Positive topological entropy of Tonelli Lagrangian flows

We study the topological entropy of the Lagrangian flow restricted to an energy level EL⁻¹(c) ⊂ TM for c >e₀(L). We prove that if the flow of the Tonelli Lagrangian L: M → 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>e₀(L) and satisfies certain open dense conditions, then there exist a smooth potential u: M→ R, with C²-norm arbitrarily small, such that the flow of the perturbed Lagrangian Lᵤ=L-u restricted to ELᵤ⁻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 (M)=2 and c > e₀(L), then L admits a C²-perturbation by a smooth potential u, such that, the perturbed flow φₜLᵤ|_ELᵤ⁻¹(c) has positive topological entropy.