2008/03/05 by Marie-Claude Arnaud, Arnaud, Marie-Claude
Mathematics · Physics and Astronomy · #37C20 #37J45 #37J50 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.0803.0626
openalex publication_date 2008/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.