Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere
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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.References
- Peter Albers, Joel W. Fish, Urs Frauenfelder, Helmut Hofer, and Otto van Koert, Global surfaces of section in the planar restricted 3-body problem, Arch. Ration. Mech. Anal. 204 (2012), no. 1, 273–284. MR 2898741, DOI 10.1007/s00205-011-0475-2
- Peter Albers, Urs Frauenfelder, Otto van Koert, and Gabriel P. Paternain, Contact geometry of the restricted three-body problem, Comm. Pure Appl. Math. 65 (2012), no. 2, 229–263. MR 2855545, DOI 10.1002/cpa.21380
- J. W. Alexander, A lemma on systems of knotted curves, Proc. Natl. Acad. Sci. USA, 9 (1923), 93–95.
- Cengiz Aydin, The mathematics behind the anomalistic and draconitic month, 2019, Master Thesis –Universität Augsburg.
- Daniel Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87–161 (French). MR 753131
- George D. Birkhoff, The restricted problem of three bodies, Rend. Circ. Mat. Palermo 39 (1915), 265–334.
- Frederic Bourgeois, A Morse-Bott approach to contact homology, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–Stanford University. MR 2703292
- F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888. MR 2026549, DOI 10.2140/gt.2003.7.799
- R. C. Churchill, G. Pecelli, and D. L. Rod, A survey of the Hénon-Heiles Hamiltonian with applications to related examples, Stochastic behavior in classical and quantum Hamiltonian systems (Volta Memorial Conf., Como, 1977) Lecture Notes in Phys., vol. 93, Springer, Berlin-New York, 1979, pp. 76–136. MR 550890
- Kai Cieliebak and Urs Adrian Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math. 239 (2009), no. 2, 251–316. MR 2461235, DOI 10.2140/pjm.2009.239.251
- C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol′d, Invent. Math. 73 (1983), no. 1, 33–49. MR 707347, DOI 10.1007/BF01393824
- Charles Conley and Eduard Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), no. 2, 207–253. MR 733717, DOI 10.1002/cpa.3160370204
- Adrian Constantin and Boris Kolev, The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math. (2) 40 (1994), no. 3-4, 193–204. MR 1309126
- Max Dörner, Hansjörg Geiges, and Kai Zehmisch, Finsler geodesics, periodic Reeb orbits, and open books, Eur. J. Math. 3 (2017), no. 4, 1058–1075. MR 3736798, DOI 10.1007/s40879-017-0158-0
- Dragomir L. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Comm. Pure Appl. Math. 57 (2004), no. 6, 726–763. MR 2038115, DOI 10.1002/cpa.20018
- Yakov Eliashberg, Contact $3$-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165–192 (English, with French summary). MR 1162559, DOI 10.5802/aif.1288
- Christian Evers, Real contact geometry, 2017. Thesis (Ph.D.)–Universität zu Köln.
- Urs Frauenfelder and Jungsoo Kang, Real holomorphic curves and invariant global surfaces of section, Proc. Lond. Math. Soc. (3) 112 (2016), no. 3, 477–511. MR 3474481, DOI 10.1112/plms/pdw003
- Urs Frauenfelder and Otto van Koert, The restricted three-body problem and holomorphic curves, Pathways in Mathematics, Birkhäuser/Springer, Cham, 2018. MR 3837531, DOI 10.1007/978-3-319-72278-8
- Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738, DOI 10.1017/CBO9780511611438
- Roberto Giambò, Fabio Giannoni, and Paolo Piccione, Multiple brake orbits in $m$-dimensional disks, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2553–2580. MR 3412384, DOI 10.1007/s00526-015-0875-5
- Adam Harris and Gabriel P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders, Ann. Global Anal. Geom. 34 (2008), no. 2, 115–134. MR 2425525, DOI 10.1007/s10455-008-9111-2
- M. Hénon, Numerical exploration of the restricted problem. V. Hill’s case: periodic orbits and their stability, Astron. Astrophys. 1 (1969), 223–238.
- Michel Hénon and Carl Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J. 69 (1964), 73–79. MR 158746, DOI 10.1086/109234
- G. W. Hill, Researches in the Lunar Theory, Amer. J. Math. 1 (1878), no. 2, 129–147. MR 1507853, DOI 10.2307/2369304
- H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515–563. MR 1244912, DOI 10.1007/BF01232679
- H. Hofer, K. Wysocki, and E. Zehnder, A characterisation of the tight three-sphere, Duke Math. J. 81 (1995), no. 1, 159–226 (1996). A celebration of John F. Nash, Jr. MR 1381975, DOI 10.1215/S0012-7094-95-08111-3
- H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995), no. 2, 270–328. MR 1334869, DOI 10.1007/BF01895669
- H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 3, 337–379 (English, with English and French summaries). MR 1395676, DOI 10.1016/s0294-1449(16)30108-1
- H. Hofer, K. Wysocki, and E. Zehnder, Unknotted periodic orbits for Reeb flows on the three-sphere, Topol. Methods Nonlinear Anal. 7 (1996), no. 2, 219–244. MR 1481697, DOI 10.12775/TMNA.1996.010
- H. Hofer, K. Wysocki, and E. Zehnder, Correction to: “A characterisation of the tight three-sphere” [Duke Math. J. 81 (1995), no. 1, 159–226 (1996); MR1381975 (97a:53043)], Duke Math. J. 89 (1997), no. 3, 603–617. MR 1470344, DOI 10.1215/S0012-7094-97-08925-0
- H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2) 148 (1998), no. 1, 197–289. MR 1652928, DOI 10.2307/120994
- H. Hofer, K. Wysocki, and E. Zehnder, A characterization of the tight $3$-sphere. II, Comm. Pure Appl. Math. 52 (1999), no. 9, 1139–1177. MR 1692144, DOI 10.1002/(SICI)1097-0312(199909)52:9<1139::AID-CPA5>3.3.CO;2-C
- H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectizations. III. Fredholm theory, Topics in nonlinear analysis, Progr. Nonlinear Differential Equations Appl., vol. 35, Birkhäuser, Basel, 1999, pp. 381–475. MR 1725579
- H. Hofer, K. Wysocki, and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2) 157 (2003), no. 1, 125–255. MR 1954266, DOI 10.4007/annals.2003.157.125
- Umberto Hryniewicz, Fast finite-energy planes in symplectizations and applications, Trans. Amer. Math. Soc. 364 (2012), no. 4, 1859–1931. MR 2869194, DOI 10.1090/S0002-9947-2011-05387-0
- Umberto Hryniewicz and Pedro A. S. Salomão, On the existence of disk-like global sections for Reeb flows on the tight 3-sphere, Duke Math. J. 160 (2011), no. 3, 415–465. MR 2852366, DOI 10.1215/00127094-1444278
- Umberto L. Hryniewicz, Systems of global surfaces of section for dynamically convex Reeb flows on the 3-sphere, J. Symplectic Geom. 12 (2014), no. 4, 791–862. MR 3333029, DOI 10.4310/JSG.2014.v12.n4.a5
- Umberto L. Hryniewicz, Joan E. Licata, and Pedro A. S. Salomão, A dynamical characterization of universally tight lens spaces, Proc. Lond. Math. Soc. (3) 110 (2015), no. 1, 213–269. MR 3299604, DOI 10.1112/plms/pdu043
- Umberto L. Hryniewicz and Pedro A. S. Salomão, Global properties of tight Reeb flows with applications to Finsler geodesic flows on $S^2$, Math. Proc. Cambridge Philos. Soc. 154 (2013), no. 1, 1–27. MR 3002580, DOI 10.1017/S0305004112000333
- Umberto L. Hryniewicz and Pedro A. S. Salomão, Elliptic bindings for dynamically convex Reeb flows on the real projective three-space, Calc. Var. Partial Differential Equations 55 (2016), no. 2, Art. 43, 57. MR 3485982, DOI 10.1007/s00526-016-0975-x
- Jungsoo Kang, Some remarks on symmetric periodic orbits in the restricted three-body problem, Discrete Contin. Dyn. Syst. 34 (2014), no. 12, 5229–5245. MR 3223870, DOI 10.3934/dcds.2014.34.5229
- Jungsoo Kang, On reversible maps and symmetric periodic points, Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1479–1498. MR 3789173, DOI 10.1017/etds.2016.71
- T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta Math. 42 (1920), no. 1, 99–144 (French). MR 1555161, DOI 10.1007/BF02404404
- J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Comm. Pure Appl. Math. 23 (1970), 609–636. MR 269931, DOI 10.1002/cpa.3160230406
- Hans-Bert Rademacher, A sphere theorem for non-reversible Finsler metrics, Math. Ann. 328 (2004), no. 3, 373–387. MR 2036326, DOI 10.1007/s00208-003-0485-y
- Hans-Bert Rademacher, The length of a shortest geodesic loop, C. R. Math. Acad. Sci. Paris 346 (2008), no. 13-14, 763–765 (English, with English and French summaries). MR 2427078, DOI 10.1016/j.crma.2008.06.001
- Joel Robbin and Dietmar Salamon, The Maslov index for paths, Topology 32 (1993), no. 4, 827–844. MR 1241874, DOI 10.1016/0040-9383(93)90052-W
- A. Schneider, Global surfaces of section for dynamically convex Reeb flows on lens spaces, Trans. Amer. Math. Soc. 373 (2020), no. 4, 2775–2803. MR 4069233, DOI 10.1090/tran/8027
- Matthias Schwarz, Cohomology operations from $S^1$-cobordisms in Floer homology, 1995. Thesis (Ph.D.)–ETH Zürich.
- H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z. 51 (1948), 197–216 (German). MR 25693, DOI 10.1007/BF01291002
- Richard Siefring, Finite-energy pseudoholomorphic planes with multiple asymptotic limits, Math. Ann. 368 (2017), no. 1-2, 367–390. MR 3651577, DOI 10.1007/s00208-016-1478-y
- P. A. Smith, Transformations of finite period, Ann. of Math. (2) 39 (1938), no. 1, 127–164. MR 1503393, DOI 10.2307/1968718
- Otto van Koert, A Reeb flow on the three-sphere without a disk-like global surface of section, Qual. Theory Dyn. Syst. 19 (2020), no. 1, Paper No. 36, 16. MR 4059964, DOI 10.1007/s12346-020-00368-3
- B. Zhou and C. Zhu, Fredholm theory for pseudoholomorphic curves with brake symmetry, Front. Math. China (2021). DOI 10.1007/s11464-021-0935-4.
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