On the divergence of Birkhoff Normal Forms

Abstract It is well known that a real analytic symplectic diffeomorphism of the $2d$ 2 d -dimensional disk ( $d\geq 1$ ≥ 1 ) admitting origin as non-resonant elliptic fixed point can be formally conjugated to its Birkhoff Normal Form, formal power series defining integrable at origin. We prove in this paper Form general divergent. This solves, any dimension, question determining which two alternatives Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) true and answers by H. Eliasson. Our result consequence fact when $d=1$ = convergence object BNF has strong dynamical consequences on Lebesgue measure set invariant circles arbitrarily small neighborhoods proof, our results, extend case diffeomorphisms annulus Diophantine torus.