Exponentially Small Estimates for Kam Theorem near an Elliptic Equilibrium Point

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 and a nondegeneracy condition is ful lled we show that in a neighborhood of radius r the measure of the complement of the KAM tori is exponentially small in r This result is obtained by putting the system in Birkho normal form up to an appropriate order and the key point relies on giving accurate estimates for its terms Introduction We consider a Hamiltonian with n degrees of freedom having the origin as an elliptic equilibrium point In suitable canonical coordinates this system can be put in the form H q p X