Periodic trajectories in the regular pentagon
نویسندگان
چکیده
The study of billiards in rational polygons and of directional flows on flat surfaces is a fast-growing and fascinating area of research. A classical construction reduces the billiard system in a rational polygon – a polygon whose angles are π-rational – to a constant flow on a flat surface with conical singularities, determined by the billiard polygon. In the most elementary case, the billiard table is a square and the surface is a flat torus obtained from four copies of the square by identifying pairs of parallel sides. We refer to [4, 6, 9, 12, 15, 16] for surveys of rational polygonal billiards and flat surfaces. It is well known that the dynamics of a constant flow on a flat torus depends on the direction: if the slope is rational then all the orbits are closed; and if the slope is irrational then all the orbits are uniformly distributed. The same dichotomy holds for the billiard flow in a square. This property is easy to deduce from the fact that a square tiles the plane by reflections in its sides. In the seminal papers [13, 14], W. Veech discovered a large class of
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