Pseudo-holomorphic dynamics in the restricted three-body problem

TL;DR

This paper extends finite energy foliation concepts to 5-dimensional analogues in the spatial circular restricted three-body problem, revealing abundant fiberwise-recurrent points and inducing Reeb flows on the 3-sphere for better understanding of the dynamics.

Contribution

It introduces a 5D analogue of finite energy foliations for SCR3BP, defines fiberwise-recurrent points, and constructs Reeb flows on the 3-sphere using pseudo-holomorphic curves.

Findings

Existence of 5D finite energy foliations in SCR3BP under convex planar dynamics

Abundance of fiberwise-recurrent points in a perturbative regime

Construction of Reeb flows on the 3-sphere approximating the original dynamics

Abstract

In this article, we identify the 5-dimensional analogue of the finite energy foliations introduced by Hofer--Wysocki--Zehnder for the study of 3-dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by Moser, and show that they exist in abundance for a perturbative regime in the SCR3BP. We then use this foliation to induce a Reeb flow on the standard 3-sphere, via the use of pseudo-holomorphic curves, to be understood as the best approximation of the given dynamics that preserves the foliation. We discuss examples, further geometric structures, and speculate on possible applications.