KAM-stability of elliptic equilibria of real-analytic Hamiltonians: Herman's conjecture beyond two degrees of freedom, and analytic examples that are neither KAM-stable nor Lyapunov unstable

Fayad and Saprykina (2017) constructed smooth ($C^\infty$) Hamiltonians on $\mathbb{R}^{2d}$, $d\geq 3$, having an elliptic equilibrium of arbitrary frequency that is not accumulated by any positive-measure set of invariant tori, i.e. not KAM-stable. The real-analytic counterparts of their result are all open, and this problem isolates that open core. Part (A) is Herman's 1998 conjecture: every elliptic equilibrium of a real-analytic Hamiltonian with a Diophantine frequency vector is KAM-stable; this is proved only for two degrees of freedom (Rüssmann 1967) and remains open for $d\geq 3$, with no analytic counterexample known in any dimension or frequency class. Part (B) asks whether there exists a real-analytic Hamiltonian with a non-resonant elliptic equilibrium that is neither KAM-stable nor Lyapunov unstable; no such example is known even for Liouville frequencies, and the available analytic machinery (successive conjugations à la Anosov–Katok) is currently inapplicable to equilibrium points. Both parts remain open despite substantial adjacent progress through 2026: analytic Lyapunov-unstable equilibria (Fayad 2023), analytic unstable quasi-periodic tori (Farré–Fayad 2022), generic divergence of Birkhoff normal forms (Krikorian 2022), and analytic pseudo-rotations of surfaces (Berger).

Background

An autonomous Hamiltonian $H$ on $\mathbb{R}^{2d}$ with canonical coordinates $(x,y)$ generates the flow of the vector field $X_H=(\partial_y H,-\partial_x H)$. If $DH(0)=0$ and the eigenvalues of the linearization are simple and purely imaginary, the origin is an elliptic equilibrium; writing $r_j=\tfrac12(x_j^2+y_j^2)$, one has $H=\langle\omega_0,r\rangle+\mathcal{O}^3(x,y)$, and the frequency vector $\omega_0\in\mathbb{R}^d$ is the vector of rotation angles of the linearized flow. The frequency is non-resonant if its coordinates are rationally independent; it is Diophantine if $|\langle k,\omega_0\rangle|\geq\gamma |k|^{-\tau}$ for all $k\in\mathbb{Z}^d\setminus\{0\}$ with some $\gamma,\tau>0$ (very well approximable and Liouville frequencies are non-Diophantine). A Lagrangian invariant torus is an invariant $d$-dimensional torus on which the flow is quasi-periodic; the equilibrium is KAM-stable if it is accumulated by invariant Lagrangian tori whose Lebesgue measure in small neighborhoods of the origin is positive, and Lyapunov stable if every neighborhood $U$ of the origin contains a neighborhood $V$ such that orbits starting in $V$ stay in $U$ for all forward time. The two stability notions do not coincide: in the Gevrey category (regularity strictly between real-analytic and $C^\infty$), Trujillo (2023) constructed an elliptic equilibrium that is KAM-stable yet Lyapunov unstable, while in the real-analytic category no example is known of an equilibrium that is not KAM-stable yet not Lyapunov unstable either.

Classical KAM theory (after Kolmogorov, Arnold, and Moser) proves KAM-stability of an elliptic equilibrium under a Kolmogorov (twist) non-degeneracy condition on the Birkhoff normal form — the formal integrable Taylor series that eliminates non-resonant terms order by order. Rüssmann (1967) proved that for two degrees of freedom no non-degeneracy condition is needed: an analytic elliptic equilibrium with Diophantine frequency is KAM-stable. For arbitrary dimension, Eliasson, Fayad, and Krikorian (2013) established KAM-stability under the Rüssmann non-planarity condition on the Birkhoff normal form (the image of the normal form's frequency map is not contained in any affine hyperplane), and (2015) proved that analytic KAM tori are never isolated; generic analytic elliptic equilibria are moreover super-exponentially stable in time (Bounemoura–Fayad–Niederman 2020), and near an analytic Diophantine torus an exponentially full relative measure of motions is effectively quasi-periodic (Bounemoura–Farré 2023). Beyond such hypotheses nothing is known: at his 1998 ICM lecture Herman conjectured that for analytic Hamiltonians, KAM-stability holds near an elliptic fixed point whenever the frequency is Diophantine, in any dimension; this remains open for $d\geq 3$. Below analytic regularity Herman also proved that smooth two-degree-of-freedom Diophantine equilibria are KAM-stable without twist, while he announced that KAM-stability fails for smooth Hamiltonians in four or more degrees of freedom. Fayad and Saprykina (2017) settled this smooth case: for every $d\geq 3$ and every frequency they built $C^\infty$ Hamiltonians with a non-KAM-stable elliptic equilibrium (for $d\geq 4$, with no invariant $d$-torus at all), via the Anosov–Katok scheme of successive conjugations of integrable models. Their paper states explicitly that in the analytic category no example is known of a non-resonant elliptic fixed point that is neither KAM-stable nor Lyapunov unstable, and their smooth method does not extend to analytic regularity near equilibria because the action-angle coordinates (the variables $(\theta,r)$ in which an integrable Hamiltonian depends only on the actions $r$) become singular on the axes $\{r_j=0\}$. Subsequent analytic progress stops at adjacent settings: Fayad (2023) produced the first real-entire Hamiltonians with Lyapunov-unstable non-resonant elliptic equilibria (any $d\geq 4$ for frequencies not of constant sign, and some $\omega_0\in\mathbb{R}^3$), but his examples can be chosen KAM-stable, and he notes that Lyapunov instability remains open for $d=3$ with Diophantine frequency and for $d=2$; Farré and Fayad (2022) built real-analytic Hamiltonians with topologically unstable quasi-periodic tori (for sufficiently Liouville frequencies, not accumulated by positive-measure families of quasi-periodic tori), but this mechanism is, in Fayad's words, inapplicable to equilibrium points; Krikorian (2022) proved that Birkhoff normal forms of analytic non-resonant elliptic fixed points are generically divergent, blocking normal-form-convergence strategies; and Berger's analytic pseudo-rotations (analytic maps with a single periodic orbit; with Delaporte's extensions) realize analytic Anosov–Katok schemes on the disk, sphere, and cylinder for surface symplectomorphisms with Liouville rotation numbers, solving the map version of Herman's problem but not the Hamiltonian-flow, equilibrium, or Diophantine cases.

Problem Statement

The problem is the analytic-regularity core left open by the smooth examples of Fayad–Saprykina (2017), in two linked parts.

Part (A) — Herman's conjecture for Hamiltonian flows beyond two degrees of freedom. Let $d\geq 3$ and let $H$ be a real-analytic Hamiltonian on a neighborhood of the origin of $\mathbb{R}^{2d}$ of the form $H=\langle\omega_0,r\rangle+\mathcal{O}^3(x,y)$ with $r_j=\tfrac12(x_j^2+y_j^2)$ and $\omega_0$ Diophantine, so that the origin is a non-resonant elliptic equilibrium with frequency $\omega_0$. Prove that the origin is KAM-stable — accumulated by invariant Lagrangian tori occupying positive Lebesgue measure in every sufficiently small neighborhood — with no non-degeneracy hypothesis on $H$; or exhibit a real-analytic $H$ (in some dimension $d$) with a Diophantine elliptic equilibrium that is not KAM-stable. The two degrees of freedom case is settled positively (Rüssmann 1967); the statement for $d\geq 3$ is open, and no analytic counterexample is known in any dimension.

Part (B) — analytic examples that are neither KAM-stable nor Lyapunov unstable. Does there exist a real-analytic Hamiltonian on $\mathbb{R}^{2d}$, for some $d\geq 2$, with a non-resonant elliptic equilibrium (of any arithmetic type, necessarily non-Diophantine if Herman's conjecture holds, e.g. Liouville) that is simultaneously not KAM-stable and not Lyapunov unstable? No such example is known in the analytic category; the known smooth examples use successive conjugations à la Anosov–Katok, whose analytic realization is presently blocked at equilibrium points, while the existing analytic constructions give either Lyapunov-unstable equilibria (Fayad 2023) or unstable quasi-periodic tori (Farré–Fayad 2022), neither of which answers the question.

An answer to either part in either direction (explicit analytic construction or general theorem) resolves the corresponding open question as stated; subsidiary open gaps recorded in the literature — Lyapunov instability for $d=3$ with Diophantine frequency and for $d=2$ — are not required for completion.

References

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