# Symplectic diffeomorphism of the cylinder moving a point to 0

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.

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

• 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. Commented Mar 6 at 14:38
• @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. Commented Mar 7 at 10:04
• 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. Commented Mar 7 at 12:17