Nekhoroshev type estimates for billiard ball maps
Gramchev, Todor ; Popov, Georgi
Annales de l'Institut Fourier, Tome 45 (1995), p. 859-895 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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},
     volume = {45},
     year = {1995},
     pages = {859-895},
     doi = {10.5802/aif.1477},
     mrnumber = {97a:58145},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 859-895. doi : 10.5802/aif.1477. http://gdmltest.u-ga.fr/item/AIF_1995__45_3_859_0/

[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 92b:58207 | Zbl 0703.70031

[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 87i:58051 | Zbl 0602.58022

[3] G. Benettin, G. Gallavotti, Stability of motions near resonances in quasi-integrable hamiltonian systems, J. Stat. Phys., 44 (1986), 293-338. | MR 88h:58042 | Zbl 0636.70018

[4] M. Berger, Sur les caustiques de surfaces en dimension 3. C. R. Acad. Sci. Paris, Série I, 331 (1990), 333-336. | MR 91h:58007 | Zbl 0713.53002

[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 85h:35058 | Zbl 0542.35022

[6] L. Boutet De Monvel, P. Kree, Pseudodifferential operators and Gevrey classes, Ann. Inst. Fourier, 17-1 (1967), 295-323. | Numdam | MR 37 #1760 | Zbl 0195.14403

[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 93e:46030 | Zbl 0738.46009

[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 90c:70026 | Zbl 0675.70027

[10] A. Giorgilli, E. Zehnder, Exponential stability for time dependent potentials, Z. angew. Math. Phys., 43 (1992), 827-855. | MR 93i:58088 | Zbl 0766.58032

[11] L. Hörmander, The Analysis of Linear Partial Differential Operators III, IV, Berlin - Heidelberg - New York, Springer ; 1985. | Zbl 0601.35001

[12] V. Kovachev, G. Popov, Invariant tori for the billiard ball map, Trans. Am. Math. Soc., 317 (1990), 45-81. | MR 90e:58050 | Zbl 0686.58037

[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 57 #17280 | Zbl 0385.46027

[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 81i:35008 | Zbl 0424.46032

[15] S. Kuksin, J. Pöschel, On the inclusion of analytic symplectic maps in analytic hamiltonian flows and its applications, preprint ETH-Zürich (1992). | Zbl 0797.58025

[16] V. Lazutkin, The existence of caustics for a billiard problem in a convex domain, Math. USSR Izv., 7, (1973), 185-214. | Zbl 0277.52002

[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 94f:58110 | Zbl 0795.58042

[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 94a:58110 | Zbl 01553442

[19] Sh. Marvizi, R. Melrose, Spectral invariants of convex planar regions, J. Diff. Geom., 17 (1982), 475-502. | MR 85d:58084 | Zbl 0492.53033

[20] R. Melrose, Equivalence of glancing hypersurfaces, Inventiones Math., 37 (1976), 165-191. | MR 55 #9173 | Zbl 0354.53033

[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 26 #5255 | Zbl 0107.29301

[22] N. Nekhoroshev, Exponential estimate of the stability time of near-integrable hamiltonian systems I, Russ. Math. Surveys, 32 (6) (1977), 1-65. | Zbl 0389.70028

[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 0668.34046

[24] T. Oshima, On analytic equivalence of glancing hypersurfaces, Sci. Papers College Gen. Ed. Univ. Tokyo, 28 (1) (1978), 51-57. | MR 58 #13231 | Zbl 0382.53026

[25] J. Pöschel, Nekhoroshev estimates for quasi-convex hamiltonian systems, Math. Z., 213 (1993), 187-217. | MR 94m:58089 | Zbl 0857.70009

[26] L. Rodino, Linear partial differential operators in Gevrey spaces, Singapore-New Jersey-London-Hong Kong, World Scientific, 1992. | Zbl 0869.35005