Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$
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| Format: | Preprint |
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2025
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| _version_ | 1866909830964314112 |
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| author | Li, Xiaorui Liu, Hui Wang, Wei |
| author_facet | Li, Xiaorui Liu, Hui Wang, Wei |
| contents | There are two long-standing conjectures in Hamiltonian dynamics concerning Reeb flows on the boundaries of star-shaped domains in $\mathbb{R}^{2n}$ ($n \geq 2$). One conjecture states that such a Reeb flow possesses either $n$ or infinitely many prime closed orbits; the other states that all the closed Reeb orbits are irrationally elliptic when the domain is convex and the flow possesses finitely many prime closed orbits. In this paper, we prove that for dynamically convex Reeb flow on the boundary of a star-shaped domain in $\mathbb{R}^{2n}$ ($n \geq 2$) with exactly $n$ prime closed orbits, at least two of them must be irrationally elliptic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_06597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$ Li, Xiaorui Liu, Hui Wang, Wei Dynamical Systems Symplectic Geometry 37J12, 57R58, 37J25 There are two long-standing conjectures in Hamiltonian dynamics concerning Reeb flows on the boundaries of star-shaped domains in $\mathbb{R}^{2n}$ ($n \geq 2$). One conjecture states that such a Reeb flow possesses either $n$ or infinitely many prime closed orbits; the other states that all the closed Reeb orbits are irrationally elliptic when the domain is convex and the flow possesses finitely many prime closed orbits. In this paper, we prove that for dynamically convex Reeb flow on the boundary of a star-shaped domain in $\mathbb{R}^{2n}$ ($n \geq 2$) with exactly $n$ prime closed orbits, at least two of them must be irrationally elliptic. |
| title | Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$ |
| topic | Dynamical Systems Symplectic Geometry 37J12, 57R58, 37J25 |
| url | https://arxiv.org/abs/2510.06597 |