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Symplectic fixed points and holomorphic spheres. (English) Zbl 0755.58022
Summary: Let \(P\) be a symplectic manifold whose symplectic form, integrated over the spheres in \(P\), is proportional to its first Chern class. On the loop space of \(P\), we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It it shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism on \(P\) is nondegenerate, then it satisfies the Morse inequalities on \(P\). We also discuss fixed point estimates for general exact diffeomorphisms.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E99 Variational problems in infinite-dimensional spaces
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
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