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    A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level

    2020, Vol. 16, no. 4, pp.  625-635

    Author(s): Damasceno J. G., Miranda J. A., Perona L. G.

    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
    Citation: Damasceno J. G., Miranda J. A.,  Perona L. G., A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level, Rus. J. Nonlin. Dyn., 2020, Vol. 16, no. 4, pp.  625-635
    DOI:10.20537/nd200407


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