalized escape orbits whose $α$ and $ω$-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture. |
| title | A counterexample to the singular Weinstein conjecture |
| topic | Symplectic Geometry Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2310.19918 |