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

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 (Commun Pure Appl Math 16:449–467, 1963) 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.