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

• Main Authors: Hryniewicz, Umberto L, Salomão, Pedro A. S
• Format: Journal Article
• Language: English
• Published: 11-05-2015
• Subjects: Mathematics - Dynamical Systems; Mathematics - Symplectic Geometry
• DOI: 10.48550/arxiv.1505.02713

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
in~\cite{conley} 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.