Exceptional Rational Directions of Zero-Entropy Finsler Tori

Definitions and mathematical statement

Let $T^2={\mathbb{R}}^2/{\mathbb{Z}}^2$, and let $F$ be a Finsler metric on $T^2$. Let $\phi_F^t$ denote its unit-speed geodesic flow. The entropy hypothesis is \[ h_{\mathrm{top}}(\phi_F^1)=0. \] Write \[ S^1_{\mathrm{rat}}=\left\{\frac{v}{\lVert v\rVert_2}:v\in{\mathbb{Z}}^2\setminus\{0\}\right\}\subset S^1. \] Thus rational directions are represented by nonzero integer vectors; opposite directions are retained separately.

The source does not specify the precise conventions for "nonminimal" or "asymptotic direction." The following makes explicit a possible interpretation, whose agreement with the source requires review. Let $\widetilde F$ be the lifted Finsler metric on ${\mathbb{R}}^2$, and let $d_{\widetilde F}(x,y)$ be its forward distance, defined by the infimum of $\widetilde F$-lengths of curves from $x$ to $y$. A complete unit-speed geodesic $c:{\mathbb{R}}\to T^2$ is called globally minimizing in the universal cover if a lift $\widetilde c:{\mathbb{R}}\to{\mathbb{R}}^2$ satisfies \[ L_{\widetilde F}\left(\widetilde c|_{[a,b]}\right) =d_{\widetilde F}\left(\widetilde c(a),\widetilde c(b)\right) \qquad\text{for every }a<b. \] Under this interpretation, it is nonminimal if this equality fails for some $a<b$. Define its forward asymptotic direction, when the displacement tends to infinity and the limit exists, by \[ \rho_+(c)=\lim_{t\to+\infty} \frac{\widetilde c(t)-\widetilde c(0)} {\lVert\widetilde c(t)-\widetilde c(0)\rVert_2}. \] These definitions are independent of the choice of lift. With these explicit conventions, put \[ E_\xi(F)=\left\{c:{\mathbb{R}}\to T^2: \begin{array}{l} c\text{ is a complete unit-speed }F\text{-geodesic},\\ c\text{ is nonminimal in the universal cover},\\ \rho_+(c)\text{ exists and equals }\xi \end{array}\right\},\qquad \xi\in S^1_{\mathrm{rat}}. \] The supplied problem asks whether, for every Finsler metric $F$ satisfying the entropy hypothesis, each of the following assertions holds:

1. There exists $\xi\in S^1_{\mathrm{rat}}$ such that $E_\xi(F)=\varnothing$.
2. The set \[ \left\{\xi\in S^1_{\mathrm{rat}}:E_\xi(F)\ne\varnothing\right\} \] is finite.

The source does not resolve whether asymptotic direction is instead intended to involve both time directions, nor does it specify regularity or reversibility assumptions on $F$. The displayed formulation is therefore conditional on the stated conventions and is not a verified reconstruction of those missing qualifications.

Short English statement

For every Finsler metric on the two-torus whose geodesic flow has zero topological entropy, must some rational direction contain no nonminimal geodesic? Must only finitely many rational directions contain such geodesics? The source leaves the precise meanings of minimality and asymptotic direction, and the metric's regularity and reversibility assumptions, unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003531$, title "Exceptional Rational Directions of Zero-Entropy Finsler Tori," source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems." The record gives proposed year 2017 and identifies no individual proposer.
• Supplied report citation: Dynamische Systeme (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/32, pp. 1987--2046. https://doi.org/10.4171/owr/2017/32. The supplied literature list also labels this reference "Oberwolfach Report 32/2017." The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: "Deformational rigidity of integrable torus metrics." https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/deformational-rigidity-of-integrable-metrics-on-the-torus/4F3C0B4C21F97A4E359EEB841FDBF9BD. No authors or further bibliographic details are supplied.
• The supplied literature assessment is dated exactly 2026-08-21 and labels the problem open. It states that classification of Riemannian metrics on the two-torus with integrable geodesic flow remains incomplete, and that known Liouville and finite-cover classifications do not settle the general classification problem. This is supplied dated curation, not independently verified literature assessment. Its relevance to the specific zero-entropy Finsler question is not established by the record.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial labels and claims, not independent verification here of the formulation or its mathematical status. The supplied curation explicitly requests independent verification of time-sensitive status and cited claims.