On the question of the stability of the periodic trajectories of Birkhoff's billiard (Q1117530)

The stability of the periodic solutions of the Birkhoff billiard are investigated. A formula, connecting the stability in the first approximation of the periodic trajectory with its Morse index, is obtained. A criterion for the hyperbolicity (ellipticity) of a periodic solution of the billiard is obtained. It is proved that to a polygon of maximal length, inscribed in the billiard curve, there corresponds a hyperbolic periodic trajectory. The connection between the obtained results and the corresponding theorems regarding closed geodesics on Riemann surfaces is discussed.