On the minimal number of periodic orbits on some hypersurfaces in 2n
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, p. 2485-2505
We study periodic orbits of the Reeb vector field on a nondegenerate dynamically convex starshaped hypersurface in 2n along the lines of Long and Zhu [24], but using properties of the S 1 - equivariant symplectic homology. We prove that there exist at least n distinct simple periodic orbits on any nondegenerate starshaped hypersurface in 2n satisfying the condition that the minimal Conley–Zehnder index is at least n-1. The condition is weaker than dynamical convexity.
Nous étudions les orbites périodiques du champ de Reeb sur les hypersurfaces non-dégénérées et dynamiquement convexes de 2n en suivant les travaux de Long et Zhu mais en utilisant l’homologie symplectique S 1 -équivariante. Nous démontrons qu’il existe au moins n orbites simples de Reeb sur toute hypersurface étoil�e et non dégénérée de 2n satisfaisant la condition que le plus petit indice de Conley–Zehnder est au moins n-1. Cette dernière condition est plus faible que celle de convexité dynamique.
Received : 2015-09-01
Revised : 2016-02-04
Accepted : 2016-03-24
Published online : 2016-10-04
DOI : https://doi.org/10.5802/aif.3069
Classification:  53D10,  37J55
Keywords: Reeb dynamics, Equivariant symplectic homology, Index jump
@article{AIF_2016__66_6_2485_0,
     author = {Gutt, Jean and Kang, Jungsoo},
     title = {On the minimal number of periodic orbits on some hypersurfaces in $\mathbb{R}^{2n}$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {6},
     year = {2016},
     pages = {2485-2505},
     doi = {10.5802/aif.3069},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2016__66_6_2485_0}
}
On the minimal number of periodic orbits on some hypersurfaces in $\mathbb{R}^{2n}$. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2485-2505. doi : 10.5802/aif.3069. https://aif.centre-mersenne.org/item/AIF_2016__66_6_2485_0/

[1] Audin, Michèle; Damian, Mihai Théorie de Morse et homologie de Floer, EDP Sciences, Les Ulis; CNRS Éditions, Paris, Savoirs Actuels (Les Ulis). [Current Scholarship (Les Ulis)] (2010), xii+548 pages

[2] Berestycki, Henri; Lasry, Jean-Michel; Mancini, Giovanni; Ruf, Bernhard Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces, Comm. Pure Appl. Math., Tome 38 (1985) no. 3, pp. 253-289 | Article

[3] Bourgeois, Frédéric; Cieliebak, Kai; Ekholm, Tobias A note on Reeb dynamics on the tight 3-sphere, J. Mod. Dyn., Tome 1 (2007) no. 4, pp. 597-613 | Article

[4] Bourgeois, Frédéric; Oancea, Alexandru Fredholm theory and transversality for the parametrized and for the S 1 -invariant symplectic action, J. Eur. Math. Soc. (JEMS), Tome 12 (2010) no. 5, pp. 1181-1229 | Article

[5] Bourgeois, Frédéric; Oancea, Alexandru S 1 -equivariant symplectic homology and linearized contact homology (2012) (http://arxiv.org/abs/1212.3731v1 )

[6] Bourgeois, Frédéric; Oancea, Alexandru The index of Floer moduli problems for parametrized action functionals, Geom. Dedicata, Tome 165 (2013), pp. 5-24 | Article

[7] Conley, Charles; Zehnder, Eduard Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., Tome 37 (1984) no. 2, pp. 207-253 | Article

[8] Cristofaro-Gardiner, Daniel; Hutchings, Michael From one Reeb orbit to two (2012) (to appear in J. Diff. Geom., http://arxiv.org/abs/1202.4839)

[9] Ekeland, Ivar Convexity methods in Hamiltonian mechanics, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 19 (1990), x+247 pages

[10] Ekeland, Ivar; Hofer, H. Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys., Tome 113 (1987) no. 3, pp. 419-469 | Article

[11] Ekeland, Ivar; Lasry, Jean-Michel On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. (2), Tome 112 (1980) no. 2, pp. 283-319 | Article

[12] Ginzburg, Viktor L.; Gören, Yusuf Iterated index and the mean Euler characterstic, J. Topol. Anal., Tome 7 (2015), pp. 453-481 | Article

[13] Ginzburg, Viktor L.; Hein, Doris; Hryniewicz, Umberto L.; Macarini, Leonardo Closed Reeb orbits on the sphere and symplectically degenerate maxima, Acta Math. Vietnam., Tome 38 (2013) no. 1, pp. 55-78 | Article

[14] Gürel, Başak Z. Perfect Reeb flows and action-index relations, Geom. Dedicata, Tome 174 (2015), pp. 105-120 | Article

[15] Gutt, Jean Generalized Conley-Zehnder index, Annales de la faculté des sciences de Toulouse, Tome 23 (2014) no. 4, pp. 907-932 | Article

[16] Gutt, Jean The positive equivariant symplectic homology as an invariant for some contact manifolds (2015) (to appear in Journal of Symplectic Geometry, http://arxiv.org/abs/1503.01443)

[17] Hofer, H.; Wysocki, K.; Zehnder, E. The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), Tome 148 (1998) no. 1, pp. 197-289 | Article

[18] Hofer, H.; Wysocki, K.; Zehnder, E. Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), Tome 157 (2003) no. 1, pp. 125-255 | Article

[19] Hutchings, Michael; Taubes, Clifford Henry The Weinstein conjecture for stable Hamiltonian structures, Geom. Topol., Tome 13 (2009) no. 2, pp. 901-941 | Article

[20] Kang, Jungsoo Equivariant symplectic homology and multiple closed Reeb orbits, Internat. J. Math., Tome 24 (2013) no. 13, 1350096 pages | Article

[21] Liu, Hui; Long, Yiming The existence of two closed characteristics on every compact star-shaped hypersurface in 4 , Acta Mathematica Sinica, English Series, Tome 32 (2016) no. 1, pp. 40-53 | Article

[22] Liu, Hui; Long, Yiming; Wang, Wei; Zhang, Ping’An Symmetric closed characteristics on symmetric compact convex hypersurfaces in 8 , Commun. Math. Stat., Tome 2 (2014) no. 3-4, pp. 393-411 | Article

[23] Long, Yiming Index theory for symplectic paths with applications, Birkhäuser Verlag, Progress in Mathematics, Tome 207 (2002), xxiv+380 pages

[24] Long, Yiming; Zhu, Chaofeng Closed characteristics on compact convex hypersurfaces in 2n , Ann. of Math. (2), Tome 155 (2002) no. 2, pp. 317-368 | Article

[25] Rabinowitz, Paul H. Periodic solutions of a Hamiltonian system on a prescribed energy surface, J. Differential Equations, Tome 33 (1979) no. 3, pp. 336-352 | Article

[26] Salamon, Dietmar Lectures on Floer homology, Symplectic geometry and topology (Park City, UT, 1997), Amer. Math. Soc., Providence (IAS/Park City Math. Ser.) Tome 7 (1999), pp. 143-229

[27] Salamon, Dietmar; Zehnder, Eduard Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., Tome 45 (1992) no. 10, pp. 1303-1360 | Article

[28] Seidel, Paul A biased view of symplectic cohomology, Current developments in mathematics, 2006, Int. Press, Somerville, MA (2008), pp. 211-253

[29] Viterbo, C. Functors and computations in Floer homology with applications. I, Geom. Funct. Anal., Tome 9 (1999) no. 5, pp. 985-1033 | Article

[30] Wang, Wei Symmetric closed characteristics on symmetric compact convex hypersurfaces in 2n , J. Differential Equations, Tome 246 (2009) no. 11, pp. 4322-4331 | Article

[31] Wang, Wei On a conjecture of Anosov, Adv. Math., Tome 230 (2012) no. 4-6, pp. 1597-1617 | Article

[32] Wang, Wei Closed characteristics on compact convex hypersurfaces in 8 (2013) (http://arxiv.org/abs/1305.4680 )

[33] Wang, Wei; Hu, Xijun; Long, Yiming Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., Tome 139 (2007) no. 3, pp. 411-462 | Article