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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)

##### 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
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