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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: A. Schneider
Journal: Trans. Amer. Math. Soc. 373 (2020), 2775-2803
MSC (2010): Primary 53DXX; Secondary 53D10, 37J55
DOI: https://doi.org/10.1090/tran/8027
Published electronically: January 28, 2020
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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})$.


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Additional Information

A. Schneider
Affiliation: Universidade Estadual do Centro-Oeste, Rua Camargo Varela de Sá, $3$, Guarapuava – PR, 85040-080 Brazil
Email: alexsandro@unicentro.br

DOI: https://doi.org/10.1090/tran/8027
Received by editor(s): April 13, 2018
Received by editor(s) in revised form: September 4, 2019
Published electronically: January 28, 2020
Additional Notes: The author was partially supported by CAPES grant 1526852 and CNPq grant 142059/2016-1
Article copyright: © Copyright 2020 American Mathematical Society