Global surfaces of section for dynamically convex Reeb flows on lens spaces
HTML articles powered by AMS MathViewer
- by A. Schneider PDF
- Trans. Amer. Math. Soc. 373 (2020), 2775-2803 Request permission
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 $(L(3,2),\xi _\mathrm {std})$.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
- Gianni Arioli and Piotr Zgliczyński, Symbolic dynamics for the Hénon-Heiles Hamiltonian on the critical level, J. Differential Equations 171 (2001), no. 1, 173–202. MR 1816799, DOI 10.1006/jdeq.2000.3835
- Kenneth Baker and John Etnyre, Rational linking and contact geometry, Perspectives in analysis, geometry, and topology, Progr. Math., vol. 296, Birkhäuser/Springer, New York, 2012, pp. 19–37. MR 2884030, DOI 10.1007/978-0-8176-8277-4_{2}
- 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
- Richard C. Churchill and David L. Rod, Pathology in dynamical systems. III. Analytic Hamiltonians, J. Differential Equations 37 (1980), no. 1, 23–38. MR 583336, DOI 10.1016/0022-0396(80)90085-6
- Naiara V. de Paulo and Pedro A. S. Salomão, On the multiplicity of periodic orbits and homoclinics near critical energy levels of Hamiltonian systems in $\Bbb R^4$, Trans. Amer. Math. Soc. 372 (2019), no. 2, 859–887. MR 3968790, DOI 10.1090/tran/7568
- Naiara V. de Paulo and Pedro A. S. Salomão, Systems of transversal sections near critical energy levels of Hamiltonian systems in $\Bbb R^4$, Mem. Amer. Math. Soc. 252 (2018), no. 1202, v+105. MR 3778568, DOI 10.1090/memo/1202
- John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418. MR 1161099, DOI 10.1007/BF02100612
- 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
- D. C. Gardiner, M. Hutchings, and D. Pomerleano, Torsion contact forms in three dimensions have two or infinitely many Reeb orbits, arXiv:1701.02262, (2017).
- 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
- 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, 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, 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 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, 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 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 Hryniewicz, Al Momin, and Pedro A. S. Salomão, A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$, Invent. Math. 199 (2015), no. 2, 333–422. MR 3302117, DOI 10.1007/s00222-014-0515-2
- U. L. Hryniewicz and P. A. S. Salomão, Global surfaces of section for reeb flows in dimension three and beyond, Proceedings of the ICM 2018, 1:937–964, 2018.
- Umberto L. Hryniewicz and Pedro A. S. Salomão, Uma introdução à geometria de contato e aplicações à dinâmica hamiltoniana, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2009. 27$^\textrm {o}$ Colóquio Brasileiro de Matemática. [27th Brazilian Mathematics Colloquium]. MR 2538220
- 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 and Pedro A. S. Salomão, Introdução à geometria Finsler, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2013 (Portuguese). 29$^\textrm {o}$ Colóquio Brasileiro de Matemática. [29th Brazilian Mathematics Colloquium]. MR 3100520
- 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
- C. Grotta Ragazzo, Nonintegrability of some Hamiltonian systems, scattering and analytic continuation, Comm. Math. Phys. 166 (1994), no. 2, 255–277. MR 1309550, DOI 10.1007/BF02112316
- Pedro A. S. Salomão, Convex energy levels of Hamiltonian systems, Qual. Theory Dyn. Syst. 4 (2003), no. 2, 439–457 (2004). MR 2129729, DOI 10.1007/BF02970869
- Richard Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math. 61 (2008), no. 12, 1631–1684. MR 2456182, DOI 10.1002/cpa.20224
- Richard Siefring, Intersection theory of punctured pseudoholomorphic curves, Geom. Topol. 15 (2011), no. 4, 2351–2457. MR 2862160, DOI 10.2140/gt.2011.15.2351
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
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 2775-2803
- MSC (2010): Primary 53DXX; Secondary 53D10, 37J55
- DOI: https://doi.org/10.1090/tran/8027
- MathSciNet review: 4069233