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In his search for closed orbits in the planar restricted three-body problem, Poincaré’s approach roughly reduces to:
(1) Finding a global surface of section;
(2) Proving a fixed-point theorem for the resulting return map.
This is the setting for the celebrated Poincaré-Birkhoff theorem. In this talk, I will discuss a generalization of this program to the spatial problem. For the first step, we obtain the existence of global hypersurfaces of section for which the return maps are Hamiltonian, valid for energies below the first critical value and all mass ratios. For the second, we prove a higher-dimensional version of the Poincaré-Birkhoff theorem, which gives infinitely many orbits of arbitrary large period, provided a suitable twist condition is satisfied. Time permitting, we also discuss a construction that associates a Reeb dynamics on a moduli space of holomorphic curves (a copy of the three-sphere), to the given dynamics, and its properties.

This is based on joint work with Otto van Koert.