A Relative Poincar\'e-Birkhoff theorem

In arXiv:2011.06562, the first author and Otto van Koert proved a generalized version of the classical Poincar\'e-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 the twist condition introduced in arXiv:2011.06562 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.