Geodesic
A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level
Russian journal of nonlinear dynamics, Tome 16 (2020) no. 4, pp. 625-635

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In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $E_{L}^{-1}(c)$ (i.e., $c > c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale property in $E_{L}^{-1}(c)$ (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_{L}^{-1}(c)$). The proof requires the use of well-known results from Aubry – Mather theory.
Keywords: Tonelli Lagrangian system, Aubry – Mather theory, static classes. 
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     author = {J. G. Damasceno and J. G. Miranda and L. G. Perona Ara\'ujo},
     title = {A {Note} on {Tonelli} {Lagrangian} {Systems} on $\mathbb{T}^2$ with {Positive} {Topological} {Entropy} on a {High} {Energy} {Level}},
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J. G. Damasceno; J. G. Miranda; L. G. Perona Araújo. A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level. Russian journal of nonlinear dynamics, Tome 16 (2020) no. 4, pp. 625-635. http://geodesic.mathdoc.fr/item/ND_2020_16_4_a6/