Jose Miranda

    Universidade Federal de Minas Gerais


    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

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