Billiards in rational polygons: Periodic trajectories, symmetries and $d$-stability

Periodic trajectories of billiards in rational polygons satisfying the Veech alternative, in particular, in right triangles with an acute angle of the form $\pi/n$ with integer $n$ are considered. The properties under investigation include: symmetry of periodic trajectories, asymptotics of the number of trajectories whose length does not exceed a certain value, stability of periodic billiard trajectories under small deformations of the polygon.