The tiered Aubry set for autonomous Lagrangian functions

[Ensemble d’Aubry étagé pour les lagrangiens autonomes]

Arnaud, Marie-Claude

Annales de l'Institut Fourier, Tome 58 (2008), p. 1733-1759 / Harvested from Numdam

Résumé

Soit $L\colon TM \rightarrow {\mathbb{R}}$ un lagrangien de Tonelli (avec $M$ compacte et connexe et $\dim M\ge 2$). L’ensemble d’Aubry (resp. de Mañé) étagé $\mathcal{A}^T(L)$ (resp. $\mathcal{N}^T(L)$) est la réunion des ensembles d’Aubry (resp. de Mañé) $\mathcal{A}(L+\lambda )$ (resp. $\mathcal{N} (L+\lambda )$) pour $\lambda $ 1-forme fermée. On montre

• 1. $\mathcal{N}^T(L)$ est fermé, connexe et si $\dim H^1(M)\ge 2$, sa trace sur chaque niveau d’énergie est connexe et transitive par chaîne ;
• 2. si $L$ est générique au sens de Mañé, les ensembles $\overline{\mathcal{A}^T(L)}$ et $\overline{\mathcal{N}^T(L)}$ sont d’intérieur vide ;
• 3. si l’intérieur de $\overline{\mathcal{A}^T(L)}$ 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 $M=\mathbb{T}^2$, $\overline{\mathcal{A}^T(L)}$ peut être différent de l’adhérence de la réunion des tores K.A.M.

Abstract

Let $L\colon TM \rightarrow {\mathbb{R}}$ be a Tonelli Lagrangian function (with $M$ compact and connected and $\dim M\ge 2$). The tiered Aubry set (resp. Mañé set) $\mathcal{A}^T(L)$ (resp. $\mathcal{N}^T(L)$) is the union of the Aubry sets (resp. Mañé sets) $\mathcal{A}(L+\lambda )$ (resp. $\mathcal{N}(L+\lambda )$) for $\lambda $ closed 1-form. Then

• 1. the set $\mathcal{N}^T(L)$ is closed, connected and if $\dim H^1(M)\ge 2$, its intersection with any energy level is connected and chain transitive;
• 2. for $L$ generic in the Mañé sense, the sets $\overline{\mathcal{A}^T(L)}$ and $\overline{\mathcal{N}^T(L)}$ have no interior;
• 3. if the interior of $\overline{\mathcal{A}^T(L)}$ 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 $M=\mathbb{T}^2$, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.

Détail

Publié le : 2008-01-01

MR 2445832 | Zbl 1152.37025

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

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