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

2018/03/16 by Alexsandro Schneider, Schneider, Alexsandro
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1803.06439

openalex publication_date 2018/03/16 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We show that a dynamically convex Reeb flow on the standard tight lens space\n(L(p, 1),\ξ_\std), p>1, admits a p-unknotted closed Reeb orbit\nP which is the binding of a rational open book decomposition with disk-like\npages. Each page is a rational global surface of section for the Reeb flow and\nthe Conley-Zehnder index of the p-th iterate of P is 3. We also check\ndynamical convexity in the H 'enon-Heiles system for low positive energies. In\nthis case the rational open book decomposition follows from the fact that the\nsphere-like component of the energy surface admits a \ℤ₃-symmetric\nperiodic orbit and the flow descends to a Reeb flow on the standard tight\n(L(3,2),\ξ_\std).\n

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