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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:
 [1] M.C. ARNAUD . - Sur les points fixes des difféomorphismes exacts symplectiques de Tn \times Rn , C.R. Acad. Sci. Paris t 309 ( 1989 ), 191-194. MR 90k:58059 | Zbl 0696.57017 · Zbl 0696.57017 [2] V. ARNOLD & A. AVEZ . - Problèmes ergodiques de la mécanique classique , Gauthier Villars ( 1967 ). MR 35 #334 | Zbl 0149.21704 · Zbl 0149.21704 [3] J. MARTINET . - Singularities of smooth functions and maps , London Math. Soc. Series 58 ( 1982 ), Cambridge U.P. MR 83i:58018 | Zbl 0522.58006 · Zbl 0522.58006 [4] J. MOSER . - Proof of a generalized form of a fixed point theorem due to G.D. Birkhoff , Lect. Notes in Math. 597 ( 1976 ), 464-494. MR 58 #13205 | Zbl 0358.58009 · Zbl 0358.58009 [5] H. POINCARÉ . - Sur un théorème de géométrie , Oeuvres de Henri Poincaré, t.1, Gauthier Villars ( 1953 ), 499-538. [6] H. POINCARÉ . - Les méthodes nouvelles de la mécanique céleste , t.1, Gauthier Villars ( 1893 ). Zbl 0651.70002 · Zbl 0651.70002 [7] C. ROBINSON . - Generic properties of conservative systems , Am. J. of Math. 92 ( 1970 ), 562-601. MR 42 #8517 | Zbl 0212.56502 · Zbl 0212.56502
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