On the divergence of Birkhoff Normal Forms

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.

• The paper establishes that for any nonresonant Diophantine frequency, a prevalent set of analytic symplectic systems exhibit divergent Birkhoff Normal Forms.
• It demonstrates that convergence of the BNF leads to significantly improved KAM estimates by exponentially reducing the measure of non-torus regions.
• The study utilizes refined KAM procedures and explicit constructions to connect formal analytic properties with observable dynamical rigidity and hyperbolic structures.

On the Divergence of Birkhoff Normal Forms: An Expert Summary

Background and Context

Birkhoff Normal Form (BNF) theory is central in the study of local (near-equilibrium) Hamiltonian and symplectic dynamics. Classical results show that, near a non-resonant elliptic equilibrium, any real-analytic symplectic diffeomorphism (or Hamiltonian system) can be formally conjugated to a simple, integrable model—its Birkhoff Normal Form—via a (generally divergent) formal power series. This "integrable" model is, in itself, a formal object and not a genuine real analytic function in general. Whether the BNF series converges has direct implications for the actual dynamical behavior close to the equilibrium.

Historically, while the divergence of the normalizing transformation is generic (as shown by Siegel), the convergence properties of the BNF itself are more subtle. Earlier work, notably by Pérez-Marco, established a dichotomy: either BNF always converges (for all analytic perturbations compatible with a given frequency) or, for a "prevalent" (infinite-dimensional full measure) set of systems, the BNF is actually divergent.

The present work by Krikorian provides, for analytic symplectic diffeomorphisms and for Hamiltonian flows, a complete answer to the question of BNF divergence versus convergence and relates this to explicit dynamical consequences.

Main Results

Prevalence of Divergent BNFs

The principal result established is that for any dimension $d \geq 1$, and for any nonresonant (in action-angle: Diophantine) frequency, the set of real analytic symplectic diffeomorphisms (or Hamiltonians) with a divergent Birkhoff Normal Form is prevalent. Specifically:

• "For any nonresonant (or Diophantine, when required) frequency vector $\omega$, there exists a prevalent set of real analytic symplectic diffeomorphisms whose BNF is divergent."
• The prevalence here is in the sense of infinite-dimensional Lebesgue measure (Pérez-Marco-type prevalence), not just Baire genericity.

This settles, in all dimensions and in both the (CC) 'Cartesian coordinates' and the (AA) 'action-angle' cases, the dichotomy previously open in Pérez-Marco's theorem, and answers a longstanding question posed by H. Eliasson.

Dynamical Consequences of BNF Convergence

Krikorian establishes that convergence of the BNF is not a "dynamically silent" property. Specifically, if the BNF converges for a real analytic symplectic twist map in dimension one, the Lebesgue measure of the set of points not lying on invariant tori (i.e., the "non-KAM" set or "holes" in KAM theory) becomes exponentially smaller than in the generic case. More precisely:

• For analytic area-preserving twist diffeomorphisms (on the disk or the cylinder), if the BNF converges, then one can significantly improve the classical KAM measure estimate for the measure of the non-torus set: it decays (in a specific scaling) much faster (see equations (1.18), (1.21) in the text).
• The generic divergent case achieves the optimal (smallest possible) exponent in the KAM estimate. These are demonstrated by explicit construction and sharp analytic estimates using the refined KAM iteration and Hamilton-Jacobi normal forms.

This provides, for the first time, a sharp link between formal properties (BNF convergence) and measurable dynamical rigidity.

Construction of Divergent BNF Examples

The proofs utilize explicit local constructions and analysis of resonant normal forms, together with measure-theoretical arguments (including "no-screening" criteria in the KAM iteration). The result is the explicit construction (using small-divisor mechanisms akin to Siegel's examples) of real analytic symplectomorphisms/Hamiltonians with divergent BNF for any prescribed nonresonant (or Diophantine) frequency.

Moreover, the arguments are robust and yield not just examples, but prevalent entire classes of divergence—in sharp contrast with the relative paucity of convergence cases.

Further Implications and Open Problems

Krikorian connects these findings to several deeper rigidity and integrability questions:

• Integrability Rigidity: If the BNF is trivial (matches the linear/quadratic term), real-analytic conjugacy to the integrable system can be expected, given a Diophantine condition on the frequency (Bruno-Rüssmann theorem and its generalizations).
• Accumulation by Integrable Models: The text raises the problem whether an analytic symplectomorphism is always a limit (in analytic topology) of integrable ones, a question interconnected with the density of analytic pseudo-rotations and Anosov-Katok-type systems.
• Relation to Global Dynamical Features: The divergence of the BNF is associated with the appearance of "hyperbolic eyes" (regions where no invariant curves persist), showing a rich interplay between formal properties and local/global hyperbolic structures.
• The question of whether real-analytic convergence of the BNF implies full analytic integrability remains open for $d = 1$.

Technical Innovations

KAM and Hamilton-Jacobi Normal Forms on Domains with Holes

The author develops a refined KAM procedure that works on holed complex domains rather than the standard full polydisk---a necessity after removing sets supporting resonant tori. This is supplemented with a "no-screening" criterion, controlling possible KAM "shielding" of certain regions.

Additionally, Hamilton-Jacobi normal forms are used to analyze dynamics near resonant "holes", and to show how possible analytic extensions are forced (or limited) by proximity to holomorphic functions, aiding in estimating measures of invariant versus non-invariant sets.

Quantitative Comparison and Extension Principles

A key innovation is the use of "comparison" and "extension" principles showing that frequencies defined via local normal forms must closely match across overlapping domains, and that convergence of the BNF in a single domain has strong implications for the frequency map in surrounding domains. These are vital in proving rigidity of convergence and for optimal measure estimates.

Explicit Construction of Hyperbolic Zones

A detailed local analysis shows, for divergent cases, that "hyperbolic eyes" (region of phase space not intersected by invariant tori) persist and have explicit size, depending on arithmetic properties of the frequency.

Table: Summary of Key Results and Concepts

Concept	Description/Result
Birkhoff Normal Form	Formal power series integrable normal form for symplectic maps; here proved to be generally divergent in the analytic category for nonresonant frequencies
Prevalence	Sets of analytic diffeomorphisms/Hamiltonians with divergent BNFs are "prevalent"—a measure-theoretic strengthening of genericity
KAM Estimate	The measure of the non-KAM set decays as $\exp(-C/t^{1+\tau})$, with a much improved exponent for convergent BNF cases
Hamilton-Jacobi (HJ) NF	Provides local integrable approximation near resonant zones; key for sharp measure estimates and matching with BNF in complex neighborhoods
No-Screening Criterion	Ensures that holed KAM domains, arising from removing resonant disks, cannot "screen" analytic propagation in the KAM iteration
Comparison Principle	Forces stability of frequency maps and ensures that analytic continuation (if available) glues up local forms in larger domains
Construction of "Eyes"	Dynamically "hyperbolic" regions where invariant tori cannot enter—provide lower bounds on the complement of the union of invariant curves (sharp for divergent BNF)

Future Directions

This work closes a set of dichotomy questions around BNF convergence and divergence for analytic symplectic systems. Several research directions follow:

• Rigidity/Integrability in Higher Dimensions: The full analytic consequences of BNF convergence for $d \geq 2$ remain open, especially concerning real analytic integrability/generalized rigidity.
• Fine Measure/Quantitative Rigidity: Further exploration of how quantitative arithmetic properties of the frequency impact the measure and topology of invariant sets, especially in more degenerate or singular cases.
• Global Dynamics and BNF: Whether one can establish analogous results for global objects (e.g., for entire phase space, not just local charts) or in presence of additional symmetries (e.g., time-reversal).
• Extension to Partially Hyperbolic or Dissipative Systems: Whether the analytic–formal dichotomy persists in more general (e.g., non-conservative) settings.
• Effective Analytical Results: Development of procedures for computing (numerically or analytically) the critical thresholds for BNF convergence/divergence.

Conclusion

This work provides definitive results on the divergence of Birkhoff Normal Forms for analytic symplectic maps and Hamiltonians, establishing that divergence is the prevalent behavior except in cases of special rigidity. Importantly, convergence of the BNF is shown to have sharp and far-reaching dynamical consequences, decreasing the "non-integrable" measure well beyond standard KAM estimates. The technical framework—refined KAM theory, analytic continuation on holed domains, and matching principles—represents a comprehensive toolkit for further advances in Hamiltonian and dynamical systems theory.

Reference:

R. Krikorian, "On the divergence of Birkhoff Normal Forms," (1906.01096).