Birkhoff conjecture and finite energy foliations in Hill's lunar problem

Abstract

We prove Birkhoff's retrograde orbit conjecture in Hill's lunar problem by showing that the retrograde orbit bounds a disk-like global surface of section for every energy below the critical value. We also obtain a global description of the dynamics through the critical energy level by constructing finite energy foliations for energies slightly above it. The binding of these foliations consists of the retrograde orbit together with the Lyapunov orbits near the critical points. As a consequence, there exist infinitely many periodic orbits and infinitely many trajectories asymptotic to the Lyapunov orbits. The proof combines pseudo-holomorphic curve techniques with a new convexity theorem for Hill's lunar problem. More precisely, we construct an explicit global symplectic change of coordinates under which the bounded regularized component becomes strictly convex up to the critical energy level. This convexity implies strong dynamical consequences, including lower bounds for the Conley-Zehnder indices of periodic orbits, and allows the application of the Hofer-Wysocki-Zehnder theory of finite energy foliations. As a result, we obtain disk-like global surfaces of section below the critical level and $2-3-2$ foliations for energies slightly above it.