A generalized Poincar\'e-Birkhoff theorem

TL;DR

This paper extends the classical Poincaré-Birkhoff theorem to higher even dimensions for Liouville domains, inspired by applications to the three-body problem, broadening the theorem's scope in dynamical systems.

Contribution

It provides a new generalized version of the Poincaré-Birkhoff theorem applicable to higher-dimensional Liouville domains, inspired by the existence of global hypersurfaces of section.

Findings

Generalization of Poincaré-Birkhoff theorem to even dimensions

Application to the spatial restricted three-body problem

Existence of global hypersurfaces of section in higher dimensions

Abstract

We prove a generalization of the classical Poincar\'e--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 (as proved by the authors in arXiv:2011.10386).