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Item A note on Tonelli Lagrangian systems on T2 with positive topological entropy on a high energy level.(2020) Damasceno, Josué Geraldo; Miranda, José Antônio Gonçalves; Araújo, Luiz Gustavo PeronaIn this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus T2 = R2/Z2. We prove that the Lagrangian flow restricted to a high energy level E−1 L (c) (i.e., c>c0(L)) has positive topological entropy if the flow satisfies the Kupka-Smale property in E−1 L (c) (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on E−1 L (c)). The proof requires the use of well-known results from Aubry – Mather theory.