Tores invariants des systèmes dynamiques hamiltoniens

Bost, Jean-Benoît

Séminaire Bourbaki : volume 1984/85, exposés 633-650, Astérisque, no. 133-134 (1986), Exposé no. 639, 45 p.

[1] R. Abraham and J. E. Marsden - Foundations of Mechanics, 2nd ed., Benjamin/ Cummings, Reading, Mass., 1978. | Zbl | MR

[2] V. I. Arnold - Méthodes mathématiques de la mécanique classique, MIR, Moscou, 1974. | Zbl | MR

[3] C. G. J. Jacobi - Vorlesungen über Dynamik, Gesammelte Werke, VIII (Supplementband), Berlin, 1884 - Chelsea, New York, 1969.

[4] H. Poincaré - Les méthodes nouvelles de la mécanique céleste, 3 vol., Gauthier-Villars, Paris, 1892-1899. | JFM

[5] E. T. Whittaker - A treatise on the analytical dynamics of particles and rigid bodies, 4th ed., Cambridge Univ. Press, Cambridge, 1959. | Zbl | MR

[6] G. D. Birkhoff - Dynamical systems, Colloq. Publ. IX, 2nd ed., Amer. Math. Soc., Providence, R.I, 1966. | MR

[7] V. V. Kozlov - Integrability and non-integrability in Hamiltonian mechanics, Russian Math. Surveys 38:1(1983), 1-76. | Zbl | MR

[8] J. Moser - Various aspects of integrable Hamiltonian systems, Progress in Math. 8, Birkhäuser, Boston (1980), 233-289. | Zbl | MR

[9] V. I. Arnold - Small denominators I. On the mapping of a circle into itself, Transl. Amer. Math. Soc. serie 2, 46(1965), 213-284. | Zbl

[10] V. I. Arnold - Small denominators II. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motion under small perturbations of the Hamiltonian, Russian Math. Surveys 18, n° 5(1963), 9-36. | Zbl | MR

[11] V. I. Arnold - Small denominators III. Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys 18, n° 6(1963), 85-193. | Zbl | MR

[12] V. I. Arnold - Chapitres supplémentaires de la théorie des équations différentielles ordinaires, MIR, Moscou, 1978. | Zbl | MR

[13] V. I. Arnold et A. Avez - Problèmes ergodiques de la mécanique classique, Gauthier-Villars, Paris, 1967. | Zbl | MR

[14] J. Bellisard, R. Lima and D. Testard - A metal-insulator transition for the almost Mathieu model, Comm. Math. Phys. 88(1983), 207-234. | Zbl | MR

[15] A. Chenciner - La dynamique au voisinage d'un point fixe elliptique conservatif : de Poincaré et Birkhoff à Aubry et Mather, Séminaire Bourbaki 1983-84, n° 622, Astérisque 121-122(1985), 147-170. | Zbl | MR | Numdam

[16] A. Chenciner - Bifurcations de points fixes elliptiques, preprint Université Paris 7 (1984), à paraître aux Publ. Math. IHES. | Zbl | Numdam

[17] L. Chierchia and G. Gallavotti - Smooth prime integrals for quasi-integrable Hamiltonian systems, Il Nuovo Cimento 67B(1982), 277-295. | MR

[18] Y. Colin De Verdière - Quasi-modes sur les variétés riemanniennes, Invent. Math. 43(1977), 15-52. | Zbl | MR

[19] E. I. Dinaburg and Ya. G. Sinai - The one-dimensional Schrödinger equation with a quasi-periodic potential, Funct. Anal. Appl. 9 (1976) , 279-289. | Zbl

[20] R. Douady - Applications du théorème des tores invariants, Thèse de 3e cycle, Université Paris 7 (juin 1982).

[21] R. Douady - Une démonstration directe de l'équivalence des théorèmes de tores invariants pour difféomorphismes et champs de vecteurs, C.R. Acad. Sci. Paris 295(1982), 201-204. | Zbl | MR

[22] R. Douady et P. Le Calvez - Exemple de point fixe elliptique non stable en dimension 4, C.R. Acad. Sci. Paris 296 (1983) , 895-898. | Zbl | MR

[23] S. M. Graff - On the continuation of hyperbolic invariant tori for Hamiltonian systems, Jour. of Diff. Equ. 15 (1974) , 1-69. | Zbl | MR

[24] M. R. Herman - Sur le groupe des difféomorphismes du tore, Ann. de l'Institut Fourier 23(2) (1973), 75-86. | Zbl | MR | Numdam

[25] M. R. Herman - Résultats récents sur la conjugaison différentiable, Proc. of the Internat. Congress of Mathematicians, Helsinki (1978), 811-820. | Zbl | MR

[26] M. R. Herman - Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES 49(1979), 5-234. | Zbl | MR | Numdam

[27] M. R. Herman - Démonstration du théorème des courbes translatées par les difféomorphismes de l'anneau ; démonstration du théorème des tores invariants, manuscrit (1980).

[28] M. R. Herman - Une méthode pour minorer les exposants de Lyapounov, Preprint Centre de Math. de l'École Polytechnique (1982).

[29] M. R. Herman - Sur les courbes invariantes par les difféomorphismes de l'anneau, Volume I : Astérisque 103-104 (1983) ; Volume II : Preprint Centre de Math. de l'Ecole Polytechnique, à paraître chez Astérisque. | Zbl | Numdam

[30] M. R. Herman - Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112(1984), 93-142. | Zbl | MR | Numdam

[31] M. R. Herman - Simple proof of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation number, Preprint, MSRI, Berkeley (1984). | MR

[32] A. N. Kolmogorov - Sur la conservation des orbites quasi-périodiques lors d' une petite perturbation du hamiltonien, (en russe), Dokl. Akad. Nauk 98 n° 4(1954), 527-530. | Zbl

[33] A. N. Kolmogorov - Théorie générale des systèmes dynamiques et mécanique classique, (en russe), Proc. of the 1954 Intern. Congress, North Holland, Amsterdam, V1 315-333. Traduction anglaise dans [1], 741-757. | Zbl | MR

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

[35] B. B. Lieberman - Existence of quasiperiodic solutions in the Three-Body-Problem, Celestial Mechanics 3(1971), 408-426. | Zbl | MR

[36] B. B. Lieberman - Quasiperiodic solutions of Hamiltonian systems, Jour. of Diff. Equ. 11(1972), 109-137. | Zbl | MR

[37] L. Markus and K. R. Meyer - Generic Hamiltonian systems are neither integrable nor ergodic, Memoirs of the Amer. Math. Soc. 144 (1974) . | Zbl | MR

[38] J. Martinet - Normalisation des champs de vecteurs holomorphes (d'après A.D. Brjuno), Séminaire Bourbaki 1980-81, n° 564, Springer-Verlag, Lect. Notes in Math. 901(1981), 55-70. | Zbl | MR | Numdam

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

[40] J. Moser - A rapidly convergent iteration method and nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa 20(1966), I : 265-315, II : 499-535. | Zbl | Numdam

[41] J. Moser - Convergent series expansions for quasi-periodic motions, Math. Ann. 169(1967), 136-176. | Zbl | MR

[42] J. Moser - Lectures on Hamiltonian systems, Memoirs Amer. Math. Soc. 81(1968), 1-60. | Zbl | MR

[43] J. Moser - On the construction of almost periodic solutions for ordinary differential equations, Proc. of the Internat. Conf. on Functional Analysis and Pelated Topics, Tokyo (1969), 60-67. | Zbl | MR

[44] J. Moser - Stable and random motions in dynamical systems, Ann. Math. Studies n° 77, Princeton Univ. Press, Princeton N.J., 1973. | Zbl | MR

[45] J. Moser - Is the solar system stable ?, Math. Intelligenoer 1(1978), 65-71. | MR

[46] J. Moser and J. Pöschel - An extension of a result by Dinaburg and Sinai on quasiperiodic potential, Comment. Math. Helvetici 59 (1984) , 39-85. | Zbl | MR

[47] J. Pöschel - Über invariante Tori in differenzierbaren Hamiltonschen Systemen, Bonner Mathematische Schriften 120(1980). | Zbl | MR

[48] J. Pöschel - Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35(1982), 653-695. | Zbl | MR

[49] R. C. Robinson - Generic properties of conservative systems, Amer. J. of Math. 92(1970), I : 562-603, II : 897-906. | Zbl | MR

[50] H. Rüssmann - Kleine Nenner I : Über invarianten Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen, Math. Phys. K1. (1970), 67-105. | Zbl | MR

[51] H. Rüssmann - Kleine Nenner II : Bemerkungen zur Newtonschen Methode, Nachr. Akad. Wiss. Göttingen, Math. Phys. K1. (1972), 1-10. | Zbl | MR

[52] H. Rüssmann - On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Lect. Notes in Physics 38(1975), 598-624. | Zbl | MR

[53] H. Rüssmann - Note on sums containing small divisors, Comm. Pure Appl. Math. 29(1976), 755-758. | Zbl | MR

[54] H. Rüssmann - On the one-dimensional Schrödinger equation with a quasi-periodic potential, Ann. New York Acad. Sci. 357(1980), 90-107. | Zbl | MR

[55] H. Rüssmann - On the existence of invariant curves of twist mappings of an annulus, Lect. Notes in Math. 1007(1983), 677-712. | Zbl | MR

[56] C. L. Siegel - Iteration of analytic functions, Ann. of Math. 43 (1942) , 607- 612. | Zbl | MR

[57] C. L. Siegel - Über die analytische Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. (1952), 21-30. | Zbl | MR

[58] C. L. Siegel and J. K. Moser - Lectures on Celestial Mechanics, Springer, Grundlehren Bd. 187(1971). | Zbl | MR

[59] B. Simon - Almost periodic Schrödinger operator : a review, Adv. Appl. Math. 3(1982), 463-490. | Zbl | MR

[60] N. V. Svanidze - Small perturbations of an integrable dynamical system with an integral invariant, Proc. Steklov Inst. of Math. n° 2(1981) , 127-151. | Zbl | MR

[61] W. Thurston - Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80(1974), 304-307. | Zbl | MR

[62] J.-C. Yoccoz - Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. E.N.S. 4e s., 17(1984), 333-359. | Zbl | MR | Numdam

[63] E. Zehnder - Generalized implicit function theorems with applications to some small divisor problems, Comm. Pure Appl. Math., I : 28(1975), 91-140, II: 29(1976), 49-113. | Zbl | MR

[64] M. L. Gromov - Smoothing and inversion of differential operators, Math. USSR Sbornik 17(1972), 381-434. | Zbl | MR

[65] R. S. Hamilton - The inverse function theorem of Nash and Moser, Preprint, Cornell University, 1974. | MR

[66] R. S. Hamilton - The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7(1982), 65-222. | Zbl | MR

[67] L. Hörmander - The boundary problem of physical geodesy, Arch. Rat. Mach. Anal. 62(1976), 1-52. | Zbl | MR

[68] L. Hörmander - Implicit function theorems, Lectures at Stanford University, Summer Quarter 1977.

[69] L. Hörmander - On the Nash-Moser implicit function theorem, Preprint. | MR

[70] S. Lojasiewicz and E. Zehnder - An inverse function theorem in Fréchet spaces, Jour. of Funct. Anal. 33(1979), 165-174. | Zbl | MR

[71] J. Moser - A new technique for the construction of solutions of non-linear differential equations, Proc. Nat. Acad. Sci. USA 47(1961), 1824-1831. | Zbl | MR

[72] J. Nash - The embedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), 20-63. | Zbl | MR

[73] F. Sergeraert - Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications, Ann. Sci. E.N.S. 4e s., 5(1972), 599-660. | Zbl | MR | Numdam

[74] E. Zehnder - Moser's implicit function theorem in the framework of analytic smoothing, Math. Annalen 219(1976), 105-121. | Zbl | MR