On the divergence of Birkhoff Normal Forms

TL;DR

This paper proves that Birkhoff Normal Forms for real analytic symplectic diffeomorphisms are generally divergent, resolving a long-standing question and revealing implications for the measure of invariant sets near elliptic fixed points.

Contribution

It demonstrates the divergence of Birkhoff Normal Forms in general, answering a key open question and extending the result to diffeomorphisms with Diophantine invariant tori.

Findings

Birkhoff Normal Form is generally divergent.

Divergence impacts the measure of invariant circles.

Results extend to real-analytic diffeomorphisms of the annulus.

Abstract

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.