Invariant tori for the billiard ball map

Kovachev, Valery; Popov, Georgi

doi:10.1090/S0002-9947-1990-0989578-5

Invariant tori for the billiard ball map
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by Valery Kovachev and Georgi Popov

Trans. Amer. Math. Soc. 317 (1990), 45-81

DOI: https://doi.org/10.1090/S0002-9947-1990-0989578-5

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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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