Abstract
The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.
Similar content being viewed by others
References
Booß-Bavnbek, B. and Zhu, C., Maslov index in symplectic Banach spaces, Mem. Amer. Math. Soc., to appear. arXiv:math/1406.1569v4 [math.SG]
Booß-Bavnbek, B. and Zhu, C., The Maslov index in weak symplectic functional analysis, Ann. Global. Anal. Geom., 44, 2013, 283–318.
Bott, R., On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9, 1956, 171–206.
Duistermaat, J. J., On the Morse index in variational calculus, Advances in Math., 21(2), 1976, 173–195.
Ekeland, I., Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)), Vol. 19, Springer-Verlag, Berlin, 1990.
Frauenfelder, U. and van Koert, O., The H¨ormander index of symmetric periodic orbits, Geom. Dedicata, 168, 2014, 197–205.
Ginzburg, V. L., The Conley conjecture, Ann. of Math. (2), 172(2), 2010, 1127–1180.
Hingston, N., Subharmonic solutions of Hamiltonian equations on tori, Ann. of Math. (2), 170(2), 2009, 529–560.
Hu, X. and Sun, S., Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit, Comm. Math. Phys., 290(2), 2009, 737–777.
Liu, C. and Tang, S., Maslov (P, ω) theory for symplectic paths, Adv. Nonlinear Stud., 15, 2015, 963–990.
Liu, C. and Zhang, D., Iteration theory of L-index and multiplicity of brake orbits, J. Differential Equations, 257(4), 2014, 1194–1245.
Liu, C. and Zhang, D., Seifert conjecture in the even convex case, Comm. Pure Appl. Math., 67(10), 2014, 1563–1604.
Long, Y., Bott formula of the Maslov-type index theory, Pacific J. Math., 187(1), 1999, 113–149.
Long, Y., Index theory for symplectic paths with applications, Progress in Mathematics, Vol. 207, Birkh¨auser, Basel, 2002.
Long, Y., Zhang, D. and Zhu, C., Multiple brake orbits in bounded convex symmetric domains, Adv. Math., 203(2), 2006, 568–635.
Long, Y. and Zhu, C., Closed characteristics on compact convex hypersurfaces in R2n, Ann. of Math (2), 155(2), 2002, 317–368.
Salamon, D. and Zehnder, E., Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45(10), 1992, 1303–1360.
Whitehead, G. W., Elements of Homotopy Theory, Graduate Texts in Mathematics, 61, Springer-Verlag, New York, Berlin, 1978.
Zhu, C., A generalized Morse index theorem, Analysis, Geometry and Topology of Elliptic Operators, World Sci. Publ., Hackensack, NJ, 2006, 493–540.
Acknowledgments
The authors would like to thank the referees for their critical reading and very helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 11221091, 11471169) and the Key Laboratory of Pure Mathematics and Combinatorics, the Ministry of Education of China.
Rights and permissions
About this article
Cite this article
Wu, L., Zhu, C. The iteration formulae of the Maslov-type index theory in weak symplectic Hilbert space. Chin. Ann. Math. Ser. B 39, 17–32 (2018). https://doi.org/10.1007/s11401-018-1048-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-018-1048-6