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 $(L(p, 1),\xi_{\mathrm{std}})$, $p>1,$ admits a $p$-unknotted closed Reeb orbit $P$ 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 $p$-th iterate of $P$ is $3$. We also check dynamical convexity in the H\'enon-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 $\mathbb{Z}_{3}$-symmetric periodic orbit and the flow descends to a Reeb flow on the standard tight $(L(3,2),\xi_{\mathrm{std}})$.