**
Below is the ascii version of the abstract for 07-135.
The html version should be ready soon.**A. Gonz\'alez-Enr\'{i}quez, R. de la Llave
Analytic smoothing of geometric maps with applications to KAM theory
(831K, pdf)
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.