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 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 ${\mathbb{Z}}_3$-symmetric periodic orbit and the flow descends to a Reeb flow on the standard tight $(L(3,2),{\xi }_{\mathrm{std}})$.

arXiv comment31 pages; 4 figures; v2 Corrected the argument in the application of our results to the Hénon-Heiles hamiltonian and removed the application to Hill's lunar problem. v3 Corrected the argument on page 22. v4 Corrects an error in the proof of the main theorem arising from the incorrect equation (4.14) in the published version. The argument has been revised, and all results remain unchanged