On the Billiard Problem of Dragović and Radnović for Two Concentric Half-Circles: A Periodicity Criterion and the Periods in Special Cases

Dragović and Radnović posed the following problem (Arnold Math. J. 1 (2015)). A billiard table is bounded by two concentric half-circles of radii R1 > R2, lying on opposite sides of a common diameter, and by the two segments of this diameter between them. Consider the trajectories tangent to a fixed concentric circle of radius r < R2, and let ρ_i = (1/π) arccos(r/R_i) be the two rotation numbers. Given ρ1 and ρ2, are these trajectories periodic, how many periodic and non-periodic regions do they form on the boundary, and what are the periods? We give a partial answer. An unfolding reduces the billiard to an explicit map of two circles, which is one of McMullen's coupled rotations; its first-return map is an exchange of three arcs of a circle, and the period of a trajectory, counted as the number of reflections off the arcs and the segments, is given by a formula in terms of the reduced orbit. Through this reduction published theorems apply; the next two statements are such applications and are not claimed as new. By Boshernitzan's theorem that minimal interval exchange transformations of rank two are uniquely ergodic (alternatively, by McMullen's theorems on measured foliations of genus two), all these trajectories are periodic if and only if ρ1 + ρ2 is rational. By the decomposition of interval exchange transformations into periodic and minimal components, there are at most two regions; if ρ1 + ρ2 is irrational, there is exactly one non-periodic region and at most one periodic region. Each region is symmetric under the mirror symmetry of the table. For irrational ρ1 + ρ2 we decide whether the periodic region exists, and find its period, when ρ1 is rational, when ρ1 + qρ2 ∈ ½Z and when pρ1 + ρ2 ∈ ½Z for integers p, q ≥ 2. For ρ1 + ρ2 = 1/2 the periods are 3s − 2 (s even) or 6s − 4 (s odd), where s is ⌊1/(2ρ2)⌋ or ⌊1/(2ρ2)⌋ + 1. All examples of Dragović and Radnović are recovered. A closed formula for the periods when ρ1 + ρ2 is rational and different from 1/2, and the existence of the periodic region when the only relations are pρ1 + qρ2 ∈ ½Z with p, q ≥ 2, remain open; tables with more arcs are not treated. This is an unrefereed note.

Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text.

Corpus identifier: AMR-044-0001 (Dragović-Radnović, Periods of pseudo-integrable billiards, Arnold Math. J. 1 (2015)).