I'm stuck with the following statement in the book Morse Theory and Floer Homology written by Audin(Proof of Proposition 6.1.6):
If $W$ is a compact symplectic manifold and $H\colon W\to\mathbb{R} $ is a Hamiltonian wichi his $C^2 $-sufficiently small, then there is a finite cover of Darboux charts such that
- Every closed orbit of period $1$ of the Hamiltonian flow $X_H $ is contained in one chart;
- We have $\Vert dX_H \Vert <2\pi $ on every chart.