Vu Ngoc's Conjecture on focusfocus singular fibers with multiple pinched points
Abstract
We classify, up to symplectomorphisms, a neighborhood of a singular fiber of an integrable system (which is proper and has connected fibers) containing $k > 1$ focusfocus critical points. Our result shows that there is a onetoone correspondence between such neighborhoods and $k$ formal power series, up to a $(\mathbb{Z}_2 \times D_k)$action, where $D_k$ is the $k$th dihedral group. The $k$ formal power series determine the dynamical behavior of the Hamiltonian vector fields $X_{f_1}, X_{f_2}$ associated to the components $f_1, f_2 \colon (M, \omega) \to \mathbb{R}$ of the integrable system on the symplectic manifold $(M,\omega)$ via the differential equation $\omega(X_{f_i}, \cdot) = \mathop{}\!\mathrm{d} f_i$, near the singular fiber containing the $k$ focusfocus critical points. This proves a conjecture of San Vu Ngoc from 2002.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.00998
 Bibcode:
 2018arXiv180300998P
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Dynamical Systems;
 53D20;
 70H06
 EPrint:
 Presentation improved and minor corrections not affecting the main results. 29 pages, 3 figures