Notes on symplectic squeezing in $T^* \mathbb T^n$ and spectra of Finsler dynamics
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912284489547776 |
|---|---|
| author | Feng, Qi Zhang, Jun |
| author_facet | Feng, Qi Zhang, Jun |
| contents | In this paper, on the one hand, we prove that for $n \geq 2$ any subbundle of $T^* \mathbb T^n$ with bounded fibers symplectically embeds into a trivial subbundle of $T^* \mathbb T^n$ where the fiber is an irrational cylinder. This not only resolves an open problem in Gong-Xue's recent work (which was stated for the 4-dimension case, that is, $n =2$) and also generalizes to any higher-dimensional situation. The proof is based on some version of Dirichlet's approximation theorem. On the other hand, we generalize a main result in Gong-Xue's work mentioned above, showing that any topologically trivial Liouville diffeomorphism on $T^*M$ (for instance, a diffeomorphism induced by an isometry on $M$) does not change the full marked length spectrum of a Finsler metric $F$ on $M$, up to a lifting of the Finsler metric $F$ to the unit codisk bundle $D^*_FM$. The proof is based on persistence module theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Notes on symplectic squeezing in $T^* \mathbb T^n$ and spectra of Finsler dynamics Feng, Qi Zhang, Jun Symplectic Geometry Dynamical Systems 53D05, 58B20 In this paper, on the one hand, we prove that for $n \geq 2$ any subbundle of $T^* \mathbb T^n$ with bounded fibers symplectically embeds into a trivial subbundle of $T^* \mathbb T^n$ where the fiber is an irrational cylinder. This not only resolves an open problem in Gong-Xue's recent work (which was stated for the 4-dimension case, that is, $n =2$) and also generalizes to any higher-dimensional situation. The proof is based on some version of Dirichlet's approximation theorem. On the other hand, we generalize a main result in Gong-Xue's work mentioned above, showing that any topologically trivial Liouville diffeomorphism on $T^*M$ (for instance, a diffeomorphism induced by an isometry on $M$) does not change the full marked length spectrum of a Finsler metric $F$ on $M$, up to a lifting of the Finsler metric $F$ to the unit codisk bundle $D^*_FM$. The proof is based on persistence module theory. |
| title | Notes on symplectic squeezing in $T^* \mathbb T^n$ and spectra of Finsler dynamics |
| topic | Symplectic Geometry Dynamical Systems 53D05, 58B20 |
| url | https://arxiv.org/abs/2401.17635 |