Elliptic bindings for dynamically convex Reeb flows on the real projective three-space

TL;DR

This paper proves the existence of special elliptic-parabolic Reeb orbits on dynamically convex contact forms on real projective three-space and relates these to open book decompositions, with applications to the three-body problem.

Contribution

It establishes the existence of elliptic-parabolic Reeb orbits with specific properties on $ r P^3$ and links periodic orbits to rational open book decompositions in lens spaces.

Findings

Existence of elliptic-parabolic Reeb orbits near convex forms on $ r P^3$

Periodic orbits form bindings of rational open book decompositions

Application to the planar circular restricted three-body problem

Abstract

The first result of this paper is that every contact form on $\mathbb{R} P^3$ sufficiently $C^\infty$-close to a dynamically convex contact form admits an elliptic-parabolic closed Reeb orbit which is $2$-unknotted, has self-linking number $-1/2$ and transverse rotation number in $(1/2,1]$. Our second result implies that any $p$-unknotted periodic orbit with self-linking number $-1/p$ of a dynamically convex Reeb flow on a lens space of order $p$ is the binding of a rational open book decomposition, whose pages are global surfaces of section. As an application we show that in the planar circular restricted three-body problem for energies below the first Lagrange value and large mass ratio, there is a special link consisting of two periodic trajectories for the massless satellite near the smaller primary -- lunar problem -- with the same contact-topological and dynamical properties of the orbits found by Conley in~\cite{conley} for large negative energies. Both periodic trajectories bind rational open book decompositions with disk-like pages which are global surfaces of section. In particular, one of the components is an elliptic-parabolic periodic orbit.