Bounemoura, Fayad and Niederman proved that a real-analytic elliptic equilibrium point of a Hamiltonian system with Diophantine frequency is, generically in the finite jet of its Birkhoff normal form, doubly exponentially stable: solutions starting at distance r from the equilibrium remain within distance 2r for times at least exp(exp(C r^{-1/(tau+1)})). Their works (arXiv:1509.00285, published as Adv. Math. 2020) explicitly pose three questions that remain open: (1) whether the genericity-producing hypothesis that the order-(n^2+4) Birkhoff jet is stably steep is necessary for such doubly exponential stability, or can be weakened or removed; (2) whether the doubly exponential lower bound on the stability time is essentially optimal in the analytic category, which hinges on constructing analytic Hamiltonians with Lyapunov-unstable elliptic equilibria and quantified instability times — not a single such example is known, and this is the crux of Arnold's 1994 instability conjecture; and (3) whether the result extends to quasi-periodic normally elliptic reducible invariant tori, of which the invariant Lagrangian Diophantine torus case was subsequently resolved positively by the same authors (Commun. Math. Phys. 2017) while the general case with nontrivial elliptic normal directions remains open. No resolution of any of these cores is known in the literature to date.

Background

A point where a Hamiltonian flow vanishes is called an elliptic equilibrium if the eigenvalues of the linearized flow are purely imaginary and simple; in suitable symplectic coordinates the Hamiltonian takes the form H(x,y)=α⋅I(x,y)+O3(x,y)H(x,y)=\alpha\cdot I(x,y)+O^3(x,y) with actions Ij=(xj2+yj2)/2I_j=(x_j^2+y_j^2)/2 and a frequency vector α∈Rn\alpha\in\mathbb{R}^n that is assumed non-resonant (k⋅α≠0k\cdot\alpha\neq 0 for all k∈Zn∖{0}k\in\mathbb{Z}^n\setminus\{0\}) and typically Diophantine, i.e. ∣k⋅α∣≥γ∣k∣1−τ|k\cdot\alpha|\ge\gamma|k|_1^{-\tau} for some γ>0\gamma>0 and τ≥n−1\tau\ge n-1. Lyapunov stability of such an equilibrium means that initial conditions sufficiently near it remain near it for all time; apart from two classical cases (definite quadratic part, and two degrees of freedom under an iso-energetic non-degeneracy condition where KAM tori — invariant tori carrying quasi-periodic motion that separate the energy levels — block escape), stability is expected to fail generically for n≥3n\ge 3 (Arnold's conjecture of 1994), yet under the non-resonance assumption above not a single example of an unstable analytic elliptic equilibrium is known. Effective stability measures how long the equilibrium behaves as if stable: T(r)T(r) is the largest time for which every solution starting within distance rr of the origin stays within distance 2r2r. Because the equilibrium sits at the origin, action-angle coordinates are singular there and the classical Nekhoroshev theory (exponential-in-time stability T(r)≳exp⁡(cr−a)T(r)\gtrsim\exp(c r^{-a}) for steep integrable systems under small perturbations) does not apply directly; steepness is a quantitative transversality condition on the gradient flow of the integrable Hamiltonian that prevents drift along resonances. Two exponential mechanisms are nevertheless available near an elliptic equilibrium: (i) analytically, the Birkhoff normal form (BNF) — the unique formal integrable approximation α⋅I+hm(I)\alpha\cdot I+h_m(I) obtained by a symplectic transformation averaging out angle-dependent terms up to a finite order KK, with m=⌊K/2⌋m=\lfloor K/2\rfloor — has remainders exponentially small in r−1r^{-1} when α\alpha is Diophantine; (ii) dynamically, if the normal form is convex or steep, Nekhoroshev-type estimates confine actions for exponentially long times (independent results of Niederman, Fassò–Guzzo–Benettin, and Pöschel around 1998–99, resolving a conjecture of Nekhoroshev for the convex case). Combining both mechanisms yields doubly exponential stability, first achieved by Giorgilli and Morbidelli for quasi-periodic Lagrangian tori under a convexity assumption that is open but far from generic. Bounemoura–Fayad–Niederman replaced convexity by genericity: a polynomial P0P_0 of degree mm is stably steep if there exist a neighborhood VV of P0P_0 and constants C,δ>0C,\delta>0 such that, for every integer l∈[1,n−1]l\in[1,n-1], every polynomial P∈VP\in V restricts, on every ll-dimensional vector subspace Λ\Lambda, to a function PΛP_\Lambda satisfying max⁡0≤η≤ξmin⁡∥x∥=η, x∈Λ∥∇PΛ(x)∥>Cξm−1\max_{0\le\eta\le\xi}\min_{\|x\|=\eta,\,x\in\Lambda}\|\nabla P_\Lambda(x)\|>C\xi^{m-1} for all 0<ξ≤δ0<\xi\le\delta; they proved (arXiv:1509.00285, 2015; published as Adv. Math. 366 (2020) 107088, with the companion Nonlinearity 33 (2020) 1–33 handling steep full normal forms) that the set SS(n,m0)SS(n,m_0) of stably steep polynomials is open, dense and of full measure in the space of possible Birkhoff jets of order K0=n2+4K_0=n^2+4, and that whenever the order-K0K_0 Birkhoff jet of HH lies in SS(n,m0)SS(n,m_0) the equilibrium is doubly exponentially stable: T(r)≥exp⁡(cr−2exp⁡(c′Δα(c′′r−1)))T(r)\ge\exp(c r^{-2}\exp(c'\Delta_\alpha(c'' r^{-1}))), where Δα\Delta_\alpha is the function controlling the small divisors of α\alpha (so for Diophantine α\alpha of exponent τ\tau, T(r)≥exp⁡(exp⁡(Cr−1/(τ+1)))T(r)\ge\exp(\exp(C r^{-1/(\tau+1)}))). In the published version this is phrased as generic super-exponential stability, generic meaning open and dense of full Lebesgue measure in the space of jets of degree between 3 and K0K_0. The quasi-periodic analogue of the setting is a normally elliptic reducible invariant torus, i.e. an invariant torus carrying quasi-periodic motion whose normal linear dynamics is elliptic and can be reduced to constants, modeled by Hamiltonians H(θ,J,x,y)=β⋅J+α⋅I(x,y)+F(θ,I,x,y)H(\theta,J,x,y)=\beta\cdot J+\alpha\cdot I(x,y)+F(\theta,I,x,y); the invariant Lagrangian (maximal-dimensional, n=0n=0 normal directions) Diophantine case of KAM theory is the special case in which the same authors later proved generic doubly exponential stability. Gevrey regularity — an intermediate class between real-analytic and C∞C^\infty with controlled growth of derivatives — is relevant because explicit instability constructions matching doubly exponential times are feasible there but have resisted all attempts in the analytic category.

Problem Statement

Let HH be a real-analytic Hamiltonian on R2n\mathbb{R}^{2n} with an elliptic equilibrium point at the origin (the eigenvalues of the linearized flow are purely imaginary and simple) and non-resonant frequency vector α\alpha, so H(x,y)=α⋅I(x,y)+O3(x,y)H(x,y)=\alpha\cdot I(x,y)+O^3(x,y) with Ij=(xj2+yj2)/2I_j=(x_j^2+y_j^2)/2 and O3O^3 denoting terms of total order at least 3, let K0=n2+4K_0=n^2+4, m0=⌊K0/2⌋m_0=\lfloor K_0/2\rfloor, let BNFK0(H)∈P2(n,m0)\mathrm{BNF}_{K_0}(H)\in\mathcal{P}_2(n,m_0) denote its Birkhoff normal form jet of order K0K_0 — the degree-m0m_0 polynomial in the actions II, with vanishing constant and linear terms, obtained by a symplectic transformation averaging out angle-dependent terms up to total order K0K_0 — and let T(r)T(r) denote the effective stability time (any solution starting within distance rr of the origin stays within distance 2r2r for ∣t∣≤T(r)|t|\le T(r)). Bounemoura–Fayad–Niederman proved that if BNFK0(H)∈SS(n,m0)\mathrm{BNF}_{K_0}(H)\in SS(n,m_0), the set of stably steep polynomials — polynomials PP for which there exist a neighborhood VV and constants C,δ>0C,\delta>0 such that, for every integer ll with 1≤l≤n−11\le l\le n-1, every polynomial in VV restricts, on every ll-dimensional vector subspace Λ\Lambda of Rn\mathbb{R}^n, to a function PΛP_\Lambda with max⁡0≤η≤ξmin⁡∥x∥=η, x∈Λ∥∇PΛ(x)∥>Cξm0−1\max_{0\le\eta\le\xi}\min_{\|x\|=\eta,\,x\in\Lambda}\|\nabla P_\Lambda(x)\|>C\xi^{m_0-1} for all 0<ξ≤δ0<\xi\le\delta — a condition satisfied by an open dense set of full Lebesgue measure in the space P2(n,m0)\mathcal{P}_2(n,m_0) of possible jets — then

T(r) ≥ exp⁡(c r−2exp⁡(c′ Δα(c′′ r−1))),T(r)\ \ge\ \exp\Big(c\,r^{-2}\exp\big(c'\,\Delta_\alpha(c''\,r^{-1})\big)\Big),

where c,c′,c′′>0c,c',c''>0 and Δα\Delta_\alpha is the small-divisor function of α\alpha, defined by Δα(x)=sup⁡{K≥1:KΨα(K)≤x}\Delta_\alpha(x)=\sup\{K\ge 1:K\Psi_\alpha(K)\le x\} with Ψα(K)=max⁡{∣k⋅α∣−1:k∈Zn, 0<∣k∣1≤K}\Psi_\alpha(K)=\max\{|k\cdot\alpha|^{-1}:k\in\mathbb{Z}^n,\ 0<|k|_1\le K\}; for Diophantine α∈DC(τ,γ)\alpha\in DC(\tau,\gamma) — meaning ∣k⋅α∣≥γ∣k∣1−τ|k\cdot\alpha|\ge\gamma|k|_1^{-\tau} for all k∈Zn∖{0}k\in\mathbb{Z}^n\setminus\{0\} — this gives T(r)≥exp⁡(exp⁡(Cr−1/(τ+1)))T(r)\ge\exp(\exp(C r^{-1/(\tau+1)})). The open problem posed by the authors consists of three linked questions: Question 1. Does the conclusion of this theorem remain true without the hypothesis BNFK0(H)∈SS(n,m0)\mathrm{BNF}_{K_0}(H)\in SS(n,m_0) — i.e., can doubly exponential effective stability at a non-resonant analytic elliptic equilibrium be proved under strictly weaker finite-jet hypotheses, or for all such Hamiltonians? Question 2. Is the doubly exponential lower bound on T(r)T(r) essentially optimal in the analytic category — in particular, does there exist a real-analytic Hamiltonian with a non-resonant elliptic equilibrium (with non-sign-definite quadratic part, n≥3n\ge 3) that is Lyapunov unstable, and if so can the instability time be quantified as an upper bound T(r)≲exp⁡(exp⁡(C′r−1/(τ+1)))T(r)\lesssim\exp(\exp(C' r^{-1/(\tau+1)})) matching the generic lower bound? Question 3. Does the result extend to quasi-periodic normally elliptic reducible invariant Diophantine tori, i.e. to real-analytic Hamiltonians of the form H(θ,J,x,y)=β⋅J+α⋅I(x,y)+F(θ,I,x,y)H(\theta,J,x,y)=\beta\cdot J+\alpha\cdot I(x,y)+F(\theta,I,x,y) with (θ,J)∈Td×Rd(\theta,J)\in\mathbb{T}^d\times\mathbb{R}^d, (x,y)∈R2n(x,y)\in\mathbb{R}^{2n} and FF of order at least 2 in II and 3 in (x,y)(x,y): if (β,α)(\beta,\alpha) is Diophantine of exponent τ\tau and (J,I(x,y))(J,I(x,y)) starts at distance rr from the torus {J=0,(x,y)=0}\{J=0,(x,y)=0\}, does (J(t),I(x(t),y(t)))(J(t),I(x(t),y(t))) remain within distance 2r2r for a time doubly exponentially large in r−1/(τ+1)r^{-1/(\tau+1)}? The common scientific objective of the three questions is the sharp characterization of generic long-time (effective) stability of analytic quasi-periodic and elliptic dynamics: Questions 1 and 2 delimit, respectively from below and above, the stability time and the hypotheses under which it holds at elliptic equilibria, and Question 3 asks whether the same mechanism governs motion near the mixed quasi-periodic-elliptic invariant tori that contain equilibria and Lagrangian tori as endpoint cases. Each of the three questions is stated independently and can be attacked on its own: a complete resolution of any one of them is a self-contained result, while the common scientific objective is attained only when all three are resolved. Of Question 3 only the Lagrangian endpoint (n=0n=0 normal directions) is settled, positively, by the same authors.

References

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