The tiered Aubry set for autonomous Lagrangian functions

Arnaud, Marie-Claude

doi:10.5802/aif.2397

The tiered Aubry set for autonomous Lagrangian functions
[Ensemble d’Aubry étagé pour les lagrangiens autonomes]

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)

Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1733-1759.

Résumé
Abstract

Soit $L : T M \to ℝ$ un lagrangien de Tonelli (avec $M$ compacte et connexe et $dim M \geq 2$ ). 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 $dim H^{1} (M) \geq 2$ , sa trace sur chaque niveau d’énergie est connexe et transitive par chaîne ;
2. si $L$ est générique au sens de Mañé, les ensembles $\bar{𝒜^{T} (L)}$ et $\bar{𝒩^{T} (L)}$ sont d’intérieur vide ;
3. si l’intérieur de $\bar{𝒜^{T} (L)}$ 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 $M = 𝕋^{2}$ , $\bar{𝒜^{T} (L)}$ peut être différent de l’adhérence de la réunion des tores K.A.M.

Let $L : T M \to ℝ$ be a Tonelli Lagrangian function (with $M$ compact and connected and $dim M \geq 2$ ). 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 $dim H^{1} (M) \geq 2$ , its intersection with any energy level is connected and chain transitive;
2. for $L$ generic in the Mañé sense, the sets $\bar{𝒜^{T} (L)}$ and $\bar{𝒩^{T} (L)}$ have no interior;
3. if the interior of $\bar{𝒜^{T} (L)}$ 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.

MR Zbl

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

Affiliations des auteurs :

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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     title = {The tiered {Aubry} set for autonomous {Lagrangian} functions},
     journal = {Annales de l'Institut Fourier},
     pages = {1733--1759},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     doi = {10.5802/aif.2397},
     mrnumber = {2445832},
     zbl = {1152.37025},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2397/}
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2397/

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