Magnetic Katok examples on the two-sphere (Q2830658)

Let \(\sigma\) be a closed 1-form on a manifold \(Q\). The cotangent bundle \(T^*Q\) can be endowed with the symplectic form NEWLINE\[NEWLINE \omega_{\sigma}= d \theta - \pi_Q^* \theta,NEWLINE\]NEWLINE where \(\theta\) is the tautological 1-form on \(T^*Q\) (so that \(d \theta\) is the canonical symplectic structure on the cotangent bundle) and \(\pi_Q:T^*Q \to Q\) is the canonical projection. If a Riemannian metric \(g\) is also given on \(Q\) one can further choose as a Hamiltonian function the associated kinetic energy \(H_g(q,p)=\frac12 |p|_q^2\), where \(|\cdot|_q\) is the norm induced by \(g\) on \(T_q^*Q\). In this case we refer to the flow of the Hamiltonian vector field generated by \(H_g\) via \(\omega_\sigma\) as the magnetic flow of the pair \((g,\sigma)\). The reason for such a terminology is that these dynamical systems describe the motion of a charged particle under the action of a stationary magnetic field. A classical problem is to study the existence of periodic orbits for a magnetic flow on the energy level \(H_g^{-1}(k)\), with \(k\) a positive number. When the manifold \(Q\) is the 2-sphere \(S^2\) and the level is of contact type (see e.g. [\textit{H. Hofer} and \textit{E. Zehnder}, Symplectic invariants and Hamiltonian dynamics. Basel: Birkhäuser (1994; Zbl 0805.58003)]) in \((T^*S^2, \omega_{\sigma})\) it is known that a number of periodic orbits is at least two. If the energy level is dynamical convex then the number of periodic orbits is either exactly two or it is infinite. We refer the reader to [\textit{H. Hofer} et al., Ann. Math. (2) 148, No. 1, 197--289 (1998; Zbl 0944.37031)] for the definition of dynamical convexity. The main result of this paper is that there exist non-exact magnetic flows on \(S^2\) having an energy level whose double cover is strictly contactomorphic to an irrational ellipsoid in \(\mathbb{C}^2\). In particular the flow has exactly two periodic orbits with such an energy level.