Lower bounds for the number of closed billiard trajectories of period 2 and 3 in manifolds embedded in Euclidean space

Let T2 be a smooth strictly convex domain bounded by a smooth curve M=T. The billiard ball is a point which moves in T along a straight line and rebounds from M making the angle of incidence equal to the angle of reflection. The classical problem by G. Birkhoff is to find the lower estimate for the...

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Published in:International mathematics research notices 2003-01, Vol.2003 (8), p.425-449
Main Author: Duzhin, Fedor S
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description Let T2 be a smooth strictly convex domain bounded by a smooth curve M=T. The billiard ball is a point which moves in T along a straight line and rebounds from M making the angle of incidence equal to the angle of reflection. The classical problem by G. Birkhoff is to find the lower estimate for the number of closed billiard trajectories with p reflection points. In this paper, we give a definition of a periodic billiard trajectory in a smooth closed m-dimensional manifold Mn, find a lower bound for the number of 3-periodic billiard trajectories, and give a new proof of P. Pushkar's estimate for 2-periodic trajectories.
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title Lower bounds for the number of closed billiard trajectories of period 2 and 3 in manifolds embedded in Euclidean space
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