Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916384290635776 |
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| author | Guggisberg, Yann Ziltener, Fabian |
| author_facet | Guggisberg, Yann Ziltener, Fabian |
| contents | We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian diffeomorphism that is arbitrarily $C^\infty$-close to the identity. As a consequence, every countably $n$-rectifiable and $n$-negligible subset of $\mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption.
The proof of our squeezing result uses folding. Potentially, our folding method can be modified to show that the Gromov width of $B^{2n}_1\setminus A$ equals $π$ for every countably $(n-1)$-rectifiable closed subset $A$ of the open unit ball $B^{2n}_1$. This means that $A$ is not a barrier. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_17444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets Guggisberg, Yann Ziltener, Fabian Symplectic Geometry Dynamical Systems 53D05 We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian diffeomorphism that is arbitrarily $C^\infty$-close to the identity. As a consequence, every countably $n$-rectifiable and $n$-negligible subset of $\mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding method can be modified to show that the Gromov width of $B^{2n}_1\setminus A$ equals $π$ for every countably $(n-1)$-rectifiable closed subset $A$ of the open unit ball $B^{2n}_1$. This means that $A$ is not a barrier. |
| title | Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets |
| topic | Symplectic Geometry Dynamical Systems 53D05 |
| url | https://arxiv.org/abs/2408.17444 |