Lower bounds on the number of closed trajectories of generalized billiards

F. S. Duzhin

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Tome 325 (2005), pp. 113-126

Résumé

The mathematical study of periodic billiard trajectories is a classical question and goes back to George Birkhoff. A billiard is a motion of a particle when a field of force is lacking. Trajectories of such a particle are geodesics. A billiard ball rebounds from the boundary of a given domain making the angle of incidence equal the angle of reflection. Let $k$ be a fixed integer. Birkhoff proved a lower estimate for the number of closed billiard trajectories of length $k$ in an arbitrary plane domain. We give a general definition of a closed billiard trajectory when the billiard ball rebounds from a submanifold of a Euclidean space. Besides, we show how in this case one can apply the Morse inequalities using the natural symmetry (a closed polygon may be considered starting at any of its vertices and with the reversed direction). Finally, we prove the following estimate. Let $M$ be a smooth closed $m$-dimensional submanifold of a Euclidean space, $p>2$ a prime integer. Then $M$ has at least $$ \frac{(B-1)((B-1)^{p-1}-1)}{2p}+\frac{mB}{2}(p-1) $$ closed billiard trajectories of length $p$.