POSITIVE TOPOLOGICAL ENTROPY OF TONELLI LAGRANGIAN FLOWS

The paper’s Theorem 2 states that there exists an open and dense subset G(n,L,c)⊆F2(M) (with G(1,L,c)=F2(M)) such that, for u∈int^{C2}H(L,c)∩G(n,L,c) and c>e0(L−u), the set Per(L,u,c) is a hyperbolic set . The proof framework is: (i) establish a Franks-type lemma for Tonelli Lagrangians via C2 potential perturbations that vanish to first order along an orbit segment (Theorem 10) ; (ii) define a generic set G(n,L,c) through a non-degeneracy condition Φn>0 ensuring controllability of the linear variational equation and show G is C2-open and (C∞-)dense (Proposition 9) ; (iii) for u∈int^{C2}H(L,c)∩G(n,L,c), build a stably hyperbolic family of linearized Poincaré maps along periodic orbits and invoke Contreras’s uniform hyperbolicity criterion for such families (Theorem 20) to conclude that Per(L,u,c) is hyperbolic (Theorem 18) . The candidate model’s solution proves the same claim but via a slightly different route: Star + Franks ⇒ a uniform spectral gap on periodic orbits ⇒ dominated splitting for the linear Poincaré cocycle on the symplectic normal bundle ⇒ uniform hyperbolicity by symplecticity. This is a standard alternative approach and aligns with the paper’s ingredients (Franks lemma, generic controllability/realizability, and a uniform hyperbolicity upgrade), though the model compresses the last step via the general fact that, in a symplectic cocycle, a dominated splitting with dim Es = dim Eu implies hyperbolicity. The only substantive gap in the model sketch is in justifying the density of G(n,L,c) “along every closed orbit”: the paper handles the global quantifier via the Φn>0 condition (orbit-independent), while the model appeals informally to geometric control and a countable intersection over orbits. Despite this minor quantifier gap in the genericity step, the core mechanism and the final conclusion match the paper’s result. Therefore, both are correct, with different proof styles.