Uniqueness of Fixed-Bounce Geodesic Loops in Billiards

Definitions and mathematical statement

Let $\Omega$ be a billiard domain and $x\in\partial\Omega$. The supplied record does not specify the ambient geometry, dimension, boundary regularity, or convexity assumptions on $\Omega$. For a positive integer $q$, write formally \[ \mathcal L_q(\Omega,x)=\left\{\gamma:\ \gamma\text{ is a billiard geodesic loop based at }x,\ \operatorname{wind}(\gamma)=1,\ N_{\mathrm{refl}}(\gamma)=q\right\}. \] Here a loop based at $x$ begins and ends at $x$; $N_{\mathrm{refl}}$ denotes the number of reflections under the intended counting convention, and $\operatorname{wind}$ denotes the intended winding number. These last two conventions are not defined in the supplied record. In the usual Euclidean interpretation, a billiard path consists of straight segments joined at boundary contacts satisfying the equal-angle reflection law wherever that law is defined. This interpretation alone does not resolve the missing hypotheses or conventions.

Question, conditional on specifying the admissible class and conventions. For every domain $\Omega$, boundary point $x$, and reflection count $q$ in the intended ranges, does there exist exactly one such loop, that is, \[ \#\mathcal L_q(\Omega,x)=1? \] If its unique member is denoted by $\gamma_{q,x}$, the proposed loop-length function is \[ \psi_q(x)=\operatorname{Length}(\gamma_{q,x}),\qquad x\in\partial\Omega. \] The assertion is an existence-and-uniqueness question, not a theorem asserted here. The source leaves unspecified the allowable values of $q$, the meaning of winding number, whether the base point counts as a reflection, whether the reflection law is imposed at the closing contact, and when two loops are identified, including under reparametrization or reversal. Consequently, the displayed formulation is a conditional formalization of the supplied target; it is not a fully specified conjecture for arbitrary billiard domains. Uniqueness of a loop would make its length unambiguous, but a common length for several loops would be a weaker property.

Short English statement

The problem asks whether each boundary point of a billiard domain supports exactly one loop with winding number $1$ and $q$ reflections, thereby defining its length $\psi_q(x)$. A precise conjecture requires additional domain hypotheses and conventions for winding number, reflection counting, and equality of loops. The supplied registry labels the problem partially solved; that status is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004735$, source problem code catalogue entry, title: Uniqueness of Fixed-Bounce Geodesic Loops in Billiards. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2021.
• Supplied report citation: Dynamische Systeme (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/33, pp. 1735--1803. https://doi.org/10.4171/owr/2021/33. The supplied background describes these reports as documenting mathematical workshops and open-problem discussions. No precise page or passage for this problem is supplied.
• Supplied literature reference, labeled "analytic symmetric domains": https://annals.math.princeton.edu/2009/170-1/p06.
• Supplied literature reference, labeled "marked-length determination": https://doi.org/10.1007/s00222-023-01191-8.
• Supplied literature reference, labeled "convex spectral invariants": https://doi.org/10.4310/jdg/1214437138.
• Literature assessment checked at 2026-08-22: this is supplied dated curation, not independently verified. It labels the problem partially solved and states that analytic symmetric and selected chaotic billiards admit marked-length or spectral determination results, while general well-definedness of the loop-length data and spectral detection of convexity remain unresolved. 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. Spectral detection of convexity is background context, not an additional assertion in the displayed loop-uniqueness question.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containedness and preservation of the mathematical target. This is supplied review metadata, not independent verification of the formulation or its mathematical correctness.