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| The existence question of global surfaces of section on energy levels of Hamiltonian systems with two degress of freedom goes back to Poincar\'e and his studies on the three body problem. In the planar restricted circular case with small mass ratio and energy, Poincar\'e considers an annular global surface of section bounded by the direct and retrograde orbits. The study of the associated first return map motivated him to state what is known today as the Poincar\'e-Birkhoff Theorem, a result that has been a strong driving force in the field of Dynamical Systems for the last century.
The above example is only one of many other important systems given by the Hamiltonian dynamics on contact-type energy levels contactomorphic to $SO(3)$ with its standard contact structure. Other such important systems are: geodesic flows of Riemannian or Finsler metrics on the two sphere, magnetic flows on the two sphere for high energy levels, two vortices moving on the two sphere etc.
Thus, it becomes of relevance to investigate under which conditions one can find annulus-like global surfaces of section of Reeb flows for the standard contact structure on $SO(3)$. The old theorem of Birkhoff on the existence of such annuli for positively curved Riemannian metrics on the two sphere serves a guide.
Our main result discussed in this talk, which is joint work with Pedro Salom\~ao and Kris Wysocki, states that if such a Reeb flow on $SO(3)$ admits two closed orbits forming a Hopf link $L$, then these orbits bound an annular global surface of section provided that: (1) all periodic trajectories have positive Conley-Zehnder indices, and (2) all periodic trajectories in the complement of $L$ are positively linked with~$L$. These conditions are easily verified for positively curved geodesic flows on the two sphere. |
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