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La dynamique au voisinage d’un point fixe elliptique conservatif: de Poincaré et Birkhoff à Aubry et Mather. (French) Zbl 0582.58013
Sémin. Bourbaki, 36e année, Vol. 1983/84, Exp. 622, Astérisque 121-122, 147-170 (1985).
[For the entire collection see Zbl 0542.00005.]
This is an exposition of recent work of Aubry and Mather showing that, for area preserving diffeomorphisms of the plane, the missing circles of K.A.M. theory give rise to invariant Cantor sets on which the dynamics is conjugate to that of a Denjoy counterexample on the circle restricted to its minimal invariant set. In this light, the invariant curve theorem (K.A.M. theory) appears as a regularity result. The method chosen here is a mixture of Aubry’s original work and Katok’s approach of Mather’s results via Birkhoff’s Lipschitz estimates on invariant curves of monotone twist mappings of the annulus. N.B. It appears that many of the results were already known to Hedlund in the slightly different context of minimising geodesics on a Riemannian torus.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37C75 Stability theory for smooth dynamical systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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