TITLE:
Exponentially small estimates for KAM theorem
near an elliptic equilibrium point

AUTHORS:
Amadeu Delshams(1) and Pere Gutierrez(2)

(1) Dept. de Matematica Aplicada I (ETSEIB), Universitat Politecnica de
    Catalunya, Diagonal 647, 08028 Barcelona (Spain).
    E-mail: amadeu@ma1.upc.es

(2) Dept. de Matematica Aplicada II, Universitat Politecnica de
    Catalunya, Pau Gargallo 5, 08071 Barcelona (Spain).
    E-mail: gutierrez@ma2.upc.es


ABSTRACT:
We give a precise statement of KAM theorem for a Hamiltonian system
in a neighborhood of an elliptic equilibrium point.
If the frequencies of the elliptic point satisfy a Diophantine condition,
with exponent $\tau$, and a nondegeneracy condition is fulfilled,
we show that in a neighborhood of radius $r$ the measure of the complement of
the KAM tori is exponentially small in $(1/r)^{1/(\tau+1)}$.
This result is obtained by putting the system in Birkhoff normal form up to 
an appropriate order, and the key point relies on giving accurate estimates 
for its terms.
