Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] V. I. Arnol′d, Mathematical methods of classical mechanics, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR 0690288
  • [3] V. I. Arnold, V. V. Kozlov and A. I. Neustadt, Mathematical aspects of classical and celestial mechanics, Current Problems in Math., Fundamental Directions $ 3$, Moscow, 1985. (Russian)
  • [4] 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
  • [5] -, Applications du theorème de tores invariants, Thèse, Univ. Paris VII, 1982.
  • [6] 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
  • [7] 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,
  • [8] 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
  • [9] Wilhelm Klingenberg, Lectures on closed geodesics, Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Vol. 230. MR 0478069
  • [10] Wilhelm Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin-New York, 1982. MR 666697
  • [11] 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
  • [12] V. F. Lazutkin, \cyr Vypuklyĭ billiard i sobstvennye funktsii operatora Laplasa, Leningrad. Univ., Leningrad, 1981 (Russian). MR 633153
  • [13] A. Magnuson, Symplectic singularities, periodic orbits of the billiard ball map, and the obstacle problem, Thesis, M.I.T., Cambridge, Mass., 1984.
  • [14] Shahla Marvizi and Richard Melrose, Spectral invariants of convex planar regions, J. Differential Geom. 17 (1982), no. 3, 475–502. MR 679068
  • [15] R. B. Melrose, Equivalence of glancing hypersurfaces, Invent. Math. 37 (1976), no. 3, 165–191. MR 0436225,
  • [16] 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) Springer, Berlin, 1977, pp. 464–494. Lecture Notes in Math., Vol. 597. MR 0494305
  • [17] G. Popov, Invariant circles and length spectrum of the billiard ball map, Preprint.
  • [18] -, Quasimodes for the Laplace operator (in preparation).
  • [19] Jürgen Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35 (1982), no. 5, 653–696. MR 668410,
  • [20] 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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F05, 58G25

Retrieve articles in all journals with MSC: 58F05, 58G25

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society