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

Periodic Reeb orbits on prequantization bundles

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.

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.

[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.

[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.

[4]

H. BerestyckiJ.-M. LasryG. 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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

V. L. GinzburgD. HeinU. 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.

[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.

[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.

[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.

[16]

H. HoferK. 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.

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

W. WangX. 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.

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.

[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.

[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.

[4]

H. BerestyckiJ.-M. LasryG. 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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

V. L. GinzburgD. HeinU. 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.

[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.

[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.

[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.

[16]

H. HoferK. 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.

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

W. WangX. 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.

Figure 1.  The function $h$. The numbers at the graph indicate the slope at this point/section
Figure 2.  The moduli space at $\rho = 0$
Figure 3.  The moduli space at $\rho>0$
[1]

Kei Irie. Dense existence of periodic Reeb orbits and ECH spectral invariants. Journal of Modern Dynamics, 2015, 9: 357-363. doi: 10.3934/jmd.2015.9.357

[2]

Peter Albers, Urs Frauenfelder. Floer homology for negative line bundles and Reeb chords in prequantization spaces. Journal of Modern Dynamics, 2009, 3 (3) : 407-456. doi: 10.3934/jmd.2009.3.407

[3]

Alessandra Celletti, Sara Di Ruzza. Periodic and quasi--periodic orbits of the dissipative standard map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 151-171. doi: 10.3934/dcdsb.2011.16.151

[4]

Alex Eskin, Maryam Mirzakhani. Counting closed geodesics in moduli space. Journal of Modern Dynamics, 2011, 5 (1) : 71-105. doi: 10.3934/jmd.2011.5.71

[5]

Corentin Boissy. Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3433-3457. doi: 10.3934/dcds.2012.32.3433

[6]

Alexandre A. P. Rodrigues. Moduli for heteroclinic connections involving saddle-foci and periodic solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3155-3182. doi: 10.3934/dcds.2015.35.3155

[7]

Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177

[8]

Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505

[9]

Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949

[10]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451

[11]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961

[12]

Dong Eui Chang, David F. Chichka, Jerrold E. Marsden. Lyapunov-based transfer between elliptic Keplerian orbits. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 57-67. doi: 10.3934/dcdsb.2002.2.57

[13]

Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331

[14]

Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032

[15]

Christopher Kumar Anand. Unitons and their moduli. Electronic Research Announcements, 1996, 2: 7-16.

[16]

Alexander Blokh, Clinton Curry, Lex Oversteegen. Density of orbits in laminations and the space of critical portraits. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2027-2039. doi: 10.3934/dcds.2012.32.2027

[17]

Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221

[18]

E. Canalias, Josep J. Masdemont. Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 261-279. doi: 10.3934/dcds.2006.14.261

[19]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

[20]

Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261

2017 Impact Factor: 0.425

Metrics

  • PDF downloads (32)
  • HTML views (159)
  • Cited by (0)

Other articles
by authors

[Back to Top]