Multiplicity of closed Reeb orbits on dynamically convex <inline-formula><tex-math id="M1">\begin{document} \mathbb{R}P^{2n-1} \end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document} n\geq2 \end{document}</tex-math></inline-formula>

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April 2022, 42(4): 1801-1816. doi: 10.3934/dcds.2021172

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 Published April 2022 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 and Continuous Dynamical Systems, 2022, 42 (4) : 1801-1816. 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.
[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.
[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.
[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.
[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.
[6]	K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser. Boston. 1993. doi: 10.1007/978-1-4612-0385-8.
[7]	D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.
[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.
[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.
[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.
[11]	I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.
[28]	C. Viterbo, Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 487-489.
[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.
[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.
[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.
[32]	A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518. doi: 10.2307/1971185.

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.
[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.
[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.
[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.
[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.
[6]	K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser. Boston. 1993. doi: 10.1007/978-1-4612-0385-8.
[7]	D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.
[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.
[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.
[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.
[11]	I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.
[28]	C. Viterbo, Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 487-489.
[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.
[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.
[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.
[32]	A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518. doi: 10.2307/1971185.

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