## Institute for Mathematical Sciences

## Preprint ims94-14

** M. Boshernitzan, G. Galperin, T. Kruger, & S. Troubetzkoy**
* Some Remarks on Periodic Billiard Orbits in Rational Polygons*

Abstract: A polygon is called rational if the angle between each pair of sides is a rational multiple of $\pi.$ The main theorem we
will prove is
Theorem 1: For rational polygons, periodic points of the
billiard flow are dense in the phase space of the billiard flow.
This is a strengthening of Masur's theorem, who has shown
that any rational polygon has ``many'' periodic billiard
trajectories; more precisely, the set of directions of the
periodic trajectories are dense in the set of velocity
directions $\S^1.$ We will also prove some refinements of
Theorem 1: the ``well distribution'' of periodic orbits in the
polygon and the residuality of the points $q \in Q$ with a
dense set of periodic directions.

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