Computation of a separatrix map and a normally hyperbolic invariant lamination for the RP3BP

Abstract

In this paper we discuss the existence of a normally hyperbolic invariant lamination (NHIL) at the Kirkwood gap $3:1$ for the Restricted Planar Elliptic 3 Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. We also show that the induced dynamics on the NHIL is a partially hyperbolic skew-shift which is of the form \[ f:(ω,I,θ)\to (σω, I+e_0 A_ω(I)\cos(θ+ψ_ω)+\mathcal{O}(e^2_0), θ+Ω_ω(I)+\mathcal{O}(e_0)),\] where $I\in [a,b],\hat{I}¸\in \mathbb{T},\ddot{I}‰\in \hat{I}\pounds =\{0,1{\}}^{\mathbb{Z}}$, the space of sequences of $0,1$'s, $\ddot{I}ƒ:\hat{I}\pounds \to \hat{I}\pounds$ is the shift in this space, $\hat{I}{\text{c◯}}_{\ddot{I}}‰$ is the shear, $A_{\ddot{I}}‰$ is an amplitude, and $e_0$ is the eccentricity of Jupiter, which is taken as a small parameter. In the companion paper arXiv:2603.19894, relying on these skew-shift, we show the existence of stochastic diffusing behavior for Asteroids belonging to the Kirkwood gap provided the eccentricity of Jupiter is $e_0$ small enough. Key ingredients to construct the NHIL are the separatrix map associated to homoclinic channels to a normally hyperbolic invariant cylinder and an isolating block construction. Some of the necessary non-degeneracy conditions are verified numerically.

58 pages, 19 figures. Minor changes to the bibliography