On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in $\mathbb{R}^{2n}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913877252374528 |
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| author | Wang, Wei |
| author_facet | Wang, Wei |
| contents | In this paper, we prove that for every dynamically convex compact star-shaped hypersurface $Σ\subset\mathbb{R}^{2n}$, there exist at least $\lfloor\frac{n+1}{2}\rfloor$ geometrically distinct closed characteristics possessing irrational mean indices provided the number of geometrically distinct closed characteristics on $Σ$ is finite, this improves Theorem 1.3 in \cite{LoZ} of Y. Long and C. Zhu by finding one more closed characteristic possessing irrational mean index when $n$ is odd. Moreover, there exist at least $\lfloor\frac{n+1}{2}\rfloor+1$ geometrically distinct closed characteristics such that the ratio of the mean indices of any two of them is a irrational number provided the number of geometrically distinct closed characteristics on $Σ$ is finite, this improves Theorem 1.2 in \cite{HuO} of X. Hu and Y. Ou when $n$ is odd. In particular, these estimates are sharp for $n=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04546 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in $\mathbb{R}^{2n}$ Wang, Wei Symplectic Geometry 58E05, 37J46, 34C25 In this paper, we prove that for every dynamically convex compact star-shaped hypersurface $Σ\subset\mathbb{R}^{2n}$, there exist at least $\lfloor\frac{n+1}{2}\rfloor$ geometrically distinct closed characteristics possessing irrational mean indices provided the number of geometrically distinct closed characteristics on $Σ$ is finite, this improves Theorem 1.3 in \cite{LoZ} of Y. Long and C. Zhu by finding one more closed characteristic possessing irrational mean index when $n$ is odd. Moreover, there exist at least $\lfloor\frac{n+1}{2}\rfloor+1$ geometrically distinct closed characteristics such that the ratio of the mean indices of any two of them is a irrational number provided the number of geometrically distinct closed characteristics on $Σ$ is finite, this improves Theorem 1.2 in \cite{HuO} of X. Hu and Y. Ou when $n$ is odd. In particular, these estimates are sharp for $n=3$. |
| title | On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in $\mathbb{R}^{2n}$ |
| topic | Symplectic Geometry 58E05, 37J46, 34C25 |
| url | https://arxiv.org/abs/2506.04546 |