Billiards and Quantitative Symplectic Geometry — Serge Tabachnikov — Problem 7

Definitions and mathematical statement

Let $\Sigma$ be a strictly convex closed hypersurface for which the intended multidimensional outer symplectic billiard construction is defined. Write $(P_\Sigma,\omega_\Sigma)$ for its symplectic phase space and $T_\Sigma:D_\Sigma\to P_\Sigma$, with $D_\Sigma\subseteq P_\Sigma$, for its billiard map. These are placeholders for the intended construction: the supplied record does not define the ambient space, the required regularity of $\Sigma$, the phase space, or the map.

If $\dim P_\Sigma=2m$, let $\mu_\Sigma$ denote the measure associated with the symplectic volume density $\lvert\omega_\Sigma^{\wedge m}\rvert/m!$. Define \[ D_\Sigma^{(3)}=\{x\in D_\Sigma:T_\Sigma x\in D_\Sigma,\ T_\Sigma^2x\in D_\Sigma\}, \] \[ F_3(\Sigma)=\{x\in D_\Sigma^{(3)}:T_\Sigma^3x=x\},\qquad E_3(\Sigma)=\{x\in F_3(\Sigma):T_\Sigma x\ne x\}. \] Thus $F_3(\Sigma)$ consists of phase points whose period divides $3$, and $E_3(\Sigma)$ consists of phase points of least period $3$.

The requested conjectural assertion is that, for every strictly convex closed hypersurface in the intended multidimensional setting, the set of $3$-periodic phase points has $\mu_\Sigma$-measure zero. Under the convention that $3$-periodic means $T_\Sigma^3x=x$, this is \[ \mu_\Sigma\left(F_3(\Sigma)\right)=0. \] Under the least-period convention, it is instead \[ \mu_\Sigma\left(E_3(\Sigma)\right)=0. \] The supplied record does not specify this convention or establish that fixed points are absent. It also does not specify the ambient dimension range meant by "multidimensional." Consequently, the displayed assertions express the target conditionally on the missing construction and conventions; they are not a complete geometric definition of the dynamics.

Short English statement

Serge Tabachnikov asks for the period-$3$ case of Ivrii's conjecture for multidimensional outer symplectic billiards associated with a strictly convex closed hypersurface: the corresponding periodic phase points should have zero symplectic volume. The supplied record leaves the precise dynamics, admissible dimensions, regularity assumptions, and period convention unspecified.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $800485$, source problem code catalogue entry; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is a supplied status, not an independently verified mathematical assessment.
• Albach, Barraud et al., Open Problems in Billiards and Quantitative Symplectic Geometry (2026), Serge Tabachnikov, Problem 7, https://arxiv.org/abs/2602.12896. The registry lists Albach, Barraud et al. under attribution and identifies Serge Tabachnikov as the contributor. Supplied access date: 2026-10-16; supplied extraction method: arxiv_latex. These provenance details are supplied metadata, not evidence of independent source consultation here.
• The supplied background reports that the conventional billiard version is proved for periods $2$ and $3$ in all dimensions, and for period $4$ in dimension $2$. It also reports planar outer billiard results for $(2n+1)$-periodic orbits of rotation number $n$, and $2n$-periodic orbits of rotation number $n-1$. These are supplied contextual claims, not independently verified literature assessments. No range for $n$ is supplied.
• Reference label [Sh] is cited in the supplied statement for these contextual results and further references, but its bibliographic entry is absent from the supplied record.