The tiered Aubry set for autonomous Lagrangian functions
Annales de l'Institut Fourier, Volume 58 (2008) no. 5, p. 1733-1759
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. Then1.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.
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 montre1.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.
DOI : https://doi.org/10.5802/aif.2397
Classification:  37J45,  37J50,  37C20
Keywords: Lagrangian dynamics, Hamiltonian dynamics, Aubry-Mather theory, Mañé set
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     pages = {1733-1759},
     doi = {10.5802/aif.2397},
     zbl = {1152.37025},
     mrnumber = {2445832},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2008__58_5_1733_0}
}
The tiered Aubry set for autonomous Lagrangian functions. Annales de l'Institut Fourier, Volume 58 (2008) no. 5, pp. 1733-1759. doi : 10.5802/aif.2397. https://aif.centre-mersenne.org/item/AIF_2008__58_5_1733_0/

[1] Arnaud, M.-C. Hyperbolic periodic orbits and Mather sets in certain symmetric cases, Ergodic Theory Dynam. Systems, Tome 26 (2006) no. 4, pp. 939-959 | Article | MR 2246586 | Zbl 1118.37030

[2] Arnaud, Marie-Claude Création de points périodiques de tous types au voisinage des tores KAM, Bull. Soc. Math. France, Tome 123 (1995) no. 4, pp. 591-603 | Numdam | MR 1373949 | Zbl 0853.58046

[3] Carneiro, M. J. Dias On minimizing measures of the action of autonomous Lagrangians, Nonlinearity, Tome 8 (1995) no. 6, pp. 1077-1085 | Article | MR 1363400 | Zbl 0845.58023

[4] Contreras, G. Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, Tome 13 (2001) no. 4, pp. 427-458 | Article | MR 1867936 | Zbl 0993.37030

[5] Contreras, Gonzalo; Delgado, Jorge; Iturriaga, Renato Lagrangian flows: the dynamics of globally minimizing orbits. II, Bol. Soc. Brasil. Mat. (N.S.), Tome 28 (1997) no. 2, pp. 155-196 | Article | MR 1479500 | Zbl 0892.58065

[6] Contreras, Gonzalo; Iturriaga, Renato Convex Hamiltonians without conjugate points, Ergodic Theory Dynam. Systems, Tome 19 (1999) no. 4, pp. 901-952 | Article | MR 1709426 | Zbl 1044.37046

[7] Contreras, Gonzalo; Iturriaga, Renato Global minimizers of autonomous Lagrangians, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 22 ∘ Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium] (1999) | Zbl 0957.37065

[8] Contreras, Gonzalo; Paternain, Gabriel P. Connecting orbits between static classes for generic Lagrangian systems, Topology, Tome 41 (2002) no. 4, pp. 645-666 | Article | MR 1905833 | Zbl 1047.37042

[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), Longman Sci. Tech., Harlow (Pitman Res. Notes Math. Ser.) Tome 243 (1992), pp. 92-112 | Zbl 0789.58037

[11] Herman, Michael-R. Sur les courbes invariantes par les difféomorphismes de l’anneau. Vol. 1, Société Mathématique de France, Paris, Astérisque, Tome 103 (1983) (With an appendix by Albert Fathi, With an English summary) | Zbl 0532.58011

[12] Mañé, Ricardo Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, Tome 9 (1996) no. 2, pp. 273-310 | Article | MR 1384478 | Zbl 0886.58037

[13] Mañé, Ricardo Lagrangian flows: the dynamics of globally minimizing orbits, International Conference on Dynamical Systems (Montevideo, 1995), Longman, Harlow (Pitman Res. Notes Math. Ser.) Tome 362 (1996), pp. 120-131 | MR 1460800 | Zbl 0870.58026

[14] Mather, John N. Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., Tome 207 (1991) no. 2, pp. 169-207 | Article | MR 1109661 | Zbl 0696.58027

[15] Mather, John N. Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., Tome 4 (1991) no. 2, pp. 207-263 | Article | MR 1080112 | Zbl 0737.58029

[16] Mather, John N. Examples of Aubry sets, Ergodic Theory Dynam. Systems, Tome 24 (2004) no. 5, pp. 1667-1723 | Article | MR 2104599 | Zbl 1090.37047