A generalized Poincaré-Birkhoff theorem

Abstract

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

Contact geometry in the restricted three-body problem

“…(5) (Long orbits) If W is a global hypersurface of section for some Reeb dynamics, with return map τ , interior periodic points with long (integer) period for τ translates into spatial Reeb orbits with long (real) period. See Appendix C in [MvK2].…”

“…We remark that the first two results are valid for arbitrary mass-ratio and are therefore nonperturbative. We also point out that the second result, while a general fixed-point theorem, hasn't so far seen an application to the SCR3BP, for which the generalized notion of a twist condition introduced in [MvK2] seems, as of yet, perhaps unsuitable. The third result, while of theoretical interest, might perhaps lead to insights on the original problem coming from 3-dimensional dynamics; this is work in progress.…”

“…The current set of notes is an attempt to put into context a series of very recent (and yet unpublished) results of the author in co-authorship with Otto van Koert [MvK,MvK2], and the spinoff [M20] by the author. They also serve the purpose of introducing and threading together a collection of basic and important notions, disseminated across the literature, with the main driving motivation coming from a very old and famous problem; namely, the three-body problem.…”

“…We further review nonperturbative modern results coming from holomorphic curve theory à-la Hofer-Wysocki-Zehnder [HWZ98]. We then introduce the main results of [MvK,MvK2,M20], which include:…”