The tiered Aubry set for autonomous Lagrangian functions

Details The tiered Aubry set for autonomous Lagrangian functions The tiered Aubry set for autonomous Lagrangian functions

2008-03-05

arxiv.org/abs/0803.0626v1

Mathematics

Dynamical Systems

28 pages; to appear in Ann. Inst. Fourier number 58 (2008)

Scientific paper

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.

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