Measure of Rational-Direction Geodesics on Finsler Tori

Definitions and mathematical statement

Let $T^2={\mathbb{R}}^2/{\mathbb{Z}}^2$, and let $F$ be a Finsler metric on $T^2$. Write \[ S_FT^2=\{v\in TT^2:F(v)=1\} \] for its unit tangent bundle, denoted by $ST^2$ in the supplied record. Let $\varphi_F^t$ be its geodesic flow and let $\mu_L$ be the associated Liouville measure on $S_FT^2$. The zero-entropy hypothesis is interpreted as \[ h_{\mathrm{top}}(\varphi_F^1)=0. \] A direction $\xi\in S^1\subset{\mathbb{R}}^2$ is rational if \[ \xi\in\mathcal R:=\left\{\frac{k}{\lVert k\rVert_2}:k\in{\mathbb{Z}}^2\setminus\{0\}\right\}. \] For $v\in S_FT^2$, let $\gamma_v$ be the unit-speed geodesic with initial velocity $v$, and let $\widetilde\gamma_v$ be a lift to ${\mathbb{R}}^2$. A precise forward-time interpretation of its asymptotic direction is \[ \rho_+(v)=\lim_{t\to+\infty} \frac{\widetilde\gamma_v(t)-\widetilde\gamma_v(0)} {\lVert\widetilde\gamma_v(t)-\widetilde\gamma_v(0)\rVert_2}, \] whenever the quotient is defined for all sufficiently large $t$ and this limit exists. This definition is independent of the chosen lift. Under this interpretation, put \[ E_\xi=\{v\in S_FT^2:\rho_+(v)\text{ exists and equals }\xi\}. \] Thus $E_\xi$ is a set of initial unit tangent vectors, which makes the measure expression meaningful, rather than a set of unparametrized geodesics.

The question is whether, for every such zero-topological-entropy Finsler metric, \[ \mu_L\!\left(\bigcup_{\xi\in\mathcal R}E_\xi\right) <\mu_L(S_FT^2). \] Unresolved qualifications. The supplied record does not define asymptotic direction or specify a forward-time versus two-sided convention. The definition above makes one interpretation precise and requires confirmation against the source. No assertion is made here that every initial vector has an asymptotic direction. The record also does not specify the regularity assumptions on the Finsler metric or whether reversibility is required; no additional restriction resolving these issues is imposed in this draft.

Short English statement

For every Finsler metric on the two-torus whose geodesic flow has zero topological entropy, must the initial unit tangent vectors generating geodesics of rational asymptotic direction occupy strictly less than the total Liouville measure? The displayed formulation uses forward asymptotic directions; the source's intended direction convention and metric hypotheses require confirmation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003530$, titled Measure of Rational-Direction Geodesics on Finsler Tori; source problem code catalogue entry. Project registry: data/unsolvedmath/problems.json; supplementary existing-record catalog: catalog/problems.json. The supplied collection is Oberwolfach Reports Open Problems. No individual proposer is named; the supplied proposed year is 2017.
• Supplied report citation: Dynamische Systeme (2018), Oberwolfach Reports, pp. 1987--2046, DOI 10.4171/owr/2017/32. https://doi.org/10.4171/owr/2017/32. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The literature list labels this reference Oberwolfach Report 32/2017.
• Glasmachers--Knieper, Geodesic flows on the two-torus. Supplied literature link: https://arxiv.org/abs/0707.0666.
• Glasmachers--Knieper, Minimal geodesic foliations. Supplied literature link: https://arxiv.org/abs/1101.1660.
• Literature assessment dated 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open and states that zero-entropy torus theory provides substantial structure and examples, while the listed measure and rational-direction questions, especially the classification of integrable Riemannian metrics, remain unresolved in general. The background explicitly calls this dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with a recorded claim of manual semantic, typographic, and LaTeX review. This is source metadata, not independent mathematical verification of the present draft.