TITLE:
Analytic smoothing of geometric maps with applications to KAM theory

AUTHORS:
Alejandra Gonzalez-Enriquez^(1) and Rafael de la Llave^(2)

(1) Dipartimento di Matematica ed Informatica,
    Universita degli Studi di Camerino,
    Via Madonna delle Carceri,
    62032 Camerino (MC) Italy.

(2) Department of Mathematics,
    University of Texas at Austin,
    1 University Station, C1200
    Austin, Texas 78712 USA. 

E-mails: alejandra.gonzalez@unicam.it, llave@math.utexas.edu

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, 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.  As an application to this result, we 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 to be either written
in action-angle variables or perturbations of integrable systems.
Our 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.

KEYWORDS:
smoothing, symplectic maps, volume-preserving maps,
contact maps, KAM tori, uniqueness, bootstrap of regularity.