On the divergence of Birkhoff Normal Forms

Raphaël Krikorian

Published 2019-06-03Version 1

It is well known that a real analytic symplectic diffeomorphism of the two-dimensional annulus admitting a real analytic invariant curve with diophantine rotation number can be {\it formally} conjugated to its Birkhoff Normal Form, a formal power series defining a {\it formal integrable} symplectic diffeomorphism. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves 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 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant curves in arbitrarily small neighborhoods of the original invariant curve: the measure of the complement of the set of invariant curves in these neighborhoods is much smaller than what it is in general. As a consequence, for any $d\geq 1$, the Birhkoff Normal Form of a symplectic real-analytic diffeomorphism of the $d$-dimensional annulus attached to an invariant real-analytic lagrangian torus with prescribed diophantine frequency vector is in general divergent.