On the symplectic geometry of $A_k$ singularities

This paper presents a complete symplectic classification of $A_k$ Hamiltonians on $\mathbb R^2$, in the analytic and smooth categories. Precisely, consider the pair $(H, \omega)$ consisting of a Hamiltonian and a symplectic structure on $\mathbb R^2$ such that $H$ has an $A_{k-1}$ singularity at the origin with $k\geq 2$. We classify such pairs near the origin, up to fiberwise symplectomorphisms, and up to $H$-preserving symplectomorphisms. The classification is obtained by bringing the pair $(H, \omega)$ to a symplectic normal form $$\big(H = \xi^2 \pm x^k, \ \omega = d (f d \xi)\big), \quad f = \sum_{i=1}^{k-1} x^i f_i(x^k),$$ modulo some relations which are explicitly given. We also show that the group of $H$-preserving symplectomorphisms of an $A_{k-1}$ singularity for $k$ odd consists of symplectomorphisms that can be included into a $C^\infty$-smooth (resp., real-analytic) $H$-preserving flow, whereas for $k$ even with $k \ge 4$ the same is true modulo the $\mathbb Z_2$-subgroup generated by the involution $Inv(x,\xi) = (-x,-\xi)$. The paper is concluded with a brief discussion of the conjecture that the symplectic invariants of $A_{k-1}$ singularities are spectrally determined.