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A Relative Poincar\'e-Birkhoff theorem

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arxiv 2408.06919 v2 pith:X4UPIOS2 submitted 2024-08-13 math.SG

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keywords versionarbitraryarxive-birkhofffloerhomologylargelength
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Poincar\'e--Birkhoff theorem for $C^0$-Hamiltonian maps

    math.SG 2025-06 reject novelty 7.0 of 10

    A C0-Hamiltonian twist map on a Liouville domain with symplectic cohomology nonzero in infinitely many degrees is claimed to have arbitrarily long interior periodic orbits, but the proof's action-growth exclusion of b...

  2. Bi-normal trajectories in the Circular Restricted Three-Body Problem

    math.SG 2024-12 conditional novelty 6.0 of 10

    This note reduces an existence question about bi-normal orbits in the spatial three-body problem to wrapped Floer homology, yielding a conditional proof that relies on an unverified twist assumption.

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