American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra
  • JMD Home
  • This Volume
  • Next Article
    Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions
  2018, 12: 123-150. doi: 10.3934/jmd.2018005

Periodic Reeb orbits on prequantization bundles

Peter Albers 1, , Jean Gutt 2, and  Doris Hein 3,

1. 

Mathematisches Institut, Universität Heidelberg, Mathematikon, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
2. 

Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
3. 

Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, D-79104 Freiburg, Germany

Received  January 17, 2017 Revised  January 16, 2018 Published  April 2018

In this paper, we prove that every graphical hypersurface in a prequantization bundle over a symplectic manifold $M$, pinched between two circle bundles whose ratio of radii is less than $\sqrt{2}$ carries either one short simple periodic orbit or carries at least cuplength $(M)+1$ simple periodic Reeb orbits.

Keywords: Periodic Reeb orbits, cuplength estimate, prequantization bundle, moduli space for a transfer map.
Mathematics Subject Classification: Primary: 53D10; Secondary: 37J45.
Citation: Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005
References:
[1]

P. Albers and D. Hein, Cuplength estimates in Morse cohomology, J. Topol. Anal., 8 (2016), 243-272.  doi: 10.1142/S1793525316500102.  Google Scholar

[2]

P. Albers and A. Momin, Cup-length estimates for leaf-wise intersections, Math. Proc. Cambridge Philos. Soc., 149 (2010), 539-551.  doi: 10.1017/S0305004110000435.  Google Scholar

[3]

M. Abreu and L. Macarini, Multiplicity of periodic orbits for dynamically convex contact forms, J. Fixed Point Theory Appl., 19 (2017), 175-204.  doi: 10.1007/s11784-016-0348-2.  Google Scholar

[4]

H. Berestycki, J.-M. Lasry, G. Mancini and B. Ruf, Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Comm. Pure Appl. Math., 38 (1985), 253-289.  doi: 10.1002/cpa.3160380302.  Google Scholar

[5]

F. Bourgeois and A. Oancea, An exact sequence for contact-and symplectic homology, Invent. Math., 175 (2009), 611-680.  doi: 10.1007/s00222-008-0159-1.  Google Scholar

[6]

F. Bourgeois, A Morse-Bott approach to contact homology, in Symplectic and Contact Topology: Interactions and Perspectives (Toronto, ON/Montreal, QC, 2001), Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, 55-77.  Google Scholar

[7]

D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.  doi: 10.4310/jdg/1452002876.  Google Scholar

[8]

I. Ekeland and J.-M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math., 112 (1980), 283-319.  doi: 10.2307/1971148.  Google Scholar

[9]

U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not., (2004), 2179-2269.  doi: 10.1155/S1073792804133941.  Google Scholar

[10]

H. Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, 2008. doi: 10. 1017/CBO9780511611438.  Google Scholar

[11]

V. L. Ginzburg and B. Gürel, Lusternik-Schnirelman theory and closed Reeb orbits, 2016, arXiv: 1601.03092. Google Scholar

[12]

V. L. Ginzburg, D. Hein, U. Hryniewicz and L. Macarini, Closed Reeb orbits on the sphere and symplectically degenerate maxima, Acta Math. Vietnam, 38 (2013), 55-78.  doi: 10.1007/s40306-012-0002-z.  Google Scholar

[13]

V. L. Ginzburg, On the existence and non-existence of closed trajectories for some Hamiltonian flows, Math. Z., 223 (1996), 397-409.  doi: 10.1007/BF02621606.  Google Scholar

[14]

J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in ${{\mathbb{R}}^{2n}}$, Ann. Inst. Fourier (Grenoble), 66 (2016), 2485-2505.  doi: 10.5802/aif.3069.  Google Scholar

[15]

J. Gutt, The positive equivariant symplectic homology as an invariant for some contact manifolds, J. Symplectic Geom., 15 (2017), 1019-1069.  doi: 10.4310/JSG.2017.v15.n4.a3.  Google Scholar

[16]

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), 270-328.  doi: 10.1007/BF01895669.  Google Scholar

[17]

H. Hofer, K. Wysocki and E. Zehnder, Polyfolds and Fredholm theory I -Basic theory in M-polyfolds, 2014, arXiv: 1407.3185. Google Scholar

[18]

J. Kang, Equivariant symplectic homology and multiple closed Reeb orbits, Internat. J. Math., 24 (2013), 1350096.  doi: 10.1142/S0129167X13500961.  Google Scholar

[19]

E. Kerman, Rigid constellations of closed Reeb orbits, Compos. Math., 153 (2017), 2394-2444.  doi: 10.1112/S0010437X17007448.  Google Scholar

[20]

Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in ${{\mathbf{R}}^{2n}}$, Ann. of Math., 155 (2002), 317-368.  doi: 10.2307/3062120.  Google Scholar

[21]

M. Schwarz, Morse Homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. doi: 10. 1007/978-3-0348-8577-5.  Google Scholar

[22]

M. Schwarz, Cohomology Operations from S1-cobordisms in Floer Homology, Ph. D. -thesis, Swiss Federal Inst. of Techn. Zurich, Diss. ETH No. 11182, 1995. Google Scholar

[23]

D. A. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45 (1992), 1303-1360.  doi: 10.1002/cpa.3160451004.  Google Scholar

[24]

C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., 11 (2007), 2117-2202.  doi: 10.2140/gt.2007.11.2117.  Google Scholar

[25]

W. Wang, X. Hu and Y. Long, Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462.  doi: 10.1215/S0012-7094-07-13931-0.  Google Scholar

show all references

References:
[1]

P. Albers and D. Hein, Cuplength estimates in Morse cohomology, J. Topol. Anal., 8 (2016), 243-272.  doi: 10.1142/S1793525316500102.  Google Scholar

[2]

P. Albers and A. Momin, Cup-length estimates for leaf-wise intersections, Math. Proc. Cambridge Philos. Soc., 149 (2010), 539-551.  doi: 10.1017/S0305004110000435.  Google Scholar

[3]

M. Abreu and L. Macarini, Multiplicity of periodic orbits for dynamically convex contact forms, J. Fixed Point Theory Appl., 19 (2017), 175-204.  doi: 10.1007/s11784-016-0348-2.  Google Scholar

[4]

H. Berestycki, J.-M. Lasry, G. Mancini and B. Ruf, Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Comm. Pure Appl. Math., 38 (1985), 253-289.  doi: 10.1002/cpa.3160380302.  Google Scholar

[5]

F. Bourgeois and A. Oancea, An exact sequence for contact-and symplectic homology, Invent. Math., 175 (2009), 611-680.  doi: 10.1007/s00222-008-0159-1.  Google Scholar

[6]

F. Bourgeois, A Morse-Bott approach to contact homology, in Symplectic and Contact Topology: Interactions and Perspectives (Toronto, ON/Montreal, QC, 2001), Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, 55-77.  Google Scholar

[7]

D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.  doi: 10.4310/jdg/1452002876.  Google Scholar

[8]

I. Ekeland and J.-M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math., 112 (1980), 283-319.  doi: 10.2307/1971148.  Google Scholar

[9]

U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not., (2004), 2179-2269.  doi: 10.1155/S1073792804133941.  Google Scholar

[10]

H. Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, 2008. doi: 10. 1017/CBO9780511611438.  Google Scholar

[11]

V. L. Ginzburg and B. Gürel, Lusternik-Schnirelman theory and closed Reeb orbits, 2016, arXiv: 1601.03092. Google Scholar

[12]

V. L. Ginzburg, D. Hein, U. Hryniewicz and L. Macarini, Closed Reeb orbits on the sphere and symplectically degenerate maxima, Acta Math. Vietnam, 38 (2013), 55-78.  doi: 10.1007/s40306-012-0002-z.  Google Scholar

[13]

V. L. Ginzburg, On the existence and non-existence of closed trajectories for some Hamiltonian flows, Math. Z., 223 (1996), 397-409.  doi: 10.1007/BF02621606.  Google Scholar

[14]

J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in ${{\mathbb{R}}^{2n}}$, Ann. Inst. Fourier (Grenoble), 66 (2016), 2485-2505.  doi: 10.5802/aif.3069.  Google Scholar

[15]

J. Gutt, The positive equivariant symplectic homology as an invariant for some contact manifolds, J. Symplectic Geom., 15 (2017), 1019-1069.  doi: 10.4310/JSG.2017.v15.n4.a3.  Google Scholar

[16]

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), 270-328.  doi: 10.1007/BF01895669.  Google Scholar

[17]

H. Hofer, K. Wysocki and E. Zehnder, Polyfolds and Fredholm theory I -Basic theory in M-polyfolds, 2014, arXiv: 1407.3185. Google Scholar

[18]

J. Kang, Equivariant symplectic homology and multiple closed Reeb orbits, Internat. J. Math., 24 (2013), 1350096.  doi: 10.1142/S0129167X13500961.  Google Scholar

[19]

E. Kerman, Rigid constellations of closed Reeb orbits, Compos. Math., 153 (2017), 2394-2444.  doi: 10.1112/S0010437X17007448.  Google Scholar

[20]

Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in ${{\mathbf{R}}^{2n}}$, Ann. of Math., 155 (2002), 317-368.  doi: 10.2307/3062120.  Google Scholar

[21]

M. Schwarz, Morse Homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. doi: 10. 1007/978-3-0348-8577-5.  Google Scholar

[22]

M. Schwarz, Cohomology Operations from S1-cobordisms in Floer Homology, Ph. D. -thesis, Swiss Federal Inst. of Techn. Zurich, Diss. ETH No. 11182, 1995. Google Scholar

[23]

D. A. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45 (1992), 1303-1360.  doi: 10.1002/cpa.3160451004.  Google Scholar

[24]

C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., 11 (2007), 2117-2202.  doi: 10.2140/gt.2007.11.2117.  Google Scholar

[25]

W. Wang, X. Hu and Y. Long, Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462.  doi: 10.1215/S0012-7094-07-13931-0.  Google Scholar

Figure 1.  The function $h$. The numbers at the graph indicate the slope at this point/section
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The moduli space at $\rho = 0$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The moduli space at $\rho>0$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[2]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090
[3]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[4]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405
[5]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[6]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
[7]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027
[8]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[9]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[10]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[11]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069
[12]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[13]

Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075
[14]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[15]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[16]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[17]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[18]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030
[19]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (205)
  • HTML views (1144)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences