Finite energy foliations and global dynamics in the restricted three-body problem

We establish a general criterion for the existence of finite energy foliations on contact three-manifolds with boundary, imposing strong global constraints on the associated Reeb flows. Our main abstract result shows that a configuration of periodic orbits, consisting of hyperbolic boundary orbits of Conley-Zehnder index $2$ and an interior orbit of index at least $3$, gives rise to a finite energy foliation, provided that no additional contractible periodic orbit satisfies a specific rotation, linking, and action condition. This identifies the precise dynamical obstruction to the existence of such foliations. These foliations organize the flow in a regime where the dynamics is typically chaotic, and imply the existence of infinitely many periodic orbits and infinitely many homoclinic/heteroclinic orbits to the Lyapunov orbits. We apply this result to the circular planar restricted three-body problem. For mass ratios sufficiently close to $\frac{1}{2}$ and energies slightly above the first Lagrange value, the regularized energy surface $\mathbb{R}P^3 \# \mathbb{R}P^3$ admits a finite energy foliation whose binding consists of the retrograde orbits around the primaries together with the Lyapunov orbit near the first Lagrange point. Moreover, the convexity of the critical energy surface provides a proof of Birkhoff's retrograde orbit conjecture for mass ratios sufficiently close to $\frac{1}{2}$ and all energies below the first Lagrange value.