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Analytic smoothing of geometric maps with applications to KAM theory. (English) Zbl 1160.37024
It is shown that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving or contact, and that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in $$C^r$$ -norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. This result is applied to give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. The main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. And the symplectic, volume-preserving and contact forms are assumed to be analytic. It is interesting that the use of generating functions is avoided here. As it is well known, generating functions may fail to be globally defined for some maps. One advantage of not using generating functions is that the result given here can be applied directly to non-twist maps.

##### MSC:
 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
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