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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere
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by Seongchan Kim PDF
Trans. Amer. Math. Soc. 375 (2022), 4107-4151 Request permission

Abstract:

We study Reeb dynamics on the three-sphere equipped with a tight contact form and an anti-contact involution. We prove the existence of a symmetric periodic orbit and provide necessary and sufficient conditions for it to bound an invariant disk-like global surface of section. We also study the same questions under the presence of additional symmetry and obtain similar results in this case. The proofs make use of pseudoholomorphic curves in symplectizations. As applications, we study Birkhoff’s conjecture on disk-like global surfaces of section in the planar circular restricted three-body problem and the existence of symmetric closed Finsler geodesics on the two-sphere. We also present applications to some classical Hamiltonian systems.
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Additional Information
  • Seongchan Kim
  • Affiliation: Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea
  • MR Author ID: 1273570
  • ORCID: 0000-0001-8437-7945
  • Email: seongchankim@kongju.ac.kr
  • Received by editor(s): January 28, 2020
  • Received by editor(s) in revised form: March 4, 2021, and June 24, 2021
  • Published electronically: March 16, 2022
  • Additional Notes: This research was supported by the grant 200021-181980/1 of the Swiss National Foundation.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4107-4151
  • MSC (2020): Primary 37J55; Secondary 53D35
  • DOI: https://doi.org/10.1090/tran/8516
  • MathSciNet review: 4419054