Invariant tori for the billiard ball map

Abstract

For an n n -dimensional domain Ω ( n ≥ 3 ) \Omega (n \geq 3) with a smooth boundary which is strictly convex in a neighborhood of an elliptic closed geodesic O \mathcal {O} , the existence of a family of invariant tori for the billiard ball map with a positive measure is proved under the assumptions of nondegeneracy and N N -elementarity, N ≥ 5 N \geq 5 , of the corresponding to O \mathcal {O} Poincaré map. Moreover, the conjugating diffeomorphism constructed is symplectic. An analogous result is obtained in the case n = 2 n=2 . It is shown that the lengths of the periodic geodesics determine uniquely the invariant curves near the boundary and the billiard ball map on them up to a symplectic diffeomorphism.