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Example of symplectic and hamiltonian diffeomorphism on $S^2$ and $T^2$
The difference between hamiltonian diffeomorphism and symplectomorphism that is isotopic to the identity is quite small and describe by the flux homomorphism (The breakthrough by Banyaga). So any symplectic isotopy (path to identity in the group of symplectiomorphisms) with zero flux is ended by an Hamiltonian diffeo. Precisely speaking, the isotopy can be made homotopic to hamiltonian isotopy having the same ends. |
Aug
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Example of symplectic and hamiltonian diffeomorphism on $S^2$ and $T^2$
Sure. More than being homotopy equivalent it deformation retracts to $SO(3)$. One can find the proof in Mu-Tao's paper mrlonline.org/mrl/2001-008-005/2001-008-005-007.pdf. But the interst here is to handle an explicit diffeomorphism like one can write for example $f:\mathbb R\longrightarrow \mathbb R, x\mapsto \frac{x}{x^2+4}\sin(x)$ and whatever. |
Aug
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Example of symplectic and hamiltonian diffeomorphism on $S^2$ and $T^2$
Thanks all for your interest to my question. First of all, sorry for not being precise. So could use on $S^2$ $\sin\varphi d\theta\wedge d\varphi)$ in spherical coordinate or $d\theta\wedge d z$ in cylindrical coordinate. Feel free to consider the one that can be helpful. Besides, for $T^2$, $d\theta\wedge d\varphi$ where $\theta, \varphi \in S^1$ are general angular coordinates. |
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