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

The tiered Aubry set for autonomous Lagrangian functions