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Tores invariants des systèmes dynamiques Hamiltoniens (d’après Kolmogorov, Arnol’d, Moser, Rüssmann, Zehnder, Herman, Pöschel,…). (Invariant tori of Hamiltonian dynamical systems). (French) Zbl 0602.58021
Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. No. 639, Astérisque 133/134, 113-157 (1986).
[For the entire collection see Zbl 0577.00004.]
Author’s introduction, excerpted and paraphrased: This is an introduction to the theory of Kolmogorov-Arnol’d-Moser concerning quasiperiodic orbits in dynamical systems and their perturbations. Above all we treat the theorem of Kolmogorov-Arnol’d-Moser about Hamiltonian systems - the theorem of invariant tori, i.e. a Hamiltonian system close to a completely integrable system has many invariant Lagrangian tori. A corollary is that a generic Hamiltonian system is not ergodic.
First, we survey Kolmogorov-Arnol’d-Moser theory. Next, we establish an implicit function theorem (following R. S. Hamilton) in Fréchet space of \(C^{\infty}\) functions. Finally we use this theorem to prove (following Herman) a version of the theorem of invariant tori.
Reviewer: W.Reddy

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion