A counterexample to the singular Weinstein conjecture
Abstract
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 $\alpha$ and $\omega$-limits both lie on Z (a generalized singular periodic orbit). This is the content of the generalized Weinstein conjecture.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2023
- DOI:
- 10.48550/arXiv.2310.19918
- arXiv:
- arXiv:2310.19918
- Bibcode:
- 2023arXiv231019918F
- Keywords:
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- Mathematics - Symplectic Geometry;
- Mathematics - Differential Geometry;
- Mathematics - Dynamical Systems
- E-Print:
- 22 pages, 11 figures, overall improvement of the paper, formulated the generalized Weinstein conjecture