Abstract:
We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a $C^\infty$-smooth Hamiltonian
circle action, which is persistent under small integrable $C^\infty$ perturbations.
We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly
equivalent (in the non-symplectic sense) to a standard model. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing that
every symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on the
connected components of the common level sets.
Keywords:Liouville integrability, parabolic orbit, circle action, structural stability, normal
forms.
References