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.


1 Answer 1


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

n-Transitivity of certain diffeomorphism groups


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

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


  • $\begingroup$ 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
  • 1
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.