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Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on $ \mathbb{R}^\text{2n} $
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China |
In this paper, we consider multiplicity of brake orbits on compact symmetric dynamically convex reversible hypersurfaces in $\mathbb{R}^{2n}$. We prove that there exist at least \([\frac{n+1}{2}]\) geometrically distinct closed characteristics on dynamically convex hypersurface $\Sigma$ in $R^{2n}$ with the symmetric and reversible conditions, i.e. $\Sigma = -\Sigma$ and $N\Sigma = \Sigma$, where $N = diag(-I_{n}, I_{n})$. For $n\geq2$, we prove that there are at least 2 symmetric brake orbits on $\Sigma$, which generalizes Kang's result in [
References:
[1] |
A. Borel, Seminar on Transformation Groups, Princeton Univ. Press, Princeton, 1960.
![]() ![]() |
[2] |
K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhauser, Basel, 1993.
doi: 10.1007/978-1-4612-0385-8. |
[3] |
H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in ${\Bbb R}^{2n}$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 65, 30 pp, arXiv: 1601.03470.
doi: 10.1007/s00526-017-1173-1. |
[4] |
I. Ekeland, Convexity methods in Hamiltonian Mechanics., Springer-Verlag. Berlin. 1990.
doi: 10.1007/978-3-642-74331-3. |
[5] |
I. Ekeland and H. Hofer,
Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys, 113 (1987), 419-467.
doi: 10.1007/BF01221255. |
[6] |
J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in ${\Bbb R}^{2n}$, Ann. Inst. Fourier (Grenoble), 66 (2016), no. 6, 2485-2505. |
[7] |
C. Liu.-G.,
Maslov-type index theory for symplectic paths with Lagrangian boundary conditions, Adv. Nonlinear Stud, 7 (2007), 131-161.
doi: 10.1515/ans-2007-0107. |
[8] |
J. Kang,
Some remarks on symmetric periodic orbits in the restricted three-body problem, Discrete Contin. Dyn. Syst., 34 (2014), 5229-5245.
doi: 10.3934/dcds.2014.34.5229. |
[9] |
C. Liu, Y. Long and C. Zhu,
Multiplicity of closed characteristics on symmetric convex hypersurfaces in ${\Bbb R}^{2n}$, Math. Ann, 323 (2002), 201-215.
doi: 10.1007/s002089100257. |
[10] |
C. Liu and D. Zhang,
Seifert conjecture in the even convex case, Comm. Pure Appl. Math, 67 (2014), 1563-1604.
doi: 10.1002/cpa.21525. |
[11] |
C. Liu and D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits, J. Differential Equations, 257 (2014), 1194–1245, arXiv: 0908.0021 [math.SG]
doi: 10.1016/j.jde.2014.05.006. |
[12] |
H. Liu, Y. Long and W. Wang,
Resonance identities for closed characteristics on compact star-shaped hypersurfaces in ${\Bbb R}^2n$, J. Funct. Anal., 266 (2014), 5598-5638.
doi: 10.1016/j.jfa.2014.02.035. |
[13] |
Y. Long, The topological structures of $\omega-subsets$ of symplectic groups, Acta Math., Sinica(English Series), 15 (1999), 255–268.
doi: 10.1007/BF02650669. |
[14] |
Y. Long, Index Theory for Symplectic Paths with Applictions, Progress in Mathematics. 2002.
doi: 10.1007/978-3-0348-8175-3. |
[15] |
Y. Long,
Bott formula of the Maslov-type index theory, Pacific J. Math, 187 (1999), 113-149.
doi: 10.2140/pjm.1999.187.113. |
[16] |
Y. Long, D. Zhang and C. Zhu,
Multiple brake orbits in bounded convex symmetric domains, Adv.of Math, 203 (2006), 568-635.
doi: 10.1016/j.aim.2005.05.005. |
[17] |
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. |
[18] |
C. Viterbo, Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci
Paris. Ser. I Math, 301 (1985), 487–489. |
[19] |
W. Wang,
Closed trajectories on symmetric convex Hamiltonian energy surfacs, Discrete. Contin. Dyn. Syst., 32 (2012), 679-701.
doi: 10.3934/dcds.2012.32.679. |
[20] |
D. Zhang,
Maslov-type index and brake orbits in nonlinear Hamiltonian systems, Science in China Series A: Mathematics, 50 (2007), 761-772.
doi: 10.1007/s11425-007-0034-3. |
[21] |
D. Zhang,
Multiple symmetric brake orbits in bounded convex symmetric domains, Advanced. Nonlinear Studies, 6 (2006), 643-652.
doi: 10.1515/ans-2006-0408. |
show all references
References:
[1] |
A. Borel, Seminar on Transformation Groups, Princeton Univ. Press, Princeton, 1960.
![]() ![]() |
[2] |
K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhauser, Basel, 1993.
doi: 10.1007/978-1-4612-0385-8. |
[3] |
H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in ${\Bbb R}^{2n}$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 65, 30 pp, arXiv: 1601.03470.
doi: 10.1007/s00526-017-1173-1. |
[4] |
I. Ekeland, Convexity methods in Hamiltonian Mechanics., Springer-Verlag. Berlin. 1990.
doi: 10.1007/978-3-642-74331-3. |
[5] |
I. Ekeland and H. Hofer,
Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys, 113 (1987), 419-467.
doi: 10.1007/BF01221255. |
[6] |
J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in ${\Bbb R}^{2n}$, Ann. Inst. Fourier (Grenoble), 66 (2016), no. 6, 2485-2505. |
[7] |
C. Liu.-G.,
Maslov-type index theory for symplectic paths with Lagrangian boundary conditions, Adv. Nonlinear Stud, 7 (2007), 131-161.
doi: 10.1515/ans-2007-0107. |
[8] |
J. Kang,
Some remarks on symmetric periodic orbits in the restricted three-body problem, Discrete Contin. Dyn. Syst., 34 (2014), 5229-5245.
doi: 10.3934/dcds.2014.34.5229. |
[9] |
C. Liu, Y. Long and C. Zhu,
Multiplicity of closed characteristics on symmetric convex hypersurfaces in ${\Bbb R}^{2n}$, Math. Ann, 323 (2002), 201-215.
doi: 10.1007/s002089100257. |
[10] |
C. Liu and D. Zhang,
Seifert conjecture in the even convex case, Comm. Pure Appl. Math, 67 (2014), 1563-1604.
doi: 10.1002/cpa.21525. |
[11] |
C. Liu and D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits, J. Differential Equations, 257 (2014), 1194–1245, arXiv: 0908.0021 [math.SG]
doi: 10.1016/j.jde.2014.05.006. |
[12] |
H. Liu, Y. Long and W. Wang,
Resonance identities for closed characteristics on compact star-shaped hypersurfaces in ${\Bbb R}^2n$, J. Funct. Anal., 266 (2014), 5598-5638.
doi: 10.1016/j.jfa.2014.02.035. |
[13] |
Y. Long, The topological structures of $\omega-subsets$ of symplectic groups, Acta Math., Sinica(English Series), 15 (1999), 255–268.
doi: 10.1007/BF02650669. |
[14] |
Y. Long, Index Theory for Symplectic Paths with Applictions, Progress in Mathematics. 2002.
doi: 10.1007/978-3-0348-8175-3. |
[15] |
Y. Long,
Bott formula of the Maslov-type index theory, Pacific J. Math, 187 (1999), 113-149.
doi: 10.2140/pjm.1999.187.113. |
[16] |
Y. Long, D. Zhang and C. Zhu,
Multiple brake orbits in bounded convex symmetric domains, Adv.of Math, 203 (2006), 568-635.
doi: 10.1016/j.aim.2005.05.005. |
[17] |
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. |
[18] |
C. Viterbo, Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci
Paris. Ser. I Math, 301 (1985), 487–489. |
[19] |
W. Wang,
Closed trajectories on symmetric convex Hamiltonian energy surfacs, Discrete. Contin. Dyn. Syst., 32 (2012), 679-701.
doi: 10.3934/dcds.2012.32.679. |
[20] |
D. Zhang,
Maslov-type index and brake orbits in nonlinear Hamiltonian systems, Science in China Series A: Mathematics, 50 (2007), 761-772.
doi: 10.1007/s11425-007-0034-3. |
[21] |
D. Zhang,
Multiple symmetric brake orbits in bounded convex symmetric domains, Advanced. Nonlinear Studies, 6 (2006), 643-652.
doi: 10.1515/ans-2006-0408. |
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