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://www.numdam.org/articles/10.5802/aif.2397/

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