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Lower bounds on the number of closed trajectories of generalized billiards. (Russian, English) Zbl 1293.37021

Zap. Nauchn. Semin. POMI 325, 113-126 (2005); translation in J. Math. Sci., New York 138, No. 3, 5691-5698 (2006).
Summary: The mathematical study of periodic billiard trajectories is a classical question that goes back to George Birkhoff. A billiard is the motion of a particle in the absence of field of force. 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, and let \(p > 2\) be 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 \(p\).

MSC:

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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