Topological Decoupling Near Planar Parabolic Orbits

Abstract

In two different three body problems, oscillatory orbits have been shown to exist for the three-body problem in Celestial Mechanics: Sitnikov, Alekseev, Moser, and McGehee considered a spatial problem which had one degree of freedom; Easton, McGehee, and Xia considered a planar problem which had at least three degrees of freedom. Both situations involve analyzing the motion as one particle with mass $m_3$ goes to infinity while the other two masses stay bounded in elliptic motion. Motion with $m_3$ at infinity corresponds to a periodic orbit in the first problem and the Hopf flow on $S^3$ in the second problem, both of which are normally degenerately hyperbolic. The proof of the existence of oscillatory orbits uses stable and unstable manifolds for these degenerate cases. In order to get the symbolic dynamics which shows the existence of oscillation, the orbits which go near infinity need to be controlled for an unbounded length of time. In this paper, we prove that the flow near infinity for the Easton–McGehee example with three degrees of freedom is topologically equivalent to a product flow, i.e., a Grobman–Hartman type theorem in the degenerate situation.