The tiered Aubry set for autonomous Lagrangian functions

TL;DR

This paper introduces the tiered Aubry and Mane sets for Tonelli Lagrangians on manifolds, analyzing their topological properties and differences, especially on the torus.

Contribution

It defines the tiered Aubry and Mane sets, proves their key properties, and provides explicit examples illustrating their differences on the torus.

Findings

Tiered Mane set is closed, connected, and chain transitive.

For generic Lagrangians, the tiered Mane set has no interior.

On the torus, the closure of the tiered Aubry set can differ from the union of K.A.M. tori.

Abstract

If L is a Tonelli Lagrangian defined on the tangent bundle of a compact and connected manifold whose dimension is at least 2, we associate to L the tiered Aubry set and the tiered Mane set (defined in the article). We prove that the tiered Mane set is closed, connected, chain transitive and that if L is generic in the Mane sense, the tiered Mane set has no interior. Then, we give an example of such an explicit generic Tonelli Lagrangian function and an example proving that when M is the torus, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different.