Billiards in ideal hyperbolic polygons (Q632346)

Given a polygon of the hyperbolic plane with ideal vertices, the paper under review considers billiards in ideal hyperbolic polygons. The authors state a nice conjecture about minimal average lengths of cyclically related trajectories and prove it in some cases using elementary arguments. It is proved that a trajectory is uniquely determined by the bi-infinite sequence of sides of the polygons that encounters the trajectory. The first main result characterizes which sequences code a trajectory. This characterization was already proved by \textit{M.-J. Giannoni} and \textit{D. Ullmo} in [Physica D 41, No. 3, 371--390 (1990; Zbl 0703.58039)], though a different geometric proof is provided here. Two periodic trajectories are cyclically related if their sequences of sides are related by a cyclic permutation. Notice that this is not the usual shift in dynamics that gives the same periodic trajectory. The average length of a periodic trajectory is the average of the lengths of all cyclically related periodic trajectories. The following conjecture is stated in the paper: for any periodic trajectory, the average length is uniquely minimized by the regular ideal hyperbolic polygon. This is a beautiful characterization of the regular polygon in the moduli space of ideal polygons (that is a cell of dimension \(n-3\), where \(n\) denotes the number of vertices). Some cases of this conjecture are proved in the paper using elementary and nice arguments of hyperbolic trigonometry, Möbius transformations and symmetry.