The tiered Aubry set for autonomous Lagrangian functions

Arnaud, Marie-Claude

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

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

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

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

• 1. the set ${𝒩}^T\left(L\right)$ is closed, connected and if $dim{H}^1\left(M\right)\ge 2$, its intersection with any energy level is connected and chain transitive;
• 2. for $L$ generic in the Mañé sense, the sets $\overline{{𝒜}^T\left(L\right)}$ and $\overline{{𝒩}^T\left(L\right)}$ have no interior;
• 3. if the interior of $\overline{{𝒜}^T\left(L\right)}$ 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.

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