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doi: 10.3934/dcds.2021172
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Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $

Hui Liu 1, and  Ling Zhang 2,,

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China
2. 

School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, Hubei, China

*Corresponding author: Ling Zhang. Supported by NSFC (No. 11701532) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan, Grant number: CUGST2)

Received  April 2021 Revised  September 2021 Early access November 2021

Fund Project: The first author is supported by NSFC (Nos. 11771341, 12022111) and the Fundamental Research Funds for the Central Universities (No. 2042021kf1059)

In this paper, we prove that there exist at least two non-contractible closed Reeb orbits on every dynamically convex $ \mathbb{R}P^{2n-1} $, and if all the closed Reeb orbits are non-degenerate, then there are at least $ n $ closed Reeb orbits, where $ n\geq2 $, the main ingredient is that we generalize some theories developed by I. Ekeland and H. Hofer for closed characteristics on compact convex hypersurfaces in $ {{\bf R}}^{2n} $ to symmetric compact star-shaped hypersurfaces. In addition, we use Ekeland-Hofer theory to give a new proof of a theorem recently by M. Abreu and L. Macarini that every dynamically convex symmetric compact star-shaped hypersurface carries an elliptic symmetric closed characteristic.

Keywords: Real projective spaces, non-contractible, closed Reeb orbits, Hamiltonian systems, dynamically convex, multiplicity.
Mathematics Subject Classification: Primary: 58E05, 37J46; Secondary: 37J39.
Citation: Hui Liu, Ling Zhang. Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021172
References:
[1]

M. Abreu and L. Macarini, Dynamical convexity and elliptic periodic orbits for Reeb flows, Math. Ann., 369 (2017), 331-386.  doi: 10.1007/s00208-017-1532-4.  Google Scholar

[2]

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

[3]

P. Albers, J. W. Fish, U. Frauenfelder, H. Hofer and O. van Koert, Global surfaces of section in the planar restricted 3-body problem, Arch. Ration. Mech. Anal., 204 (2012), 273-284.  doi: 10.1007/s00205-011-0475-2.  Google Scholar

[4]

P. Albers, U. Frauenfelder, O. van Koert and G. P. Paternain, Contact geometry of the restricted three-body problem, Commun. Pure Appl. Math., 65 (2012), 229-263.  doi: 10.1002/cpa.21380.  Google Scholar

[5]

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

[6]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser. Boston. 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[7]

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

[8]

G. Dell'Antoio, B. D'Onofrio and I. Ekeland, Les systèmes hamiltoniens convexes et pairs ne sont pas ergodiques en général, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 1413-1415.   Google Scholar

[9]

H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in R2n, Calc. Var. Partial Differential Equations, 56 (2017), Art. 65, 30 pp. doi: 10.1007/s00526-017-1173-1.  Google Scholar

[10]

H. Duan, H. Liu, Y. Long and W. Wang, Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in R2n, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1-18.  doi: 10.1007/s10114-016-6019-9.  Google Scholar

[11]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[12]

I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys., 113 (1987), 419-469.  doi: 10.1007/BF01221255.  Google Scholar

[13]

V. L. Ginzburg and B. Z. G$\ddot{u}$rel, Lusternik-Schnirelmann theory and closed Reeb orbits, Math. Z., 295 (2020), 515-582.  doi: 10.1007/s00209-019-02361-2.  Google Scholar

[14]

V. L. Ginzburg, B. Z. G$\ddot{u}$rel and L. Macarini, Multiplicity of closed Reeb orbits on prequantization bundles, Israel J. Math., 228 (2018), 407-453.  doi: 10.1007/s11856-018-1769-y.  Google Scholar

[15]

M. Girardi, Multiple orbits for Hamiltonian systems on starshaped surfaces with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 285-294.  doi: 10.1016/S0294-1449(16)30423-1.  Google Scholar

[16]

J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in R2n, Ann. Inst. Fourier (Grenoble), 66 (2016), 2485-2505.  doi: 10.5802/aif.3069.  Google Scholar

[17]

H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math., 148 (1998), 197-289.  doi: 10.2307/120994.  Google Scholar

[18]

H. Hofer, K. Wysocki and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math., 157 (2003), 125-255.  doi: 10.4007/annals.2003.157.125.  Google Scholar

[19]

U. L. Hryniewicz and P. A. S. Salomão, Elliptic bindings for dynamically convex Reeb flows on the real projective three-space, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 43, 57 pp. doi: 10.1007/s00526-016-0975-x.  Google Scholar

[20]

X. Hu and Y. Long, Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n, Sci. China Ser. A, 45 (2002), 1038-1052.  doi: 10.1007/BF02879987.  Google Scholar

[21]

H. Liu and Y. Long, Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces, Calc. Var. Partial Differential Equations, 54 (2015), 3753-3787.  doi: 10.1007/s00526-015-0921-3.  Google Scholar

[22]

H. Liu, Y. Long and W. Wang, Resonance identities for closed characteristics on compact star-shaped hypersurfaces in R2n, J. Funct. Anal., 266 (2014), 5598-5638.  doi: 10.1016/j.jfa.2014.02.035.  Google Scholar

[23]

H. Liu and Y. Xiao, Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, Adv. Math., 318 (2017), 158-190.  doi: 10.1016/j.aim.2017.07.024.  Google Scholar

[24]

Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154 (2000), 76-131.  doi: 10.1006/aima.2000.1914.  Google Scholar

[25]

Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math., 207, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[26]

Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in R2n, Ann. of Math., 155 (2002), 317-368.  doi: 10.2307/3062120.  Google Scholar

[27]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.  Google Scholar

[28]

C. Viterbo, Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 487-489.   Google Scholar

[29]

C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 621-655.  doi: 10.1090/S0002-9947-1989-0978370-5.  Google Scholar

[30]

W. Wang, Closed trajectories on symmetric convex Hamiltonian energy surfaces, Discrete Contin. Dyn. Syst., 32 (2012), 679-701.  doi: 10.3934/dcds.2012.32.679.  Google Scholar

[31]

W. Wang, Closed characteristics on compact convex hypersurfaces in R8, Adv. Math., 297 (2016), 93-148.  doi: 10.1016/j.aim.2016.03.044.  Google Scholar

[32]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.  doi: 10.2307/1971185.  Google Scholar

show all references

References:
[1]

M. Abreu and L. Macarini, Dynamical convexity and elliptic periodic orbits for Reeb flows, Math. Ann., 369 (2017), 331-386.  doi: 10.1007/s00208-017-1532-4.  Google Scholar

[2]

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

[3]

P. Albers, J. W. Fish, U. Frauenfelder, H. Hofer and O. van Koert, Global surfaces of section in the planar restricted 3-body problem, Arch. Ration. Mech. Anal., 204 (2012), 273-284.  doi: 10.1007/s00205-011-0475-2.  Google Scholar

[4]

P. Albers, U. Frauenfelder, O. van Koert and G. P. Paternain, Contact geometry of the restricted three-body problem, Commun. Pure Appl. Math., 65 (2012), 229-263.  doi: 10.1002/cpa.21380.  Google Scholar

[5]

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

[6]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser. Boston. 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[7]

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

[8]

G. Dell'Antoio, B. D'Onofrio and I. Ekeland, Les systèmes hamiltoniens convexes et pairs ne sont pas ergodiques en général, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 1413-1415.   Google Scholar

[9]

H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in R2n, Calc. Var. Partial Differential Equations, 56 (2017), Art. 65, 30 pp. doi: 10.1007/s00526-017-1173-1.  Google Scholar

[10]

H. Duan, H. Liu, Y. Long and W. Wang, Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in R2n, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1-18.  doi: 10.1007/s10114-016-6019-9.  Google Scholar

[11]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[12]

I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys., 113 (1987), 419-469.  doi: 10.1007/BF01221255.  Google Scholar

[13]

V. L. Ginzburg and B. Z. G$\ddot{u}$rel, Lusternik-Schnirelmann theory and closed Reeb orbits, Math. Z., 295 (2020), 515-582.  doi: 10.1007/s00209-019-02361-2.  Google Scholar

[14]

V. L. Ginzburg, B. Z. G$\ddot{u}$rel and L. Macarini, Multiplicity of closed Reeb orbits on prequantization bundles, Israel J. Math., 228 (2018), 407-453.  doi: 10.1007/s11856-018-1769-y.  Google Scholar

[15]

M. Girardi, Multiple orbits for Hamiltonian systems on starshaped surfaces with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 285-294.  doi: 10.1016/S0294-1449(16)30423-1.  Google Scholar

[16]

J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in R2n, Ann. Inst. Fourier (Grenoble), 66 (2016), 2485-2505.  doi: 10.5802/aif.3069.  Google Scholar

[17]

H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math., 148 (1998), 197-289.  doi: 10.2307/120994.  Google Scholar

[18]

H. Hofer, K. Wysocki and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math., 157 (2003), 125-255.  doi: 10.4007/annals.2003.157.125.  Google Scholar

[19]

U. L. Hryniewicz and P. A. S. Salomão, Elliptic bindings for dynamically convex Reeb flows on the real projective three-space, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 43, 57 pp. doi: 10.1007/s00526-016-0975-x.  Google Scholar

[20]

X. Hu and Y. Long, Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n, Sci. China Ser. A, 45 (2002), 1038-1052.  doi: 10.1007/BF02879987.  Google Scholar

[21]

H. Liu and Y. Long, Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces, Calc. Var. Partial Differential Equations, 54 (2015), 3753-3787.  doi: 10.1007/s00526-015-0921-3.  Google Scholar

[22]

H. Liu, Y. Long and W. Wang, Resonance identities for closed characteristics on compact star-shaped hypersurfaces in R2n, J. Funct. Anal., 266 (2014), 5598-5638.  doi: 10.1016/j.jfa.2014.02.035.  Google Scholar

[23]

H. Liu and Y. Xiao, Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$, Adv. Math., 318 (2017), 158-190.  doi: 10.1016/j.aim.2017.07.024.  Google Scholar

[24]

Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math., 154 (2000), 76-131.  doi: 10.1006/aima.2000.1914.  Google Scholar

[25]

Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math., 207, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[26]

Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in R2n, Ann. of Math., 155 (2002), 317-368.  doi: 10.2307/3062120.  Google Scholar

[27]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184.  doi: 10.1002/cpa.3160310203.  Google Scholar

[28]

C. Viterbo, Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 487-489.   Google Scholar

[29]

C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 621-655.  doi: 10.1090/S0002-9947-1989-0978370-5.  Google Scholar

[30]

W. Wang, Closed trajectories on symmetric convex Hamiltonian energy surfaces, Discrete Contin. Dyn. Syst., 32 (2012), 679-701.  doi: 10.3934/dcds.2012.32.679.  Google Scholar

[31]

W. Wang, Closed characteristics on compact convex hypersurfaces in R8, Adv. Math., 297 (2016), 93-148.  doi: 10.1016/j.aim.2016.03.044.  Google Scholar

[32]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.  doi: 10.2307/1971185.  Google Scholar

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