Lower Bounds for the Number of Periodic Billiard Trajectories in Manifolds Embedded in Euclidean Space.

摘要

The authors shall study the lower bounds for the number of the periodic billiard trajectories in manifolds embedded in Euclidean space. A p-periodic billiard trajectory is a closed polygon consisting of p segments, all of whose vertices belong to the given manifold and, at every vertex, the two angles formed by the line and the manifold are equal (the exact definition will be given later). The first who considered this problem was George Birkhoff. He proved the following fact. Suppose p>2 is an odd prime; M is a strictly convex smooth closed curve then there exist in M at least two periodic billiard trajectories for each rotation number from 1 to (p-1)/2. In Section 2 the authors show how one can apply Morse theory to study periodic billiard trajectories. Section 3 contains the generalized Birkhoff theorem. In Section 4 the authors prove the Farber-Tabachnikov estimate for generic small perturbations of the standard round n-sphere for any n. In Section 5 the authors study 3-periodic billiard trajectories of the standard round n-sphere for any n. In Section 5 the authors study 3-periodic billiard trajectories in a two-dimensional sphere. At last in Section 6 the authors find an estimate for the number of 3-periodic billiard trajectories in any two-dimensional manifold.