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

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**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