Billiards and Quantitative Symplectic Geometry — Johanna Bimmermann — Problem 2

Definitions and mathematical statement

Let $N$ denote the ambient space intended by the source, and let $\mathcal B(N)$ denote its intended class of embedded billiard tables. These are placeholders: the supplied record does not specify $N$, its geometric structure, or the admissibility conditions for billiard tables. For $\Omega\in\mathcal B(N)$, retain the source notation $\Omega\text{-}\mathrm{sys}$ for its unspecified numerical invariant; its definition is absent from the supplied record.

Subject to those unresolved definitions, the question is whether \[ \exists (\Omega_i)_{i\geq 1}\in\mathcal B(N)^{\{1,2,\ldots\}} \quad\text{such that}\quad \Omega_i\text{-}\mathrm{sys}\longrightarrow+\infty \quad\text{as }i\to\infty. \] Here divergence to $+\infty$ means that, for every real $M>0$, there exists an integer $i_0\geq1$ such that $\Omega_i\text{-}\mathrm{sys}>M$ for every integer $i\geq i_0$, provided the intended invariant admits these comparisons. No nesting condition on the tables is stated.

The source further says that a positive answer would, to the best of the author's knowledge, give the first example of a disk cotangent bundle with infinite Hofer--Zehnder capacity. This is a supplied, author-qualified consequence and novelty assertion; the record does not identify the relevant bundle or provide the hypotheses or argument linking the question to that consequence.

Short English statement

Does the intended ambient space $N$ contain a sequence of embedded billiard tables whose source-designated invariant $\Omega_i\text{-}\mathrm{sys}$ tends to infinity? The ambient geometry, admissible tables, and invariant are unspecified, so this remains a conditional formulation requiring clarification. The registry labels the problem open; that label is supplied metadata, not an independently verified mathematical assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800471$, source problem code catalogue entry; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record lists Albach, Barraud et al. as proposers, gives the year 2026, and labels the problem open.
• Albach, Barraud et al., Open Problems in Billiards and Quantitative Symplectic Geometry (2026), source item "Johanna Bimmermann --- Problem 2." Supplied reference: https://arxiv.org/abs/2602.12896. Contributor: Johanna Bimmermann.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This is a supplied access record, not a claim of independent source consultation or verification. The assertion about a first disk cotangent bundle example with infinite Hofer--Zehnder capacity retains the source author's knowledge qualification.