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Suppose that we have the symplectic manifold $(M,\omega)$ where $M$ is a certain annulus, let's say of minimum radius $r_1$ and maximum radius $r_2$, and $\omega=dr\wedge d\theta$. There exists a symplectomorphism $\psi:M\rightarrow M$ that switches the boundaries, given by $\psi(r,\theta)=(r_2-r+r_1,-\theta)$. Now I am wondering if this could be an Hamiltonian diffeomorphism ? Does there exist a Hamiltonian function that could give us this symplectomorphism ?

More generally suppose I have a circle of radius $\tilde r$ with $\tilde r<r_1+\frac{r_2-r_1}{2}$. Could there exist a Hamiltonian symplectomorphism that displaces this circle from itself ?

Any insight is appreciated.

Someone
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  • I'm a bit confused. Is the annulus open, or does it have boundary instead? If it has boundary, then by Hamiltonian diffeomorphism, do you mean an Hamiltonian flow which doesn't move the boundary? tangent to it? something else? – Rei Henigman Feb 27 '23 at 16:47
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    Hamiltonian implies isotopic to the identity. But then it cannot swap the boundary circles. – Moishe Kohan Feb 27 '23 at 16:58
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    Regarding the second point, based on Moishe's comment, you can show that for every $r_1<r<r_2$, the circle of height $r$ is non-displacable since the area below it must be preserved under symplectomorphisms, and that wouldn't be true if the circle is sent "strictly above/below" its original height. – Rei Henigman Feb 27 '23 at 17:20

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