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 : https://doi.org/10.5802/aif.2397
Classification : 37J45,  37J50,  37C20
Mots clés : dynamiques lagrangiennes, dynamiques hamiltoniennes, théorie d’Aubry-Mather, ensemble de Mañé
@article{AIF_2008__58_5_1733_0,
     author = {Arnaud, Marie-Claude},
     title = {The tiered Aubry set for autonomous Lagrangian functions},
     journal = {Annales de l'Institut Fourier},
     pages = {1733--1759},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     doi = {10.5802/aif.2397},
     mrnumber = {2445832},
     zbl = {1152.37025},
     language = {en},
     url = {www.numdam.org/item/AIF_2008__58_5_1733_0/}
}
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/item/AIF_2008__58_5_1733_0/

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