Wrapped Floer Homology in the Circular Restricted Three-Body Problem

Applications of symplectic geometry to the Three-Body Problem have slowly begun appearing in the past few years, allowing one to approach the problem with tools from Floer theory. In this thesis, we introduce a new variant of Floer theory: Local Wrapped Floer Homology, which generalises the previously-existing 'Wrapped Floer Homology' to degenerate settings. We use this new machinery to prove a generalisation of the famous Poincaré-Birkhoff theorem to open-ended paths with exact Lagrangian ends in a Liouville domain, assuming a twist condition first stated in [arXiv:2011.10386]. We then proceed to improve the applicability of our theorem to real-world problems, by replacing the constraining 'twist condition' mentioned above by a 'Weakened Twist Condition', and by adapting the setup to degenerate Liouville domains. Finally, we deduce applications to the Three-Body Problem: first to prove existence of infinitely many trajectories of collision, and then trajectories bi-normal to the xz-plane; under the assumption of the Weakened Twist Condition.