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