Geodesic
Lower bounds on the number of closed trajectories of generalized billiards
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Tome 325 (2005), pp. 113-126 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $p$. 
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F. S. Duzhin. Lower bounds on the number of closed trajectories of generalized billiards. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Tome 325 (2005), pp. 113-126. http://geodesic.mathdoc.fr/item/ZNSL_2005_325_a6/

[1] G. Birkhoff, Dynamical Systems, New York, 1927; Dzh. Birkgof, Dinamicheskie sistemy, Moskva, 1942

[2] M. Morse, S. S. Cairns, Critical Points Theory in Global Analysis and Differential Topology, Academic Press, New York, 1969 | MR | Zbl

[3] F. Takens, J. White, “Morse theory of double normals of immersions”, Indiana Univ. Math. J., 21:1 (1971), 11–17 | DOI | MR | Zbl

[4] P. Pushkar, “Diameters of immersed manifolds and of wave fronts”, C. R. Acad. Sci. Paris, 326:2 (1998), 201–205 | MR | Zbl

[5] M. Farber, S. Tabachnikov, “Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards”, Topology, 41:3 (2002), 553–589 | DOI | MR | Zbl

[6] M. Farber, S. Tabachnikov, “Periodic trajectories in 3-dimensional convex billiards”, Manuscripta Math., 108:4 (2002), 431–437 | DOI | MR | Zbl

[7] F. S. Duzhin, “Lower bounds for the number of closed billiard trajectories of period $2$ and $3$ in manifolds embedded in Euclidean space”, IMRN, 2003, no. 8, 425–449 | DOI | MR | Zbl