Positive topological entropy of Tonelli Lagrangian flows

Abstract: We study the topological entropy of the Lagrangian flow restricted to an energy level $E_{L}^{-1}(c) \subset TM$ for $ c >e_0(L)$. We prove that if the flow of the Tonelli Lagrangian $ L: M \to \mathbb{R}$, on a closed manifold of dimension $ n+1$, has a non-hyperbolic closed orbit or an infinite number of closed orbits with energy $ c>e_0(L)$ and satisfies certain open dense conditions, then there exist a smooth potential $ u: M\to \mathbb{R} $, with $ C^2$-norm arbitrarily small, such that the flow of the perturbed Lagrangian $ L_u=L-u$ restricted to $E_{L_u}^{-1}(c)$ has positive topological entropy. The proof of this result is based on an analog version of the Franks' Lemma for Lagrangian flows and Ma~n\'e's techniques on dominated splitting. As an application, we show that if $\dim (M)=2$ and $c > e_0(L)$, then $ L$ admits a $C^2$-perturbation by a smooth potential $u$, such that, the perturbed flow $\phi_t^{L_u}\big{|}{E{L_u}^{-1}(c)}$ has positive topological entropy.