The tiered Aubry set for autonomous Lagrangian functions
[Ensemble d’Aubry étagé pour les lagrangiens autonomes]
Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1733-1759.

Soit L:TM un lagrangien de Tonelli (avec M compacte et connexe et dimM2). L’ensemble d’Aubry (resp. de Mañé) étagé 𝒜 T (L) (resp. 𝒩 T (L)) est la réunion des ensembles d’Aubry (resp. de Mañé) 𝒜(L+λ) (resp. 𝒩(L+λ)) pour λ 1-forme fermée. On montre

  • 1. 𝒩 T (L) est fermé, connexe et si dimH 1 (M)2, sa trace sur chaque niveau d’énergie est connexe et transitive par chaîne ;
  • 2. si L est générique au sens de Mañé, les ensembles 𝒜 T (L) ¯ et 𝒩 T (L) ¯ sont d’intérieur vide ;
  • 3. si l’intérieur de 𝒜 T (L) ¯ est non vide, il contient une partie dense de points périodiques.

On donne ensuite un exemple explicite satisfaisant 2 et un exemple montrant que si M=𝕋 2 , 𝒜 T (L) ¯ peut être différent de l’adhérence de la réunion des tores K.A.M.

Let L:TM be a Tonelli Lagrangian function (with M compact and connected and dimM2). The tiered Aubry set (resp. Mañé set) 𝒜 T (L) (resp. 𝒩 T (L)) is the union of the Aubry sets (resp. Mañé sets) 𝒜(L+λ) (resp. 𝒩(L+λ)) for λ closed 1-form. Then

  • 1. the set 𝒩 T (L) is closed, connected and if dimH 1 (M)2, its intersection with any energy level is connected and chain transitive;
  • 2. for L generic in the Mañé sense, the sets 𝒜 T (L) ¯ and 𝒩 T (L) ¯ have no interior;
  • 3. if the interior of 𝒜 T (L) ¯ is non empty, it contains a dense subset of periodic points.

We then give an example of an explicit Tonelli Lagrangian function satisfying 2 and an example proving that when M=𝕋 2 , the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.

DOI : 10.5802/aif.2397
Classification : 37J45, 37J50, 37C20
Keywords: Lagrangian dynamics, Hamiltonian dynamics, Aubry-Mather theory, Mañé set
Mot clés : dynamiques lagrangiennes, dynamiques hamiltoniennes, théorie d’Aubry-Mather, ensemble de Mañé
Arnaud, Marie-Claude 1

1 Université d’Avignon et des Pays de Vaucluse Laboratoire d’Analyse non linéaire et Géométrie (EA 2151) 84 018 Avignon (France)
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Arnaud, Marie-Claude. The tiered Aubry set for autonomous Lagrangian functions. Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1733-1759. doi : 10.5802/aif.2397. http://archive.numdam.org/articles/10.5802/aif.2397/

[1] Arnaud, M.-C. Hyperbolic periodic orbits and Mather sets in certain symmetric cases, Ergodic Theory Dynam. Systems, Volume 26 (2006) no. 4, pp. 939-959 | DOI | MR | Zbl

[2] Arnaud, Marie-Claude Création de points périodiques de tous types au voisinage des tores KAM, Bull. Soc. Math. France, Volume 123 (1995) no. 4, pp. 591-603 | EuDML | Numdam | MR | Zbl

[3] Carneiro, M. J. Dias On minimizing measures of the action of autonomous Lagrangians, Nonlinearity, Volume 8 (1995) no. 6, pp. 1077-1085 | DOI | MR | Zbl

[4] Contreras, G. Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, Volume 13 (2001) no. 4, pp. 427-458 | DOI | MR | Zbl

[5] Contreras, Gonzalo; Delgado, Jorge; Iturriaga, Renato Lagrangian flows: the dynamics of globally minimizing orbits. II, Bol. Soc. Brasil. Mat. (N.S.), Volume 28 (1997) no. 2, pp. 155-196 | DOI | MR | Zbl

[6] Contreras, Gonzalo; Iturriaga, Renato Convex Hamiltonians without conjugate points, Ergodic Theory Dynam. Systems, Volume 19 (1999) no. 4, pp. 901-952 | DOI | MR | Zbl

[7] Contreras, Gonzalo; Iturriaga, Renato Global minimizers of autonomous Lagrangians, 22 ∘ Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999 | Zbl

[8] Contreras, Gonzalo; Paternain, Gabriel P. Connecting orbits between static classes for generic Lagrangian systems, Topology, Volume 41 (2002) no. 4, pp. 645-666 | DOI | MR | Zbl

[9] Fathi, Albert Weak K.A.M. theorems, Book in preparation

[10] Herman, M.-R. On the dynamics of Lagrangian tori invariant by symplectic diffeomorphisms, Progress in variational methods in Hamiltonian systems and elliptic equations (L’Aquila, 1990) (Pitman Res. Notes Math. Ser.), Volume 243, Longman Sci. Tech., Harlow, 1992, pp. 92-112 | Zbl

[11] Herman, Michael-R. Sur les courbes invariantes par les difféomorphismes de l’anneau. Vol. 1, Astérisque, 103, Société Mathématique de France, Paris, 1983 (With an appendix by Albert Fathi, With an English summary) | Numdam | Zbl

[12] Mañé, Ricardo Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, Volume 9 (1996) no. 2, pp. 273-310 | DOI | MR | Zbl

[13] Mañé, Ricardo Lagrangian flows: the dynamics of globally minimizing orbits, International Conference on Dynamical Systems (Montevideo, 1995) (Pitman Res. Notes Math. Ser.), Volume 362, Longman, Harlow, 1996, pp. 120-131 | MR | Zbl

[14] Mather, John N. Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., Volume 207 (1991) no. 2, pp. 169-207 | DOI | MR | Zbl

[15] Mather, John N. Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., Volume 4 (1991) no. 2, pp. 207-263 | DOI | MR | Zbl

[16] Mather, John N. Examples of Aubry sets, Ergodic Theory Dynam. Systems, Volume 24 (2004) no. 5, pp. 1667-1723 | DOI | MR | Zbl

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