07-135 A. Gonz\'alez-Enr\'{i}quez, R. de la Llave
Analytic smoothing of geometric maps with applications to KAM theory (831K, pdf) Jun 1, 07
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Abstract. We prove that finitely differentiable diffeomorphisms preserving a geometric structure can be quantitatively approximated by analytic diffeomorphisms preserving the same geometric structure. More precisely, we show 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. We prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. As an application to this result, we give a proof of the existence, local uniqueness and bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed to be written either in action-angle variables or as perturbations of integrable ones.

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