Billiards and Quantitative Symplectic Geometry — Serge Tabachnikov — Problem 4

Definitions and mathematical statement

A domain is a nonempty connected open set. A domain $\Omega$ in a Euclidean space is convex if $(1-t)x+ty\in\Omega$ for all $x,y\in\Omega$ and $t\in[0,1]$. Its boundary is $C^1$-smooth if it is locally the graph of a continuously differentiable function.

Write $T_\Omega$ for the symplectic billiard dynamics associated with $\Omega$, using the ambient symplectic space, phase space, and billiard convention intended by the source. These are not defined in the supplied record. The source asks: \[ \text{Does there exist a convex domain }\Omega\text{ with }C^1\text{-smooth boundary such that }T_\Omega\text{ is chaotic?} \] The term chaotic is not defined in the supplied record. Thus this is a formulation template requiring a specified dynamical property before it becomes a precise existence statement. The ambient dimension, any boundedness requirement, and the definition and well-definedness of the intended dynamics at the stated regularity also require clarification; no additional hypotheses are imposed here.

The source identifies the stadium as a natural candidate and reports numerical evidence suggesting chaotic behavior. It supplies neither a precise chaos criterion nor details of that numerical evidence. This suggestion is not asserted as a mathematical result.

Short English statement

The question asks whether a convex domain with a $C^1$-smooth boundary can have chaotic symplectic billiard dynamics. The stadium is suggested as a candidate on the basis of reported numerical evidence. A precise statement still requires the intended definition of chaos and the ambient and dynamical conventions.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json and the supplied additions record: permanent ID $800482$, source problem code catalogue entry; title: Billiards and Quantitative Symplectic Geometry --- Serge Tabachnikov --- Problem 4. The registry attributes the collection to Albach, Barraud et al., with proposed year 2026. Its supplied status is open; this label is not an independently verified mathematical or literature assessment.
• Albach, Barraud et al., Open Problems in Billiards and Quantitative Symplectic Geometry (2026), source item Serge Tabachnikov --- Problem 4; contributor: Serge Tabachnikov. Supplied source URL: https://arxiv.org/abs/2602.12896. The record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source was not independently accessed or verified for this draft.