Density of Finite-Order Absolutely Periodic Billiard Orbits

Definitions and mathematical statement

The supplied record does not determine a fully specified mathematical problem. Its retained assertion is the following source claim:

For every $n\in{\mathbb{N}}$ and every strictly convex domain $\Omega\subset{\mathbb{R}}^2$ of the source's stated regularity $\Omega\in C^r({\mathbb{R}}^2)$, there exists a strictly convex domain $\widetilde\Omega\in C^{r-1}({\mathbb{R}}^2)$, arbitrarily close to $\Omega$ in the $C^{r-1}$ topology, having an absolutely periodic orbit of order $n$.

Here "arbitrarily close" expresses a neighborhood quantifier: every neighborhood of $\Omega$ in the intended $C^{r-1}$ topology must contain such a $\widetilde\Omega$. A domain conventionally means a nonempty connected open set. However, the source does not specify boundedness, the admissible values of $r$, the meaning of regularity for these domains, or the topology used to compare them. The notation $\Omega\in C^r({\mathbb{R}}^2)$ is retained as source notation; a domain is not itself a function, and interpreting this as boundary regularity requires clarification.

The billiard dynamics, the term "absolutely periodic orbit," and the meaning of its "order" are not defined in the supplied record. In particular, order $n$ must not silently be identified with least period $n$, number of reflections, or an order of vanishing. The convention for ${\mathbb{N}}$, including whether it contains zero, is also unspecified.

The supplied statement-verification field says that the official report presents this density construction as Theorem 1, rather than as an open problem. It marks the statement as unrecoverable and supplies no replacement open question. Accordingly, the displayed assertion is preserved for provenance and is not promoted here to an open conjecture or independently verified theorem.

Short English statement

The retained claim says that every strictly convex planar domain of the stated regularity can be approximated in $C^{r-1}$ by strictly convex domains carrying an absolutely periodic billiard orbit of any prescribed order. The orbit terminology and regularity hypotheses remain unspecified. Although the registry labels the entry open, its supplied verification identifies this assertion as Theorem 1 and provides no recoverable open question.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30004743$, exact source code catalogue entry, title "Density of Finite-Order Absolutely Periodic Billiard Orbits." Collection: Oberwolfach Reports Open Problems. The proposed year is recorded as 2021; no proposer is supplied.
• Supplied report citation: Dynamische Systeme (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/33, pp. 1735--1803. https://doi.org/10.4171/owr/2021/33. This is also the supplied literature-source link labeled "Oberwolfach question." The supplied verification attributes the displayed assertion to Theorem 1; this attribution has not been independently checked.
• Supplied literature-source link labeled "rigidity of Lagrangian products": https://www.intlpress.com/site/pub/files/_fulltext/journals/jsg/2019/0017/0005/JSG-2019-0017-0005-a007.pdf. No authors or fuller bibliographic citation are supplied.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the entry open and states that many special Lagrangian products, including bidisks, $\ell^p$ sums, and symmetric polygonal examples, are symplectomorphic to toric domains, while no general characterization or equivalence with Minkowski-billiard integrability is known. The background describes this as dated literature triage derived from supplied OWR curation and requests independent verification of time-sensitive status and cited claims. Its relationship to the displayed density assertion is unresolved.