A note on Tonelli Lagrangian systems on ${\mathbb{T}}^2$ with positive topological entropy on high energy level

Abstract

In this work we study the dynamical behavior 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 propriety in $E_L^{-1}(c)$ (i.e, all closed orbit 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 in Aubry-Mather's Theory.

11 pages. Some mistakes have been corrected