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Invariant tori for the billiard ball map

Authors: Valery Kovachev and Georgi Popov
Journal: Trans. Amer. Math. Soc. 317 (1990), 45-81
MSC: Primary 58F05; Secondary 58G25
MathSciNet review: 989578
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Abstract: For an $ n$-dimensional domain $ \Omega (n \geq 3)$ with a smooth boundary which is strictly convex in a neighborhood of an elliptic closed geodesic $ \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$-elementarity, $ N \geq 5$, of the corresponding to $ \mathcal{O}$ Poincaré map. Moreover, the conjugating diffeomorphism constructed is symplectic. An analogous result is obtained in the case $ 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.

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