Abstract
In this paper, we define mean index for non-periodic orbits in Hamiltonian systems and study its properties. In general, the mean index is an interval in ℝ which is uniformly continuous on the systems. We show that the index interval is a point for a quasi-periodic orbit. The mean index can be considered as a generalization of rotation number defined by Johnson and Moser in the study of almost periodic Schrödinger operators. Motivated by their works, we study the relation of Fredholm property of the linear operator and the mean index at the end of the paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnol’d, V. I.: On a characteristic class entering into conditions of quantization. (Russian) Funkcional. Anal. i Priložen., 1, 1–14 (1967)
Ben-Artzi, A., Gohberg, I., Kaashoek, M. A.: Invertibility and dichotomy of differential operators on a half-line. Journal of Dynamics and Differential Equations, 5(1), 1–36 (1993)
Cappell, S. E., Lee, R., Miller, E. Y.: On the Maslov index. Comm. Pure Appl. Math., 47(2), 121–186 (1994)
Ekeland, Ivar.: Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990
Furstenberg, H.: Strict ergodicity and transformation of the torus. American Journal of Mathematics, 83(4), 573–601 (1961)
Hale, J. K.: Ordinary Differential Equations, R. E. Krieger Pub. Co., Malabar, Florida, 1980
Hu, X., Portaluri, A.: Index theory for heteroclinic orbits of Hamiltonian systems. Calc. Var. Partial Differ. Equ., 56(6), paper No. 167, 24 pp. (2017)
Hu, X., Sun, S.: Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit. Comm. Math. Phys., 290(2), 737–777 (2009)
Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Communications in Mathematical Physics, 84(3), 403–438 (1982)
Liu, C., Tang, S.: Maslov (P, ω)-index theory for symplectic paths. Adv. Nonlinear Stud., 15(4), 963–990 (2015)
Long, Y.: Index theory for symplectic paths with applications, Progress in Mathematics, Vol. 207. Birkhäuser Verlag, Basel, 2002
Long, Y., Zhu, C.: Maslov-type index theory for symplectic paths and spectral flow II. Chinese Ann. Math. Ser. B, 21(1), 89–108 (2000)
Palmer, K. J.: Exponential dichotomies and Fredholm operators. Proceedings of the American Mathematical Society, 104(1), 149–156 (1988)
Peter, W.: An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982
Robbin, J., Salamon, D.: The Maslov index for paths. Topology, 32(4), 827–844 (1993)
Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bulletin of the London Mathematical Society, 27(1), 1–33 (1995)
Zhou, Y., Wu, L., Zhu, C.: Härmander index in finite-dimensional case. Frontiers of Mathematics in China, 13, 725–761 (2018)
Zhu, C., Long, Y.: Maslov-type index theory for symplectic paths and spectral flow I. Chinese Ann. Math. Ser. B, 20(4), 413–424 (1999)
Acknowledgements
The first author sincerely thanks Yingfei Yi for the suggestion of using index theory to study the quasi-periodic orbits. Both the author sincerely thanks Jiangong You for helpful discussion of rotation number.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is partially supported by National Key R&D Program of China (Grant No. 2020YFA0713300) and NSFC (Grant Nos. 12071255, 11790271); the second author is partially supported by NSFC (Grant Nos. 12071255, 11425105)
Rights and permissions
About this article
Cite this article
Hu, X.J., Wu, L. Mean Index for Non-periodic Orbits in Hamiltonian Systems. Acta. Math. Sin.-English Ser. 38, 291–310 (2022). https://doi.org/10.1007/s10114-022-0507-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-022-0507-x