Consecutive collision orbits in the restricted three-body problem above the first critical energy value

Abstract

In this paper, we study the planar circular restricted three-body problem for energy levels slightly above the first critical value. We first observe that the energy hypersurfaces in the Birkhoff regularization corresponding to these energy levels are of contact type. Then, using a version of Rabinowitz Floer homology, we establish the existence of either a periodic symmetric collision orbit or infinitely many symmetric consecutive collision orbits. Furthermore, by an analytic continuation argument, for generic mass ratios and energy levels, we prove that there is no periodic symmetric collision orbit with odd number of collisions. This in turn implies the existence of at least two symmetric consecutive collision orbits.