On the divergence of Birkhoff Normal Forms

TL;DR

This work establishes that for real analytic symplectic diffeomorphisms with a nonresonant elliptic fixed point, the Birkhoff Normal Form (BNF) is generically divergent, answering the Eliasson–Perez-Marco question in any dimension. The authors develop a comprehensive KAM-type framework on domains with holes, introducing Adapted KAM Normal Forms and Hamilton-Jacobi Normal Forms, and solve cohomological equations to compare BNFs across multiple normalizations. They connect the convergence of BNFs to strong dynamical rigidity via the measure of invariant circles, showing that convergence would sharply constrain the system, while divergence is prevalent in the analytic setting. The results extend to real analytic diffeomorphisms of the annulus with a Diophantine invariant torus and provide detailed extension properties and residue computations that quantify how far the dynamics can be pushed toward integrability. Overall, the paper advances the understanding of formal versus actual integrability in near-integrable Hamiltonian and symplectic systems, with precise quantitative tools for analyzing invariant structures near elliptic equilibria.

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.

Figures (3)

• Figure 1: Adapted KAM Normal Forms ($\omega_{0}$ diophantine) in the complex $r$-plane. The triple $(U^{(\rho)},\widehat{D}\smallsetminus (1/10)\widehat{D},\mathbb{D}(0,\rho^{b_{\tau})}))$ is $C_{b}|\ln\rho|^{-1}$-good
• Figure 2: Adapted KAM Normal Forms (CC Case) in the complex $r$-plane. The triple $(U^{n},\widehat{U}^{n}, \mathbb{D}(0,q_{n+1}^{-6}))$ is $1/(10|\ln \rho_{n}|)$-good.
• Figure 3: Invariant graphs cannot intersect the interior of $V$.

Theorems & Definitions (140)

• Theorem : Birkhoff
• Theorem : Pérez-Marco PM
• Theorem : Moser
• Theorem A
• Theorem B
• Theorem A'
• Theorem B'
• Theorem C
• Theorem 1.1
• Theorem D