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

TL;DR

This paper proves the existence of special closed Reeb orbits forming rational open book decompositions on lens spaces, providing new insights into the dynamics of convex Reeb flows and their global surfaces of section.

Contribution

It establishes the existence of p-unknotted closed Reeb orbits forming rational open books on lens spaces, linking dynamical convexity with global surfaces of section.

Findings

Existence of p-unknotted closed Reeb orbit as binding of rational open book

Each page of the open book is a rational global surface of section

Dynamical convexity checked for low-energy Hénon-Heiles system

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 $(L(3,2),\xi_{\mathrm{std}})$.