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

Salamon, Dietmar; Zehnder, Eduard

Commun. Pure Appl. Math. 45, No. 10, 1303-1360 (1992).

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)

MSC:

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

Keywords:

Hamiltonian systems; Fredholm index; periodic solutions; Morse theory; Maslov index

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