Flexibility of the adjoint action of the group of Hamiltonian diffeomorphisms
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929636980555776 |
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| author | Buhovsky, Lev Stokić, Maksim |
| author_facet | Buhovsky, Lev Stokić, Maksim |
| contents | On a closed and connected symplectic manifold, the group of Hamiltonian diffeomorphisms has the structure of an infinite-dimensional Fréchet Lie group, where the Lie algebra is naturally identified with the space of smooth and zero-mean normalized functions, and the adjoint action is given by pullbacks. We show that this action is flexible: for every non-zero smooth and zero-mean normalized function $ u $, any other smooth and zero-mean function $ f $ can be written as a finite sum of elements in the orbit of $u$ under the adjoint action. Additionally, the number of elements in this sum is dominated by the uniform norm of $f$. This result can be interpreted as a (bounded) infinitesimal version of Banyaga's theorem on the simplicity of the group of Hamiltonian diffeomorphisms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_04106 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Flexibility of the adjoint action of the group of Hamiltonian diffeomorphisms Buhovsky, Lev Stokić, Maksim Symplectic Geometry 53D05, 22E65, 58D19 On a closed and connected symplectic manifold, the group of Hamiltonian diffeomorphisms has the structure of an infinite-dimensional Fréchet Lie group, where the Lie algebra is naturally identified with the space of smooth and zero-mean normalized functions, and the adjoint action is given by pullbacks. We show that this action is flexible: for every non-zero smooth and zero-mean normalized function $ u $, any other smooth and zero-mean function $ f $ can be written as a finite sum of elements in the orbit of $u$ under the adjoint action. Additionally, the number of elements in this sum is dominated by the uniform norm of $f$. This result can be interpreted as a (bounded) infinitesimal version of Banyaga's theorem on the simplicity of the group of Hamiltonian diffeomorphisms. |
| title | Flexibility of the adjoint action of the group of Hamiltonian diffeomorphisms |
| topic | Symplectic Geometry 53D05, 22E65, 58D19 |
| url | https://arxiv.org/abs/2303.04106 |