Josué Damasceno
Publications:
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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
2020, Vol. 16, no. 4, pp. 625-635
Abstract
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.
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