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 \(R_1>R_2\), 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 \(r<R_2\), and let \(\rho_i\)\({}=\frac1\pi\arccos(r\)\(/R_i)\) be the two rotation numbers. Given \(\rho_1\) and \(\rho_2\), 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 \(\rho_1+\rho_2\) is rational. By the decomposition of interval exchange transformations into periodic and minimal components, there are at most two regions; if \(\rho_1+\rho_2\) 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 \(\rho_1+\rho_2\) we decide whether the periodic region exists, and find its period, when \(\rho_1\) is rational, when \(\rho_1\)\({}+q\rho_2\)\({}\in\frac12\mathbb{Z}\) and when \(p\rho_1\)\({}+\rho_2\)\({}\in\frac12\mathbb{Z}\) for integers \(p,q\ge2\). For \(\rho_1\)\({}+\rho_2\)\({}=\frac12\) the periods are \(3s-2\) (\(s\) even) or \(6s-4\) (\(s\) odd), where \(s\) is \(\lfloor 1\)\(/(2\rho_2)\rfloor\) or \(\lfloor 1\)\(/(2\rho_2)\rfloor\)\({}+1\). All examples of Dragović and Radnović are recovered. A closed formula for the periods when \(\rho_1+\rho_2\) is rational and different from \(\frac12\), and the existence of the periodic region when the only relations are \(p\rho_1\)\({}+q\rho_2\)\({}\in\frac12\mathbb{Z}\) with \(p,q\ge2\), remain open; tables with more arcs are not treated. This is an unrefereed note.