We provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by antisymplectic involutions (e.g. the restricted three-body problem). Such cylinders induce continuous paths which do not cross the bifurcation locus of suitable GIT quotients of the symplectic group, which are branched manifolds whose topology provide the desired obstructions. Namely, the complement of the corresponding loci consist of several connected components which we enumerate and explicitly describe; by construction these cannot be joined by a path induced by an orbit cylinder. Our construction extends the notions from Krein theory (which only applies for elliptic orbits), to allow also for the case of symmetric orbits which are hyperbolic. This gives a general theoretical framework for the study of stability and bifurcations of symmetric orbits, with a view towards practical and numerical implementations within the context of space mission design. We shall explore this in upcoming work. CONTENTS 1. Introduction 2. Geometric and dynamical setup 3. The symplectic group, symmetries, and GIT quotients 4. The characteristic polynomial 5. Normal forms 6. Bifurcations and stability Appendix A. The GIT quotient Appendix B. Krein theory and strong stability for Hamiltonian systems References

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We prove a generalization of the classical Poincaré-Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem [MvK]. CONTENTS 1. Introduction 1 2. Motivation and background 5 3. Preliminaries on symplectic homology 6 4. Proof of the Generalized Poincaré-Birkhoff Theorem 11 Appendix A. Hamiltonian twist maps: examples and non-examples 16 Appendix B. Symplectic homology of surfaces 19 Appendix C. On symplectic return maps 20 Appendix D. Strong convexity implies strong index-positivity 24 Appendix E. Strongly index-definite symplectic paths 25 References 27

The intention of this article is to illustrate the use of invariants coming from Floer-type theories, as well as global topological methods, for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g. the three-body problem). In this paper, we illustrate the use of the GIT sequence introduced in [10] by the first and third authors, consisting of a sequence of spaces and maps between them. Roughly speaking, closed orbits of an arbitrary Hamiltonian system induce points in these spaces, and so their topology imposes restrictions on the existence of regular orbit cylinders, as well as encodes information on all types of bifurcations. We also consider the notion of the SFT-Euler characteristic, as the Euler characteristic of the local Floer homology groups associated to a closed orbit of a Hamiltonian system. This number stays invariant before and after a bifurcation, and can be computed in explicit terms from the eigenvalues of the reduced monodromy matrix. This makes it useful for practical applications, e.g. for predicting the existence of orbits for optimization and space mission design. For the case of symmetric orbits (i.e. preserved by an antisymplectic involution), which may be thought both as open or closed strings, we further consider the notion of the real Euler characteristic, as the Euler characteristic of the corresponding Lagrangian Floer homology group. We illustrate the practical use of these invariants, together with the GIT sequence, via numerical work based on the cell-mapping method as described in [16], for the Jupiter-Europa and the Saturn-Enceladus systems. These are currently systems of interest, as they fall in the agenda of space agencies like NASA, as these icy moons are considered candidates for harbouring conditions suitable for extraterrestrial life. CONTENTS 1. Introduction 2. The SFT-Euler characteristic of a periodic orbit 3. The real Euler characteristic of a symmetric orbit 4. Non-symmetric orbits and branching structure 5. Symmetric subtle division 6. Numerics Appendix A. Reduced vs nonreduced monodromy matrices Appendix B. Planar orbits and spatial vs. planar bifurcations Appendix C. Basis changes References

C'est avec l'intuition qu'on trouve, c'est avec la logique qu'on prouve. H. Poincaré.ABSTRACT. We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of an S 1 -family of global hypersurfaces of section for the regularized dynamics, which are copies of (D * S 2 , ω), where ω is deformation equivalent to the standard symplectic form. This is the page of an open book in S * S 3 = S 3 × S 2 , with binding S * S 2 = RP 3 , and the situation at the binding reduces to the planar problem. The first return map is Hamiltonian, restricting to the boundary as the time-1 map of a positive reparametrization of the Reeb flow in RP 3 giving the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly. CONTENTS 1. Introduction 2. The circular restricted three-body problem 3. Stark-Zeeman systems 4. Moser regularization 5. Contact topology and dynamics 6. First order estimates 7. The case of the restricted three-body problem 8. Second order estimates 9. A global hypersurface of section for the connected sum 10. Summary of the proof of the main results Appendix A. Return map for the rotating Kepler problem Appendix B. Symplectic monodromy and return maps References