A relative Poincaré–Birkhoff theorem

Abstract A. Moreno and Otto van Koert proved a generalised version of the classical Poincaré–Birkhoff theorem, for Liouville domains of any dimension. In this article, we prove a relative version for Lagrangians with Legendrian boundary. This gives interior chords of arbitrary large length, provided that the twist condition introduced by Moreno and van Koert is satisfied. The motivation comes from finding spatial consecutive collision orbits of arbitrary large length in the spatial circular restricted three‐body problem, which are relevant for gravitational assist in the context of orbital mechanics. This is an application of a local version of wrapped Floer homology, which we introduce as the open string analogue of local Floer homology for closed strings.