The contact property for symplectic magnetic fields on S2

In this paper we give some positive and negative results about the contact property for the energy levels Σcof a symplectic magnetic field on S2. In the first part we focus on the case of the area form on a surface of revolution. We state a sufficient condition for an energy level to be of contact type and give an example where the contact property fails. If the magnetic curvature is positive, the dynamics and the action of invariant measures can be numerically computed. The collected data hint at the conjecture that an energy level of a symplectic magnetic field with positive magnetic curvature should be of contact type. In the second part we show that, for a small energy c, there exist a convex hypersurface Nc in C2 and a double cover Nc !Σc such that the pull-back of the characteristic distribution on Σc is the standard characteristic distribution on Nc. As a corollary, we prove that there are either two or infinitely many periodic orbits on Σc. The second alternative holds if there exists a contractible prime periodic orbit.