Closed trajectories for quadratic differentials with an application to billiards (Q1821231)

It is proved that for any given holomorphic quadratic differential q on a compact Riemann surface of genus \(g\geq 2\), there exists a dense set of \(\theta\) for which \(e^{i\theta}q\) has a closed regular vertical trajectory. This result is relative to a problem of the dynamical system of a polygonal table with angles, rational multiples of \(\pi\), the so called rational billiards. In a paper of \textit{C. Boldrighini, C. Keane} and \textit{F. Marchetti} [Ann. Prob. 6, 532-540 (1978; Zbl 0377.28014)] the question was raised whether every rational billiard has a periodic orbit. As an application of the above result of the present paper, it is proved that for any rational billiard table there is a dense set of directions each with a periodic orbit.