Some remarks on periodic billiard orbits in rational polygons

Details Some remarks on periodic billiard orbits in rational polygons Some remarks on periodic billiard orbits in rational polygons

1994-08-26

arxiv.org/abs/math/9408217v1

Mathematics

Dynamical Systems

Scientific paper

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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