Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Relative Poincaré-Birkhoff theorem

Agustin Moreno, Arthur Limoge

TL;DR

This work proves a relative Poincaré-Birkhoff theorem for Liouville domains with Lagrangians having Legendrian boundary, guaranteeing interior chords of arbitrarily large order under a Hamiltonian twist condition. It introduces local wrapped Floer cohomology $HW_ ext{loc}^{*}(x)$ for open strings and a local-to-global spectral sequence converging to the global wrapped Floer cohomology $HW^{*}(\,\widehat{L}_0,\widehat{L}_1)$, enabling analysis even for degenerate Hamiltonians. The framework is applied to the SCR3BP, recasting consecutive collision orbits as chords between collision Lagrangians on an open book, and proving the existence of infinitely many spatial collision orbits in the convexity regime. The results connect Floer-theoretic invariants to celestial mechanics and suggest new gravitational-assist paths via long interior chords.

Abstract

In arXiv:2011.06562, the first author and Otto van Koert proved a generalized 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 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.

A Relative Poincaré-Birkhoff theorem

TL;DR

This work proves a relative Poincaré-Birkhoff theorem for Liouville domains with Lagrangians having Legendrian boundary, guaranteeing interior chords of arbitrarily large order under a Hamiltonian twist condition. It introduces local wrapped Floer cohomology for open strings and a local-to-global spectral sequence converging to the global wrapped Floer cohomology , enabling analysis even for degenerate Hamiltonians. The framework is applied to the SCR3BP, recasting consecutive collision orbits as chords between collision Lagrangians on an open book, and proving the existence of infinitely many spatial collision orbits in the convexity regime. The results connect Floer-theoretic invariants to celestial mechanics and suggest new gravitational-assist paths via long interior chords.

Abstract

In arXiv:2011.06562, the first author and Otto van Koert proved a generalized 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 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.
Paper Structure (8 sections, 15 theorems, 43 equations, 1 figure)

This paper contains 8 sections, 15 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on wrapped Floer cohomology
  3. Setup.
  4. Local wrapped Floer cohomology: Definition
  5. Local wrapped Floer cohomology: Properties
  6. Collision orbits in the circular restricted three-body problem
  7. Proof of Theorem \ref{['thm:longintchords']}
  8. Collision orbits in the rotating Kepler problem

Key Result

Theorem A

Suppose that $\tau$ is an exact symplectomorphism of a connected Liouville domain $(W,\lambda)$. Let $\alpha:=\lambda\vert_{\partial W}$, and $L\subset (W,\lambda)$ be an exact, spin, Lagrangian with Legendrian boundary. Assume the following: Then $\tau$ admits infinitely many interior chords with respect to $L$, of arbitrary large order, and which are not sub-chords of any periodic chord.

Figures (1)

  • Figure 1: In the subcritical case of the SCR3BP, for suitably chosen page, the spatial collision locus corresponds to one of the Lagrangian thimbles of the standard Lefschetz fibration on $\mathbb{D}^*\mathbb{S}^2$ (shown above). Its boundary, the corresponding vanishing cycle, is the planar collision locus; this is a circle which can be identified with the zero section of a regular fiber. In the case where the planar problem admits an adapted open book (e.g. when the planar dynamics is dynamically convex HSW), the existence of an adapted Lefschetz fibration as above was shown in Mo. In this case, this vanishing cycle can be thought of as the zero section of an annuli-like global surface of section for the planar problem.

Theorems & Definitions (31)

  • Definition 1.1
  • Remark 1.2
  • Theorem A: Long interior chords
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Corollary 2.4
  • Proposition 2.5
  • ...and 21 more