A Multidimensional Birkhoff Theorem for Recurrent Lagrangian Submanifolds by a Tonelli Hamiltonian
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913949717364736 |
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| author | Charfi, Skander |
| author_facet | Charfi, Skander |
| contents | Consider a closed manifold $M$ and a time-periodic Tonelli Hamiltonian $H : \mathbb{R}/\mathbb{Z} \times T^*M \to \mathbb{R}$ with flow $ϕ_H$. Let $\mathcal{L} \subset T^*M$ be a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $ϕ_H^n(\mathcal{L})$ admits convergent subsequences in both positive and negative times, in the Hausdorff topology and with control on the Liouville primitives, to two Lagrangian submanifolds, then $\mathcal{L}$ is a graph over the zero section $0_{T^*M}$ of $T^*M$. Furthermore, we show that $\mathcal{L}$ is recurrent in both positive and negative times for the same type of convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14561 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Multidimensional Birkhoff Theorem for Recurrent Lagrangian Submanifolds by a Tonelli Hamiltonian Charfi, Skander Dynamical Systems Symplectic Geometry Consider a closed manifold $M$ and a time-periodic Tonelli Hamiltonian $H : \mathbb{R}/\mathbb{Z} \times T^*M \to \mathbb{R}$ with flow $ϕ_H$. Let $\mathcal{L} \subset T^*M$ be a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $ϕ_H^n(\mathcal{L})$ admits convergent subsequences in both positive and negative times, in the Hausdorff topology and with control on the Liouville primitives, to two Lagrangian submanifolds, then $\mathcal{L}$ is a graph over the zero section $0_{T^*M}$ of $T^*M$. Furthermore, we show that $\mathcal{L}$ is recurrent in both positive and negative times for the same type of convergence. |
| title | A Multidimensional Birkhoff Theorem for Recurrent Lagrangian Submanifolds by a Tonelli Hamiltonian |
| topic | Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2507.14561 |