Herman conjectured (ICM 1998) that an elliptic equilibrium, or an invariant torus, of a real-analytic Hamiltonian system whose frequency vector is Diophantine is KAM-stable: every neighborhood contains real analytic invariant KAM-tori whose union has Lebesgue density one at the point (and hence positive Lebesgue measure). The conjecture is known only for two degrees of freedom (Rüssmann). Eliasson, Fayad and Krikorian proved that an analytic elliptic fixed point with Diophantine frequency is always accumulated by complex analytic KAM-tori, and that real positive-measure KAM-stability holds whenever the Birkhoff normal form satisfies a Rüssmann transversality condition; but when the Birkhoff normal form is degenerate at an intermediate level they obtain only a complex-analytic foliation by KAM-tori of fixed frequency, with no guarantee that these tori are real. This leaves precisely two linked open questions: Herman's conjecture itself in d >= 3 degrees of freedom, and whether the complex KAM-tori of the degenerate regime can be realized as real analytic invariant tori — the exact obstruction separating Eliasson–Fayad–Krikorian's theorem from the full conjecture. The conjecture is open for d >= 3: the only claimed proofs (Garay and Garay–van Straten) are not accepted by the KAM community, while partial results delimit the arithmetic hypotheses (Liouville counterexamples exist without nondegeneracy, and non-degenerate Liouville tori are KAM-stable).

Background

A Hamiltonian flow on the symplectic space R2d\mathbb{R}^{2d} with coordinates (x,y)(x,y) is generated by a real-analytic function HH via x˙=∂H/∂y\dot{x}=\partial H/\partial y, y˙=−∂H/∂x\dot{y}=-\partial H/\partial x. An equilibrium (fixed point) at the origin is called elliptic if the linearized flow is conjugate to a rotation with purely imaginary eigenvalues ±iω0,j\pm i\omega_{0,j}, j=1,…,dj=1,\dots,d; the vector ω0∈Rd\omega_0\in\mathbb{R}^d is the frequency of the equilibrium. A frequency vector is Diophantine if for some constants γ>0\gamma>0, τ>d\tau>d it satisfies ∣⟨k,ω0⟩∣≥γ∣k∣−τ|\langle k,\omega_0\rangle|\ge\gamma|k|^{-\tau} for every nonzero integer vector k∈Zdk\in\mathbb{Z}^d — a quantitative condition of being poorly approximated by resonances, satisfied by almost every vector in the measure sense but excluding Liouville vectors, which are extremely well approximated by resonances. A KAM-torus is an invariant dd-dimensional torus on which the flow is analytically conjugate to a rigid rotation t↦t+ωt\mapsto t+\omega (quasi-periodic motion); following the convention adopted by Eliasson–Fayad–Krikorian, the frequency vector ω\omega of a KAM-torus is required to lie in some Diophantine set DC(κ,τ)DC(\kappa,\tau), i.e., to satisfy ∣⟨k,ω⟩∣≥κ∣k∣−τ|\langle k,\omega\rangle|\ge\kappa|k|^{-\tau} for all 0≠k∈Zd0\ne k\in\mathbb{Z}^d with some κ,τ>0\kappa,\tau>0. KAM theory (Kolmogorov–Arnold–Moser) establishes persistence of such tori under small perturbations of integrable systems, but classically only under a nondegeneracy (twist) condition ensuring that the local frequency map covers a positive-measure set of admissible frequencies. A compact invariant set is called KAM-stable if every neighborhood of it contains invariant KAM-tori whose union has Lebesgue density one at the set — the definition adopted by Eliasson–Fayad–Krikorian ('a set of KAM tori whose Lebesgue density in the phase space tend to one in the neighborhood' of the set); density one implies in particular positive Lebesgue measure of the union in every neighborhood, and a weaker variant of the notion asks only for this positive-measure property.

Near an elliptic equilibrium, an integrable approximation is provided by the Birkhoff normal form (BNF): a formal, unique, integrable series NH(r)N_H(r) in the actions rj=(xj2+yj2)/2r_j=(x_j^2+y_j^2)/2 to which HH can be formally conjugated when ω0\omega_0 is non-resonant (no integer relation ⟨k,ω0⟩=0\langle k,\omega_0\rangle=0). The Rüssmann transversality condition — the requirement that the image of the gradient map r↦∇NH(r)r\mapsto\nabla N_H(r) is not contained in any proper affine subspace of Rd\mathbb{R}^d — is the weakest classical condition under which this frequency map sweeps out a positive-measure set of Diophantine vectors, and Rüssmann (1967) showed it is necessary and sufficient for KAM-stability in the non-singular perturbative setting. The neighborhood of an elliptic fixed point is, however, a singular perturbation problem: in the natural variables the perturbation of the integrable normal form is not small, so classical KAM theorems do not apply directly, and the analytic category genuinely matters — in the C∞C^\infty or Gevrey categories, Eliasson–Fayad–Krikorian constructed (for d≥4d\ge4) Hamiltonians with invariant tori of arbitrary frequency that are not accumulated by any positive-measure set of invariant tori.

In this landscape, Herman conjectured in his 1998 ICM lecture that for real-analytic Hamiltonians, Diophantine frequency alone — with no nondegeneracy or transversality hypothesis — forces KAM-stability of an elliptic equilibrium or an invariant torus. The conjecture is known to be true for two degrees of freedom (d=2d=2; Rüssmann's theorem as cited by Eliasson–Fayad–Krikorian), and remains open for all d≥3d\ge3. Eliasson–Fayad–Krikorian (2013) proved for arbitrary dd that an analytic elliptic equilibrium with Diophantine ω0\omega_0 is always accumulated by complex analytic KAM-tori, via the following alternative, where the BNF is called jj-degenerate if there exist exactly jj orthonormal vectors γ1,…,γj∈Rd\gamma_1,\dots,\gamma_j\in\mathbb{R}^d such that ⟨∂rNH(r),γi⟩=0\langle\partial_r N_H(r),\gamma_i\rangle=0 for every rr near the origin — equivalently, NHN_H itself is constant along those directions — and no j+1j+1 orthonormal vectors have this property (since ω0≠0\omega_0\ne0, necessarily 0≤j≤d−10\le j\le d-1): (i) if NHN_H satisfies Rüssmann transversality (00-degenerate), the equilibrium is accumulated by real analytic KAM-tori of positive measure and density one, and Diophantine frequency can be weakened to mere irrationality; (ii) if NHN_H is (d−1)(d-1)-degenerate, Rüssmann's theorem gives a full neighborhood foliated by real analytic KAM-tori with frequency in Rω0\mathbb{R}\omega_0; (iii) if NHN_H is jj-degenerate with 1≤j≤d−21\le j\le d-2, they construct an analytic subvariety of complex dimension d+jd+j through the origin, foliated by complex analytic KAM-tori carrying the fixed frequency ω0\omega_0 — but their method guarantees reality only of the squared actions c2c^2, not of cc itself, so the tori cannot be certified to meet the real phase space. Cases (i) and (ii) exhaust all possibilities when d=2d=2, which is exactly why the conjecture is open precisely for d≥3d\ge3: the unresolved window is the intermediate degeneracy 1≤j≤d−21\le j\le d-2, where the gap between complex analytic KAM-tori and real KAM-stability remains open. In the companion invariant-torus case (Eliasson–Fayad–Krikorian 2015) the same alternative holds with real tori in the degenerate regime, so the reality obstruction is specific to the fixed-point setting.

Problem Statement

Prove or refute Herman's conjecture (1998) in three or more degrees of freedom: let d≥3d\ge3 and let HH be a real-analytic Hamiltonian defined near the origin of R2d\mathbb{R}^{2d}, having the origin as an elliptic equilibrium with Diophantine frequency vector ω0\omega_0 (linear part 12∑j=1dω0,j(xj2+yj2)\tfrac{1}{2}\sum_{j=1}^d\omega_{0,j}(x_j^2+y_j^2) with ∣⟨k,ω0⟩∣≥γ∣k∣−τ|\langle k,\omega_0\rangle|\ge\gamma|k|^{-\tau} for all 0≠k∈Zd0\ne k\in\mathbb{Z}^d and some γ>0\gamma>0, τ>d\tau>d). Is the origin necessarily KAM-stable — i.e., does every neighborhood of the origin contain real analytic invariant KAM-tori (with Diophantine frequency vectors, per the convention above) whose union has Lebesgue density one at the origin, and hence positive Lebesgue measure? The same question is posed, for d≥3d\ge3, for an analytic invariant Lagrangian torus with Diophantine frequency of a real-analytic Hamiltonian; in the torus case the degenerate-regime tori constructed by Eliasson–Fayad–Krikorian (2015) are already known to be real, and what remains open there is whether the real invariant KAM-tori near the torus fill a set of Lebesgue density one (hence positive measure) in every neighborhood of it. Herman's conjecture is open in both of these settings — the elliptic equilibrium and the invariant Lagrangian torus; the open core identified below is the remaining obstruction in the elliptic-equilibrium case.

The specific open core left by the source paper (Eliasson–Fayad–Krikorian 2013), which is the single obstruction separating their main theorem from the conjecture for elliptic equilibria: when the Birkhoff normal form NHN_H is jj-degenerate with 1≤j≤d−21\le j\le d-2, decide whether the complex analytic KAM-tori of fixed frequency ω0\omega_0 that they construct (foliating an analytic subvariety of complex dimension d+jd+j through the origin) can be realized as real analytic invariant tori in R2d\mathbb{R}^{2d}. A positive answer, together with the measure and density estimates for the resulting family of real tori, would prove Herman's conjecture for analytic elliptic equilibria in all dimensions; an explicit real-analytic counterexample — a Hamiltonian with a Diophantine elliptic equilibrium not accumulated by any positive-measure set of real analytic KAM-tori — would refute it. A negative answer to the open-core question — the constructed complex analytic KAM-tori cannot be realized as real analytic invariant tori — answers only that sub-question: it shows that the degenerate-regime complex-analytic foliation does not by itself yield real KAM-tori, and neither proves nor refutes Herman's conjecture.

References

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