The contact property for nowhere vanishing magnetic fields on the two-sphere

In this paper we give some positive and negative results about the contact property for the energy levels $\Sigma_c$ of a symplectic magnetic field on $S^2$. 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. This hints 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 small energies, there exists a convex hypersurface $N_c$ in $\mathbb C^2$ and a covering map $p_c:N_c \rightarrow \Sigma_c$ such that the pull-back via $p_c$ of the characteristic distribution on $\Sigma_c$ is the standard characteristic distribution on $N_c$. As a corollary we prove that there are either two or infinitely many periodic orbits on $\Sigma_c$. The second alternative holds if there exists a contractible prime periodic orbit.