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Closed trajectories for quadratic differentials with an application to billiards. (English) Zbl 0616.30044
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 C. Boldrighini, C. Keane and 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.
Reviewer: Li Zhong

##### MSC:
 30F30 Differentials on Riemann surfaces 30F99 Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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##### References:
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