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.