Zero Entropy Systems as Limits of Integrable Systems

Definitions and mathematical statement

Scope and unresolved qualifications. The supplied statement asks whether every conservative dynamical system of zero topological entropy in low dimensions is a limit of integrable systems. Here zero topological entropy means $h_{\mathrm{top}}(f)=0$ for a discrete-time system $f$. The record does not specify the dimensions, phase spaces, whether systems are maps or flows, the meaning of conservative or integrable, or the topology of approximation. Consequently, this broad question does not yet determine a unique quantified mathematical statement.

Disk specialization: a conditional formulation. To display the conservative approximation requirement, adopt the conventional closed-disk interpretation \[ D=\{(x,y)\in{\mathbb{R}}^2:x^2+y^2\leq 1\},\qquad \omega_0={\,\mathrm{d}} x\wedge{\,\mathrm{d}} y. \] Let $\operatorname{Diff}^{\infty}(D)$ denote diffeomorphisms smooth up to the boundary, with smooth inverses, and put \[ \operatorname{Diff}^{\infty}_{\omega_0}(D) =\{f\in\operatorname{Diff}^{\infty}(D):f^*\omega_0=\omega_0\}. \] Such maps carry the boundary to itself; no pointwise boundary-fixing condition is imposed here. Under the customary interpretation used for this template, a disk pseudo-rotation is a map $f\in\operatorname{Diff}^{\infty}_{\omega_0}(D)$ whose periodic-point set consists of exactly one interior fixed point: \[ \operatorname{Per}(f):=\bigcup_{k\geq1}\{z\in D:f^k(z)=z\}=\{z_f\},\qquad z_f\in\operatorname{int}(D). \] These disk and pseudo-rotation conventions are explanatory choices requiring confirmation against the source; they are not explicit hypotheses in the supplied wording.

For $\theta\in{\mathbb{R}}$, let $R_\theta$ be rotation through angle $2\pi\theta$: \[ R_\theta(x,y)=\left(x\cos(2\pi\theta)-y\sin(2\pi\theta),\ x\sin(2\pi\theta)+y\cos(2\pi\theta)\right). \] If rational-rotation approximations means smooth conjugates of rational rigid rotations, the disk question has the following template. For every such pseudo-rotation $f$, do there exist integers $p_n\in{\mathbb{Z}}$, positive integers $q_n$, and maps $h_n\in\operatorname{Diff}^{\infty}(D)$, indexed by $n\geq1$, such that \[ g_n=h_n^{-1}\circ R_{p_n/q_n}\circ h_n,\qquad g_n^*\omega_0=\omega_0,\qquad g_n\longrightarrow f\quad\text{in the intended topology }\mathcal T? \] Each displayed approximant satisfies $g_n^{q_n}=\operatorname{id}_D$. The preservation condition concerns the standard area form $\omega_0$; this template does not additionally require the conjugacies $h_n$ themselves to preserve it.

The topology $\mathcal T$, the intended class of conjugacies, the precise pseudo-rotation definition, and any additional boundary or rotation-number requirements remain unspecified. The displayed question is therefore conditional on resolving those choices, rather than a uniquely determined formalization of the source. The supplied dated assessment says that periodic approximation is known but that the cited primary theorem does not settle approximation preserving the standard area form. This is recorded as supplied curation, not as an independently verified theorem or status assessment.

Short English statement

The broad question asks whether conservative systems with zero topological entropy in low dimensions can be approximated by integrable systems. Its disk specialization asks whether rational-rotation approximants to every smooth area-preserving disk pseudo-rotation can also preserve the standard area form. The source leaves the convergence topology and several defining conventions unspecified, so the displayed disk formulation is conditional and requires mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30001833$, title Zero Entropy Systems as Limits of Integrable Systems, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied report citation: Dynamische Systeme (2012), Oberwolfach Reports, pp. 1913--1970, DOI 10.4171/owr/2011/34; https://doi.org/10.4171/owr/2011/34. This is also the supplied literature-source link labelled "original Oberwolfach report." The record gives proposed year 2011 and a 2011 year tag, while the citation title contains 2012; this discrepancy is unresolved.
• Supplied literature-source link labelled "Bramham periodic approximations": https://doi.org/10.4007/annals.2015.181.3.4. The record supplies no fuller bibliographic citation or theorem statement.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that periodic approximation is known, while the cited primary theorem does not settle the stronger standard-area-preserving approximation requirement. This assessment is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement "exact" and reports that the official report was checked and confirmed the wording. These are supplied source-verification claims; no independent source visit or verification is asserted here.