Billiards and Quantitative Symplectic Geometry — Bernhard Albach, Umberto Hryniewicz, and Alexandru Oancea — Problem 3

Definitions and mathematical statement

For a positive integer $n$, the standard symplectic torus is conventionally \[ (\mathbb T^{2n},\omega_0)=\left({\mathbb{R}}^{2n}/{\mathbb{Z}}^{2n},\sum_{j=1}^{n}{\,\mathrm{d}} x_j\wedge{\,\mathrm{d}} y_j\right). \] For a closed symplectic manifold $(M,\omega)$, let $\operatorname{Ham}(M,\omega)$ denote the Hamiltonian diffeomorphisms: time-one maps of flows generated by smooth time-dependent Hamiltonians on $M$. Here closed means compact without boundary.

The supplied question asks whether every $\varphi\in\operatorname{Ham}(\mathbb T^{2n},\omega_0)$ satisfies \[ \liminf_{T\to+\infty}\frac{\log P_\varphi(T)}{\log T}\geq 2, \] and whether the same assertion holds for every Hamiltonian diffeomorphism of an arbitrary aspherical closed symplectic manifold. The notation $P_\varphi(T)$ merely makes the presumed dependence of the source's $P(T)$ on $\varphi$ explicit; it does not supply a definition.

Unresolved qualifications. The complete supplied record does not define $P(T)$. In particular, it does not identify what is counted, any equivalence relation used in counting, the meaning of the parameter $T$, or whether $T$ ranges over integers or real numbers. The displayed logarithm requires a positive finite value of $P_\varphi(T)$, unless some extended-value convention is specified; none is supplied. Thus the displayed question cannot yet be made into a fully specified mathematical assertion. The torus formulation above uses the conventional standard torus; the source does not specify its dimension or explicitly state the quantification over dimensions.

The meaning of aspherical is also unspecified. Possible hypotheses include $\pi_2(M)=0$, vanishing of $\int_{S^2}u^*\omega$ for every smooth $u:S^2\to M$, or the latter condition together with $\langle c_1(TM),u_*[S^2]\rangle=0$ for every such $u$, where $c_1(TM)$ is computed using an $\omega$-compatible almost complex structure. These hypotheses must not be silently identified; the intended one requires clarification from the source.

Short English statement

The question asks whether the quantity denoted by $P(T)$ has lower logarithmic growth exponent at least $2$ for every Hamiltonian diffeomorphism of a standard symplectic torus, and more generally of an aspherical closed symplectic manifold. The supplied record leaves $P(T)$ undefined and does not specify the asphericity hypothesis, so this is an incomplete statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800490$, exact source code catalogue entry, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is supplied curation, not an independently verified mathematical or literature assessment.
• Supplied source attribution: Albach, Barraud et al., Open Problems in Billiards and Quantitative Symplectic Geometry (2026), https://arxiv.org/abs/2602.12896. The record attributes the item and contribution to Bernhard Albach, Umberto Hryniewicz, and Alexandru Oancea. It records access on 2026-10-16 and extraction from arXiv LaTeX; these are supplied provenance notes, not a claim of independent access or verification.
• The background and title identify the source item as Problem 3, while the extracted statement begins "Problem 2." The supplied evidence does not resolve this discrepancy.