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.