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I am currently reading though part of Zehnder's Lectures on Dynamical Systems. In Chapter VII, I have found myself in the following situation:

  • $Z(1)$ is a subset of standard symplectic space $(\mathbb{R}^{2n},\omega_0)$ with symplectic coordinates $(x,y)= (x_1,\ldots,x_n,y_1,\ldots,y_n)$ defined by $Z(1) = \{(x,y)\in\mathbb{R}^{2n}:x_1^2 +y_1^2 <1\}$.
  • $H:Z(1) \to \mathbb{R}$ is a smooth function.
  • There is an open $U$ and compact $K$ with $U\subseteq K\subseteq Z(1)$ such that $H = 0$ on $U$ and $H$ is also constant on $Z(1) - K$ with value $m(H)$.
  • $0 \leq H(x) \leq m(H)$ for all $x\in Z(1)$.

With this setup, the author would like to be a situation where $U$ in fact contains the point $0$. To this end he suggests (shortly after the statement of Theorem VII.15) that we can precompose $H$ with a symplectic diffeomorphism of $\mathbb{R}^{2n}$ with compact support in the cylinder $Z(1)$ sending $0$ to a point in $U$.

My question is, how do we know such a diffeomorphism exists? If we did not require the differomorphism to be symplectic, I think I could come up with one myself without much difficulty. However, a symplectic diffeomorphism in particular preserves volumes, so the construction of such a diffeomorphism would require a bit more care it seems.

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1 Answer 1

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The symplectic group acts (n)-transitively on connected manifolds of dim $\geq 2$. See Michor and Vizman

n-Transitivity of certain diffeomorphism groups

https://arxiv.org/pdf/dg-ga/9406005.pdf

(1)-transitivity is according to them due to Hatakeyama

Some notes on the groups of automorphisms of contact and symplectic structures

https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-18/issue-3/Some-notes-on-the-group-of-automorphisms-of-contact-and/10.2748/tmj/1178243425.pdf

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  • $\begingroup$ Thanks, this is a helpful resource. I was hoping that there would be a very elementary solution that I was not seeing, given the brevity of the author's remark but perhaps that is not the case. $\endgroup$
    – user14334
    Mar 6 at 14:38
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    $\begingroup$ @user14334: I think you can do this in this case explicitly, by writing down explicit hamiltonians whose flow moves points around how you want. But it will be a bit messy. $\endgroup$
    – Thomas Rot
    Mar 7 at 10:04
  • $\begingroup$ This is exactly what I was looking for! I can just pick a $H$ on e.g. a cylinder containing $0$ and a point in $p\in U$ such that $dH = -\omega(p, \cdot)$. Then I can extend $H$ by smoothly interpolating to zero not so far outside the cylinder. That way $X_H = p$ along the line segment from $0$ to $p$, and the time 1 flow is compactly supported. $\endgroup$
    – user14334
    Mar 7 at 12:17

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