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.
