Existence of a smooth Hamiltonian circle action near parabolic orbits

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. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the integrals of motion is Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.

References in corpus (2)

• Hidden toric symmetry and structural stability of singularities in integrable systems
• (Vanishing) Twist in the Saddle-Centre and Period-Doubling Bifurcation

Cited by in corpus (5)

• Hidden toric symmetry and structural stability of singularities in integrable systems
• Structurally stable non-degenerate singularities of integrable systems
• Constructions of b-semitoric systems
• Bifurcations of magnetic geodesic flows on surfaces of revolution
• Symplectic classification for universal unfoldings of $A_n$ singularities in integrable systems