Zhang–Si–Si: Corrected Resonant Almost-Periodic Poincaré–Treshchev PersistenceResearch Paper
## Motivation The classical persistence problem asks which invariant tori of an integrable Hamiltonian survive a small perturbation. A resonance makes some internal angular frequencies vanish, and an **almost-periodic perturbation** introduces infinitely many additional forcing frequencies. Zhang, Si, and Si state a Poincaré–Treshchev persistence theorem for this combined regime: most lower-dimensional resonant tori associated with nondegenerate relative equilibria persist as infinite-dimensional invariant tori [in Theorem 2.7](https://doi.org/10.3934/dcds.2026043). The formal mission targets that headline result in corrected form. It retains the paper’s nonresonance, spatial-shell, and averaged-critical-point hypotheses, while making explicit the full frequency twist and reduced-frequency parameter chart used later in the paper but absent from the printed theorem statement. ## Setting Let $n,m>0$ and $d=n+m$. The internal variables are $x\in\mathbb T^d$ and $y\in G\subset\mathbb R^d$, where $G$ is bounded and closed. The external forcing angles and their conjugate actions are $\theta\in\mathbb T^{\mathbb Z}$ and $J\in\ell^1(\mathbb Z;\mathbb R)$. The **suspended Hamiltonian** is $$ \mathcal H_\epsilon(\theta,J,x,y)= \sum_{j\in\mathbb Z}\omega_jJ_j+N(y)+\epsilon P(\theta,x,y,\epsilon). $$ Here $N$ is real analytic near $G$. A primitive rank-$m$ resonance lattice is represented by an integer matrix $K_0=(K_1,K_2)$ with $\det K_0=1$, where the last $m$ columns generate the lattice. The adapted angles are $(\psi,\phi)=K_0^{\mathsf T}x$, with $\psi\in\mathbb T^n$ and $\phi\in\mathbb T^m$. Writing $\nabla N$ for the actual derivative of $N$, define $$ O(g,G)=\{y\in G:K_2^{\mathsf T}\nabla N(y)=0\}, \qquad \Omega(y)=K_1^{\mathsf T}\nabla N(y). $$ The **averaged potential** $h_0(\phi,y)$ is the zero external and zero $\psi$ Fourier coefficient of $P$ at $\epsilon=0$. A pair $(\phi,y)$ is associated and nondegenerate when $$ \nabla_\phi h_0(\phi,y)=0, \qquad \det D_\phi^2h_0(\phi,y)\ne0. $$ Let $O_0$ be the subset of $O(g,G)$ admitting such a $\phi$, let $\Omega_0=\Omega(O_0)$, and trim in reduced-frequency coordinates: $$ \Omega_\xi=\{\eta\in\Omega_0:\operatorname{dist}(\eta,\partial\Omega_0)\ge\xi\}, \qquad O_\xi=O_0\cap\Omega^{-1}(\Omega_\xi). $$ The perturbation has the paper’s **spatial-shell Fourier expansion** over finite shells $A$ in a spatial structure $\mathcal S$. With $[A]=1+\sum_{j\in A}\log^{\varrho}(1+|j|)$ for $\varrho>2$, each shell has a uniform analytic Fourier bound $B_A$, and $\sum_AB_Ae^{s[A]}<\infty$ for some $s>0$. Every nonzero admissible finite external mode $k$ satisfies $$ |\langle k,\omega\rangle|\ge \frac{\gamma}{\Delta([[k]])\Delta(|k|_1)}, $$ where $\gamma>0$, $[[k]]$ is the minimum shell weight containing its support, and $\Delta$ is a nondecreasing approximation function with $\Delta(0)=1$, $\log\Delta(t)/t\downarrow0$, and $\int_0^\infty\log\Delta(t)t^{-2}\,dt<\infty$. ## Target Assume that $O_0$ is nonempty. On every sufficiently small trim $O_\xi$, require compactness and positive $n$-dimensional measure in the $\Omega$ chart, injectivity of $D(\nabla N)(y)$, an analytic lower-Lipschitz diffeomorphism $\Omega:O_\xi\to\Omega_\xi$, and nondegeneracy of every averaged critical point retained by the conclusion. Then, for every $0<\xi\le\xi_*$, there are $0<\epsilon_0\le1$, a rate $c(\epsilon)\to0$ as $\epsilon\downarrow0$, and closed measurable nonempty sets $\Lambda_\epsilon\subset O_\xi$ for $0<\epsilon\le\epsilon_0$ such that $$ \operatorname{vol}_n\bigl(\Omega(O_\xi\setminus\Lambda_\epsilon)\bigr) \longrightarrow0\qquad(\epsilon\downarrow0). $$ For every $y\in\Lambda_\epsilon$ and every associated nondegenerate $\phi$, construct a topological embedding $$ \iota_{\epsilon,y,\phi}:\mathbb T^{\mathbb Z}\times\mathbb T^n \longrightarrow (\mathbb T^{\mathbb Z}\times\ell^1)\times (\mathbb T^d\times\mathbb R^d) $$ that is analytic almost periodic with the same shell structure, is the image of the standard resonant torus under a local canonical transformation, is $c(\epsilon)$-close to that torus, and is invariant under $\mathcal H_\epsilon$ with rotation vector $(\omega,\Omega(y))$. ## Significance The result combines three features that are usually separated: a positive-rank internal resonance, an infinite external frequency vector, and persistence for an asymptotically full-measure parameter set. It identifies the surviving object in the full suspended phase, including the external actions, rather than only in the finite internal fibre. The paper contains a proof of its printed theorem; the mission’s open work is a Lean proof of the corrected target. The correction exposes the parameter twist and chart assumptions on which the reduction and measure assertion depend, so a completed formalization would distinguish the theorem’s stated content from hypotheses needed to make that content mathematically controlled. ## Difficulty Finite-dimensional Diophantine notation does not control infinitely many external modes: admissibility, support weights, and the decay of $\Delta$ must interact without leaving an empty mode class. Resonance also removes $m$ internal frequencies and replaces the original parameter set by a lower-dimensional surface. Its ambient Euclidean boundary is therefore unsuitable for a positive trim, and ordinary ambient volume is unsuitable for the majority statement. Finally, persistence must be expressed simultaneously as an invariant solution of the actual Hamilton equations, an analytic almost-periodic embedding, and a canonical equivalence in the suspended phase; any one of these conditions alone admits objects that do not express the theorem. ## Formalization scope The Lean representation uses finitely supported integer external modes, a covering spatial structure, the literal shell-indexed norm, complex-neighborhood coefficient analyticity, and a separate absolute-summability guard for every infinite Fourier sum. Unit modes are provably admissible. The averaged potential, its gradient and Hessian, the internal frequency $\nabla N$, the resonant set, and the reduced-frequency map are transparent definitions. Nonemptiness of $O_0$, positive volume of every allowed $\Omega_\xi$, and nonempty $\Lambda_\epsilon$ rule out empty-set and zero-measure trivializations. The formal target strengthens the printed assumptions only where the source’s reduction requires missing control: $D(\nabla N)$ is injective on the retained trim, and $\Omega$ is an analytic diffeomorphism there with a uniform lower Lipschitz bound. The boundary distance and excluded volume are both taken in the $n$-dimensional reduced-frequency chart. The word “Cantor” is represented by closedness, measurability, nonemptiness, and asymptotically full measure; perfectness and total disconnectedness are not additional targets. The torus lives in a genuine $\ell^1$ external-action space. Its action component has one weighted-$\ell^1$-valued shell expansion, and the Hamiltonian pairing and external action velocity carry convergence guards. A local conjugacy is a homeomorphism between open neighborhoods, fixes $\theta$, is differentiable along all $\ell^1$ action directions and finitely supported external-angle directions, and preserves $\sum_jd\theta_j\wedge dJ_j+\sum_i dx_i\wedge dy_i$ on those cylinder directions. Reusable contributions include spatial-shell Fourier classes, approximation functions, reduced-manifold measure interfaces, weighted infinite canonical forms, and coordinatewise Hamiltonian invariance. ## Selected references - Yuan Zhang, Wen Si, and Jianguo Si, *Poincaré–Treshchev Mechanism in Integrable Hamiltonian Systems Under Almost-Periodic Perturbations*, Discrete and Continuous Dynamical Systems **52** (2026), 32–69. [DOI: 10.3934/dcds.2026043](https://doi.org/10.3934/dcds.2026043). Main result: Theorem 2.7, journal p. 39 (PDF p. 8); spatial and nonresonance definitions: Definitions 2.2–2.4 and equations (5)–(7), journal pp. 35–38 (PDF pp. 4–7); reduction exposing the additional twist and parameter-change requirements: Lemma 3.2, journal pp. 41–43 (PDF pp. 10–12).