Global surfaces of section for dynamically convex Reeb flows on lens spaces

Abstract

We show that a dynamically convex Reeb flow on the standard tight lens space ( L ( p , 1 ) , ξ s t d ) (L(p, 1),\xi _\mathrm {std}) , p > 1 , p>1, admits a p p -unknotted closed Reeb orbit P P which is the binding of a rational open book decomposition with disk-like pages. Each page is a rational global surface of section for the Reeb flow and the Conley-Zehnder index of the p p th iterate of P P is 3 3 . We also check dynamical convexity in the Hénon-Heiles system for low positive energies. In this case the rational open book decomposition follows from the fact that the sphere-like component of the energy surface admits a Z 3 \mathbb {Z}_3 -symmetric periodic orbit and the flow descends to a Reeb flow on the standard tight ( L ( 3 , 2 ) , ξ s t d ) (L(3,2),\xi _\mathrm {std}) .