Elliptic bindings for dynamically convex Reeb flows on the real projective three-space (Q283583)

In this paper, Hamiltonian flows restricted to dynamically convex energy levels diffeomorphic to \(\mathbb RP^3\) are studied. The first main result is the following Theorem: Let \(\lambda\) be any dynamically convex contact form on \(\mathbb RP^3=L(2,1)\). There exists \(T>0\) and a \(C^{\infty}\)-neighborhood \(U\) of \(\lambda\) such that: every contact form in \(U\) admits a 2-unknotted elliptic-parabolic periodic Reeb trajectory with self-linking number \(-1/2\) and prime period smaller than \(T\). Its transverse rotation number associated to the prime period belongs to \((1/2,1]\). The second main 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. Applications to celestial mechanics and to Finsler geodesic flows are obtained. Methods come from symplectic dynamics, where the theory of pseudo-holomorphic curves plays a central role.