Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. (English) Zbl 0766.58023

Let \((M,\omega)\) be a compact symplectic manifold, \(H:\mathbb{R}\times M\to\mathbb{R}\) a time-dependent Hamiltonian function, \(X_ H:\mathbb{R}\times M\to TM\) the associated Hamiltonian vector field. Let the classes \([\omega]\), \(c_ 1\in H_ 2(M,\mathbb{Z})\) (the first Chern class to an almost complex structure \(J\) on \(TM)\) vanish over \(\pi_ 2(M)\). Assuming the periodicity \(H(t+1,x)=H(t,x)\), we shall consider the contractible \(\tau\)-periodic \((\tau\in\mathbb{Z})\) solutions of the system \(dx/dt=X_ H(t,x)\) under the weak nondegeneracy hypothesis: at least one Floquet multiplier is not equal to 1.
Theorem A: there exist infinitely many contractible periodic solutions having integer periods.
Theorem B: \(p_ k(H,\tau)-p_{k-1}(H,\tau)+\dots\geq b_{n+k}-b_{n+k- 1}+\dots\) (in particular \(p_ k(H,\tau)\geq b_{n+k})\), where \(p_ k(H,\tau)\) is the number of contractible \(\tau\)-periodic solutions with the Maslov index \(k\), and \(b_ k=\text{rank} H_ k(M,\mathbb{Z}/2\mathbb{Z})\).
The proofs are highly nontrivial and based on rather advanced results and concepts, especially on infinite-dimensional Morse theory with a generalized Maslov index substituted for the common Morse index, the Floer homologies related to the elliptic boundary value problem \(\partial u/\partial s+J(u)\partial u/\partial t+\nabla H=0\) \((u:\mathbb{R}^ 2\to M)\) standing for the common gradient vector field, and the index formula for the relevant Fredholm operator involving the Maslov index.
As a very particular case, the result cover both the classical Morse inequalities and the Lefschetz fixed point theorem for a special choice of \(H\).
Reviewer: J.Chrastina (Brno)


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37G99 Local and nonlocal bifurcation theory for dynamical systems
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[1] Arnold, C. R. Acad. Sci. Paris 261 pp 3719– (1965)
[2] Mathematical Methods in Classical Mechanics, Appendix 9, Springer-Verlag, New York, 1978.
[3] Aronszajn, J. Math. Pures Appl. 36 pp 235– (1957)
[4] Atiyah, Proc. Symp. Pure Math. 48 pp 285– (1988)
[5] Atiyah, Math. Proc. Camb. Phil. Soc. 79 pp 71– (1976)
[6] Lecture at the University of Wisconsin, 6 April 1984.
[7] Conley, Inv. Math. 73 pp 33– (1983)
[8] Conley, Comm. Pure Appl. Math. 37 pp 207– (1984)
[9] Conley, Proc. Symp. Pure Math. 45 pp 283– (1986)
[10] Duistermaat, Adv. in Math. 21 pp 173– (1976)
[11] Floer, J. Diff. Geom. 28 pp 513– (1988)
[12] Floer, Comm. Pure Appl. Math. 41 pp 393– (1988)
[13] Floer, Comm. Pure Appl. Math. 41 pp 775– (1988)
[14] Floer, Comm. Math. Phys. 118 pp 215– (1988)
[15] Floer, J. Diff. Geom. 30 pp 207– (1989)
[16] Floer, Comm. Math. Phys. 120 pp 575– (1989)
[17] and , Symplectic homology I: Preliminaries and recollections, Ruhr Universität Bochum, 1991, preprint.
[18] and , Coherent orientations for periodic orbit problems in symplectic geometry, Ruhr Universität Bochum, 1991, preprint.
[19] Franzosa, Trans. AMS 311 pp 561– (1989)
[20] Franks, Invent. Math.
[21] Gelfand, Uspekhi Math. Nak. 10 pp 3– (1955)
[22] AMS Translations Series 2 8 pp 143– (1958) · Zbl 0079.10905
[23] Gromov, Invent. Math. 82 pp 307– (1985)
[24] Hofer, Ann. Henri Poincaré Analyse Nonlinéaire 5 pp 465– (1988)
[25] Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1976.
[26] and , Elliptic operators on noncompact manifolds, Ann. Sci. Normale Sup. Pisa 4–12, 1985, pp. 409–446.
[27] and , Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, pp. 528–563 in: Stochastic Processes. Physics and Geometry, World Scientific, 1990.
[28] McDuff, Bull. AMS 23 pp 311– (1990)
[29] Lectures on the h-Cobordism Theorem, Math. Notes 1, Princeton University Press, 1965.
[30] Moeckel, Ergod. Th. Dynam. Sys. 8 pp 227– (1988)
[31] Moser, Trans. AMS 120 pp 286– (1965)
[32] Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff, pp. 464–494 in: Geometry and Topology, Springer Lecture Notes in Mathematics, Vol. 597, 1977.
[33] Notes sur les pages 316 à 322 de l’article de M. Gromov ”Pseudoholomorphic curves in symplectic manifolds,” preprint, Ecole Polytechnique, Palaiseau, 1988.
[34] Sacks, Ann. Math. 113 pp 1– (1981)
[35] Salamon, Bull. LMS 22 pp 113– (1990)
[36] and , Floer homology, the Maslov index and periodic orbits of Hamiltonian equations, pp. 573–600 in: Analysis et cetera. and , eds., Academic Press, New York, 1990.
[37] Smale, Bull. AMS 66 pp 43– (1960)
[38] Smale, Am. J. Math. 87 pp 213– (1973)
[39] Viterbo, Bull. Soc. Math. France 115 pp 361– (1987)
[40] Viterbo, Invent. math. 100 pp 301– (1990)
[41] Witten, J. Diff. Geom. 17 pp 661– (1982)
[42] Wolfson, J. Diff. Geom. 23 pp 383– (1988)
[43] The Arnold conjecture for fixed points of symplectic mappings and periodic solutions of Hamiltonian systems, pp. 1237–1246 in: Proceedings of the International Congress of Mathematics, Berkeley, 1988.
[44] A Poincaré-Birkhoff-type result in higher dimensions, pp. 119–146 in Aspects Dynamiques et Topologiques des Groups Infinis de Transformation de la Mécanique, P. Dazard, N. Desolnaux-Moulis, and J. M. Marvan, eds., Travaux en cours 25, Hermann, Paris, 1987.
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