Fayad (J. Amer. Math. Soc. 2023) constructed the first real-analytic Hamiltonians with Lyapunov-unstable elliptic equilibria: in d≥4d\geq 4 degrees of freedom for arbitrary mixed-sign frequency vectors (Diophantine included), and in d=3d=3 for one super-Liouville frequency. His paper explicitly leaves open three existence questions that jointly delineate the remaining analytic regimes: (i) real-analytic instability in two degrees of freedom with non-resonant frequency, where only Gevrey examples (Trujillo 2023) are known; (ii) three degrees of freedom with Diophantine frequency; and (iii) instability coexisting with a convergent Birkhoff normal form, solved for invariant tori (Farré–Fayad 2022) but open for equilibria. The paper additionally remarks that its constructions do not produce an orbit accumulating on the origin — a behavior known for smooth maps (Fayad–Marco–Sauzin 2020) but not for analytic flows — and lists Herman's stability conjecture; both are related questions outside the scope of this task. All three questions remain open.

Background

An autonomous Hamiltonian H:R2d→RH:\mathbb{R}^{2d}\to\mathbb{R} in canonical coordinates (x,y)(x,y) generates the flow ΦHt\Phi^t_H of the vector field x˙=∂H/∂y\dot{x}=\partial H/\partial y, y˙=−∂H/∂x\dot{y}=-\partial H/\partial x, which preserves both the energy HH and the symplectic form. The origin is an elliptic equilibrium if ∇H(0)=0\nabla H(0)=0 and the linearized flow is a pure rotation with eigenvalues ±iωj\pm i\omega_j, ωj≠0\omega_j\neq 0; the vector ω=(ω1,…,ωd)\omega=(\omega_1,\dots,\omega_d) is the frequency vector, and ω\omega is non-resonant when no integer vector k∈Zd∖{0}k\in\mathbb{Z}^d\setminus\{0\} satisfies ⟨k,ω⟩=0\langle k,\omega\rangle=0. The origin is Lyapunov stable if for every ε>0\varepsilon>0 there exists δ>0\delta>0 such that every trajectory starting within δ\delta of the origin stays within ε\varepsilon for all positive times; otherwise it is Lyapunov unstable. Conservation of HH immediately gives stability when the quadratic part of HH at the origin is definite (equivalently, when all coordinates of ω\omega have the same sign), so instability can only be sought at frequency vectors of mixed sign.

Two classical frameworks constrain — but do not settle — instability. First, KAM theory (after Kolmogorov, Arnold and Moser): a non-resonant elliptic equilibrium is typically accumulated by quasi-periodic invariant Lagrangian tori, on which the motion is conjugate to a rigid rotation; when these tori have relative density tending to one near the origin one says the equilibrium is KAM-stable. Eliasson, Fayad and Krikorian (2013) proved that KAM stability holds under a non-degeneracy condition on the Birkhoff normal form (BNF) — the formal power series in the actions Ij=(xj2+yj2)/2I_j=(x_j^2+y_j^2)/2 obtained by formally eliminating the angle variables order by order, a series that may converge or diverge. KAM stability is a positive-measure statement and does not imply Lyapunov stability: orbits may still drift through the complement of the invariant tori. Second, effective stability: for Diophantine frequency vectors — those satisfying ∣⟨k,ω⟩∣≥γ∣k∣−τ|\langle k,\omega\rangle|\geq\gamma|k|^{-\tau} for some γ,τ>0\gamma,\tau>0 and all k∈Zd∖{0}k\in\mathbb{Z}^d\setminus\{0\}, as opposed to Liouville vectors, which admit extremely small denominators — Nekhoroshev-type results confine orbits near the equilibrium over doubly exponential time scales, but only for finite time, so Lyapunov stability is again not decided.

In the smooth category, unstable elliptic examples are classical: Douady (1988) built C∞C^\infty symplectic diffeomorphisms with Lyapunov-unstable elliptic fixed points in every d≥2d\geq 2, using chains of heteroclinic intersections in the spirit of Arnold diffusion, and the Anosov–Katok successive-conjugation method — which builds examples as C∞C^\infty-limits of conjugates of a simple integrable model by a sequence of increasingly fine conjugacies — yields smooth examples with unstable elliptic equilibria: in two degrees of freedom for non-Diophantine frequencies, and in three or more degrees of freedom with arbitrary frequencies (Fayad–Saprykina 2017). Arnold conjectured (1994) that, apart from the sign-definite case and — generically — the case d=2d=2, a generic real-analytic elliptic equilibrium is Lyapunov unstable; until Fayad's work not even one real-analytic example was known, in any dimension. On the stability side, Herman conjectured (ICM 1998) that every real-analytic elliptic equilibrium with Diophantine frequency is accumulated by a positive-measure set of KAM tori, with no non-degeneracy assumption; this conjecture is also open.

Fayad (2023) produced the first real-analytic examples, via a new diffusion mechanism starting in the neighborhood of an elliptic equilibrium: real entire Hamiltonians on R2d\mathbb{R}^{2d}, d≥4d\geq 4, with a Lyapunov-unstable elliptic equilibrium of any prescribed mixed-sign frequency vector (Diophantine ones included), and, on R6\mathbb{R}^6, one example with a non-resonant but super-Liouville (extremely well approximable) frequency. In all these examples with non-resonant frequencies the BNF diverges; variants with non-degenerate BNF are simultaneously KAM-stable and Lyapunov unstable. The same paper constructs, for every d≥2d\geq 2 and every non-resonant frequency, real entire Hamiltonians with divergent BNF — but without instability. What this construction cannot reach is exactly the content of the present problem.

Problem Statement

The common objective is to complete the existence picture of Lyapunov instability for elliptic equilibria of real-analytic Hamiltonian flows: to determine, in each regime left open by Fayad's construction, whether an analytic example can exist. Three precise questions, each answerable positively by an explicit construction or negatively by a general theorem. Each is an independent existence question: an answer to any one of (P1)–(P3) settles that question and leaves the other two open, while completing the existence picture requires answers to all three.

(P1) Two degrees of freedom. Does there exist a real-analytic Hamiltonian H:R4→RH:\mathbb{R}^4\to\mathbb{R} such that the origin is an elliptic equilibrium of the flow ΦHt\Phi^t_H with non-resonant frequency vector ω∈R2\omega\in\mathbb{R}^2 of mixed sign (ω1ω2<0\omega_1\omega_2<0), and the origin is Lyapunov unstable: for some ε>0\varepsilon>0 and every δ>0\delta>0 there exist zz with ∣z∣<δ|z|<\delta and t>0t>0 such that ∣ΦHt(z)∣≥ε|\Phi^t_H(z)|\geq\varepsilon? (Resonant frequencies are not part of the open question: whenever kω1+lω2=0k\omega_1+l\omega_2=0 for some k,l≥1k,l\geq 1 with k+l>2k+l>2, Fayad's Corollary 1 gives explicit polynomial — hence real entire — Hamiltonians on R4\mathbb{R}^4 whose elliptic equilibrium of frequency (ω1,ω2)(\omega_1,\omega_2) is Lyapunov unstable, and his diffusion mechanism is built on these resonant two-degrees-of-freedom examples.)

(P2) Three degrees of freedom, Diophantine frequencies. Does there exist a real-analytic Hamiltonian on R6\mathbb{R}^6 with a Lyapunov-unstable elliptic equilibrium whose frequency vector ω∈R3\omega\in\mathbb{R}^3 (necessarily of mixed sign) is Diophantine, i.e. satisfies ∣⟨k,ω⟩∣≥γ∣k∣−τ|\langle k,\omega\rangle|\geq\gamma|k|^{-\tau} for some γ,τ>0\gamma,\tau>0 and all k∈Z3∖{0}k\in\mathbb{Z}^3\setminus\{0\}? (For d≥4d\geq 4 this is answered positively by Fayad; in d=3d=3 only a super-Liouville frequency is covered.)

(P3) Convergent Birkhoff normal form. Does there exist a real-analytic Hamiltonian, in some number d≥2d\geq 2 of degrees of freedom, with a Lyapunov-unstable elliptic equilibrium whose Birkhoff normal form at the equilibrium has positive radius of convergence? (For invariant quasi-periodic tori the analogous question is answered positively by Farré–Fayad; for equilibria every known analytic unstable example has divergent BNF.)

The following two related questions are out of scope for this task, which asks only (P1)–(P3): First, the source remarks that its constructions do not obtain an orbit accumulating on the origin, while Fayad–Marco–Sauzin (2020) built C∞C^\infty symplectic diffeomorphisms of R6\mathbb{R}^6 whose non-resonant elliptic fixed point attracts an orbit; whether a real-analytic Hamiltonian flow in the mixed-sign setting, where instability is not excluded by the energy obstruction, can have an elliptic equilibrium at the origin with an orbit z(t)z(t) satisfying z(t)→0z(t)\to 0 as t→+∞t\to+\infty — or whether such an orbit can be ruled out — is a related open question that the source mentions only as a limitation of its constructions. Second, the source lists Herman's conjecture on the stability side (positive-measure accumulation of KAM tori at every analytic Diophantine elliptic equilibrium); that is a distinct named conjecture.

References

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