no. 3
Nekhoroshev type estimates for billiard ball maps
Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 859-895.

Cet article est consacré aux estimations de la stabilité effective (du type de Nekhoroshev) du flot du billard pour les domaines bornés strictement convexes de dimension quelconque avec des bords analytiques. Le résultat principal est que toute trajectoire du billard, avec des conditions initiales δ- près de la variété glissante, reste près de la variété glissante dans un intervalle exponentiellement grand par rapport à 1/δ. La démonstration est basée sur une forme normale de l’application du billard dans les classes de Gevrey. Plus généralement, on prouve des estimations de la stabilité effective pour l’application du billard associée à une paire d’hypersurfaces glissantes analytiques dont la variété glissante est compacte.

This paper is devoted to the effective stability estimates (of Nekhoroshev’s type) of the billiard flow for strictly convex bounded domains with analytic boundaries in any dimensions. The main result is that any billiard trajectory with initial data which are δ - close to the glancing manifold remains close to the glancing manifold in an exponentially large time interval with respect to 1/δ. The proof is based on a normal form of the billiard ball map in Gevrey classes. More generally, we prove effective stability estimates for the billiard ball map associated with any pair of analytic glancing hypersurfaces with a compact glancing manifold.

@article{AIF_1995__45_3_859_0,
     author = {Gramchev, Todor and Popov, Georgi},
     title = {Nekhoroshev type estimates for billiard ball maps},
     journal = {Annales de l'Institut Fourier},
     pages = {859--895},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {3},
     year = {1995},
     doi = {10.5802/aif.1477},
     mrnumber = {97a:58145},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1477/}
}
TY  - JOUR
AU  - Gramchev, Todor
AU  - Popov, Georgi
TI  - Nekhoroshev type estimates for billiard ball maps
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 859
EP  - 895
VL  - 45
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1477/
DO  - 10.5802/aif.1477
LA  - en
ID  - AIF_1995__45_3_859_0
ER  - 
%0 Journal Article
%A Gramchev, Todor
%A Popov, Georgi
%T Nekhoroshev type estimates for billiard ball maps
%J Annales de l'Institut Fourier
%D 1995
%P 859-895
%V 45
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1477/
%R 10.5802/aif.1477
%G en
%F AIF_1995__45_3_859_0
Gramchev, Todor; Popov, Georgi. Nekhoroshev type estimates for billiard ball maps. Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 859-895. doi : 10.5802/aif.1477. https://aif.centre-mersenne.org/articles/10.5802/aif.1477/

[1] A. Bazzani, S. Marmi, G. Turchetti, Nekhoroshev estimate for isochronous non resonant symplectic maps, Celestial Mech. and Dynamical Astronomy, 47 (1990), 333-359. | MR | Zbl

[2] G. Benettin, L. Galgani, A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable hamiltonian systems, Celestial Mech., 37 (1985), 1-25. | MR | Zbl

[3] G. Benettin, G. Gallavotti, Stability of motions near resonances in quasi-integrable hamiltonian systems, J. Stat. Phys., 44 (1986), 293-338. | MR | Zbl

[4] M. Berger, Sur les caustiques de surfaces en dimension 3. C. R. Acad. Sci. Paris, Série I, 331 (1990), 333-336. | MR | Zbl

[5] P. Bolley, J. Camus, G. Métivier, Régularité Gevrey et itérés pour une classe d'opérateurs hypoelliptiques, Rend. Sem. Mat. Univ. Polit. Torino, 41 (1983), 51-74. | MR | Zbl

[6] L. Boutet De Monvel, P. Kree, Pseudodifferential operators and Gevrey classes, Ann. Inst. Fourier, 17-1 (1967), 295-323. | Numdam | MR | Zbl

[7] J. Bonet, R. Braun, R. Meise, B. Taylor, Whitney's extension theorem for nonquasianalytic classes of ultradifferentiable functions, Studia Math., 99, (1991), 155-184. | MR | Zbl

[8] T. Gramchev, G. Popov, Gevrey normal forms of glancing hypersurfaces (in preparation).

[9] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani, C. Simó, Effective stability for hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem, J. Diff. Eq., 77 (1989), 167-198. | MR | Zbl

[10] A. Giorgilli, E. Zehnder, Exponential stability for time dependent potentials, Z. angew. Math. Phys., 43 (1992), 827-855. | MR | Zbl

[11] L. Hörmander, The Analysis of Linear Partial Differential Operators III, IV, Berlin - Heidelberg - New York, Springer ; 1985. | Zbl

[12] V. Kovachev, G. Popov, Invariant tori for the billiard ball map, Trans. Am. Math. Soc., 317 (1990), 45-81. | MR | Zbl

[13] H. Komatsu, Ultradistributions. II. The kernel theorem and ultradistributions with support on submanifold, J. Fac. Sci. Univ. Tokyo, Sect. IA 24 (1977), 607-628. | MR | Zbl

[14] H. Komatsu, An analogy of the Cauchy-Kowalevsky theorem for ultradifferentiable functions and a division theorem for ultradistributions as its dual, J. Fac. Sci. Univ. Tokyo, Sect. IA 26 (1979), 239-254. | MR | Zbl

[15] S. Kuksin, J. Pöschel, On the inclusion of analytic symplectic maps in analytic hamiltonian flows and its applications, preprint ETH-Zürich (1992). | Zbl

[16] V. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Math. USSR Izv., 7, (1973), 185-214. | Zbl

[17] P. Lochak, Canonical perturbation theory : an approach based on joint approximations (Russian), Uspekhi Mat. Nauk 47, (6), (1992), 59-140 ; translation in : Russian Math. Surveys 47, (6), (1992), 57-133. | MR | Zbl

[18] P. Lochak, A.I. Neishtadt, Estimates of stability time in nearly integrable systems with a quasiconvex Hamiltonian, Chaos, 2, (4) (1992), 495-499. | MR | Zbl

[19] Sh. Marvizi, R. Melrose, Spectral invariants of convex planar regions, J. Diff. Geom., 17 (1982), 475-502. | MR | Zbl

[20] R. Melrose, Equivalence of glancing hypersurfaces, Inventiones Math., 37 (1976), 165-191. | MR | Zbl

[21] J. Moser, On invariant curves of area preserving mappings of an annulus., Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II, (1962) 1-20. | MR | Zbl

[22] N. Nekhoroshev, Exponential estimate of the stability time of near-integrable hamiltonian systems I, Russ. Math. Surveys, 32 (6) (1977), 1-65. | Zbl

[23] N. Nekhoroshev, Exponential estimate of the stability time of near-integrable hamiltonian systems II, Trudy Sem. Petrovs., 5 (1979), 5-50 (in russian). | Zbl

[24] T. Oshima, On analytic equivalence of glancing hypersurfaces, Sci. Papers College Gen. Ed. Univ. Tokyo, 28 (1) (1978), 51-57. | MR | Zbl

[25] J. Pöschel, Nekhoroshev estimates for quasi-convex hamiltonian systems, Math. Z., 213 (1993), 187-217. | MR | Zbl

[26] L. Rodino, Linear partial differential operators in Gevrey spaces, Singapore-New Jersey-London-Hong Kong, World Scientific, 1992. | Zbl

Cité par Sources :