Generic Finite Complexity in Multidimensional Dispersing Billiards

Definitions and mathematical statement

Consider the source's space of finite-horizon dispersing billiard configurations in a flat $d$-torus. In a standard interpretation, particles move freely between smooth strictly convex scatterers and reflect specularly at collisions; finite horizon gives a uniform bound on free-flight length. The number, curvature and separation assumptions on scatterers and the precise configuration space are not supplied.

The question is whether configurations satisfying the report's finite-complexity condition form a generic subset in the $C^3$ topology. The source also gives the following as an equivalent formulation: a typical configuration should have no phase point whose trajectory undergoes more than $2d-2$ tangential collisions.

At a collision, tangency means that the incoming velocity is tangent to the scatterer boundary, equivalently its scalar product with the normal is zero. If $T(q,x)$ counts the intended tangential collision events for initial phase point $x$ in configuration $q$, the second formulation asks for generic $q$ satisfying \[ \forall x\text{ in the intended phase domain}:\qquad T(q,x)\le2d-2. \] The quantifier is every phase point in each generic configuration, not merely almost every phase point with respect to billiard measure. The time range, treatment of singular trajectories, continuation at a grazing collision, and counting convention must be recovered from the report.

The term generic might mean residual (containing a countable intersection of open dense sets), open dense, or another stated convention; the source does not specify which. The $C^3$ topology also requires a fixed parametrization or equivalence of obstacle boundaries. Neither a probability law on configurations nor a stronger open-dense assertion is inserted. The detailed finite-complexity condition is absent, so its asserted equivalence with the global tangency bound remains a supplied claim requiring review.

The dated assessment says the conjecture is proved for $d=2$ in the report but the multidimensional argument remains incomplete. This is retained as source context, without independently establishing the dimension-two result or narrowing the general question to a specific higher dimension.

Short English statement

The question asks whether generic finite-horizon dispersing billiards satisfy finite complexity, expressed by a uniform $2d-2$ bound on tangencies along every trajectory. The configuration space, meaning of generic, singular-trajectory rules, and exact finite-complexity equivalence are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000792, code catalogue entry, "Generic Finite Complexity in Multidimensional Dispersing Billiards." Proposed year 2007; proposer unspecified.
• Dynamische Systeme (2008), Oberwolfach Reports, pp. 1913--1956, DOI 10.4171/owr/2007/33; https://doi.org/10.4171/owr/2007/33. The displayed report year 2008 differs from the DOI and proposed year 2007. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says official-report content was checked and wording/notation normalized without changing the source claim.
• Literature triage checked 2026-08-21: partially solved. It reports a proof for $d=2$ in the cited report and explicitly incomplete higher-dimensional work. Exact hypotheses and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.