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

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.

References

V. I. Arnol′d, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 91–192 (Russian). MR 0170705
V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391

Mathematical aspects of classical and celestial mechanics

Raphaël Douady, Une démonstration directe de l’équivalence des théorèmes de tores invariants pour difféomorphismes et champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, 201–204 (French, with English summary). MR 676353

Applications du theorème de tores invariants

Victor Guillemin and Richard Melrose, A cohomological invariant of discrete dynamical systems, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 672–679. MR 661107
Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222. MR 656198, DOI 10.1090/S0273-0979-1982-15004-2
Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
Wilhelm Klingenberg, Lectures on closed geodesics, Grundlehren der Mathematischen Wissenschaften, Vol. 230, Springer-Verlag, Berlin-New York, 1978. MR 0478069
Wilhelm Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin-New York, 1982. MR 666697
V. Ch. Kovachev and G. S. Popov, Existence of invariant tori for the billiard ball map near an elliptic periodic geodesic, C. R. Acad. Bulgare Sci. 41 (1988), no. 9, 19–22. MR 971828
V. F. Lazutkin, Vypuklyĭ billiard i sobstvennye funktsii operatora Laplasa, Leningrad. Univ., Leningrad, 1981 (Russian). MR 633153

Symplectic singularities, periodic orbits of the billiard ball map, and the obstacle problem

Shahla Marvizi and Richard Melrose, Spectral invariants of convex planar regions, J. Differential Geom. 17 (1982), no. 3, 475–502. MR 679068
R. B. Melrose, Equivalence of glancing hypersurfaces, Invent. Math. 37 (1976), no. 3, 165–191. MR 436225, DOI 10.1007/BF01390317
J. Moser, Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 464–494. MR 0494305

Invariant circles and length spectrum of the billiard ball map

Quasimodes for the Laplace operator

Jürgen Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35 (1982), no. 5, 653–696. MR 668410, DOI 10.1002/cpa.3160350504
N. V. Svanidze, Existence of invariant tori for a three-dimensional billiard, which are concentrated in the vicinity of a “closed geodesic on the boundary region”, Uspekhi Mat. Nauk 33 (1978), no. 4(202), 225–226 (Russian). MR 510686