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Types of fixed points of symplectic diffeomorphisms on \(\mathbb{T}^ n\times \mathbb{R}^ n\). (Type des points fixes des difféomorphismes symplectiques de \(\mathbb{T}{}^ n\times{}\mathbb{R}{}^ n\).) (French) Zbl 0758.58009
The exact symplectic \(C^ r\)-diffeomorphisms (\(1\leq r < \infty\)) of the \(2n\)-dimensional annulus \(\mathbb{T}^ n\times \mathbb{R}^ n\) which are generic in the \(C^ r\)-topology and \(C^ r\)-close to a completely integrable weakly monotone diffeomorphism, are studied by the author [C. R. Acad. Sci., Paris, Ser. I 309, No. 3, 191-194 (1989; Zbl 0696.57017)]. This paper studies the types of the fixed points of these symplectic diffeomorphisms. It is proved that the obtained results depend on the torsion of the perturbed completely integrable diffeomorphism. Finally the obtained results are applied using the Birkhoff-Lewis theorem to the periodic points accumulating on an elliptic point of every generic symplectic \(c^ 4\)-diffeomorphism of a symplectic manifold.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
57R50 Differential topological aspects of diffeomorphisms
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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References:
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