Global normally hyperbolic invariant cylinders in Lagrangian systems

In this paper, we study Tonelli Lagrangian $L \in C^r (T \, \mathbb{T}^2 , \mathbb{R})$ with $r \geq 5$. For a generic perturbation of Lagrangian $L \to L + P$ where $P \in C^r (\mathbb{T}^2 , \mathbb{R})$, we get simultaneous hyperbolicity of a family of minimal periodic orbits which share the same first homology class. Consequently, these periodic orbits make up one or more pieces of normally hyperbolic invariant cylinder in $ T \, \mathbb{T}^2$.

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Published 8 July 2016