Positive topological entropy of Tonelli Lagrangian flows

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 Mañé’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 φ_t^{L_u}\big{|}_{E_{L_u}^{-1}(c)} has positive topological entropy.