A note on Tonelli Lagrangian systems on \mathbbT² with positive topological entropy on high energy level

2020/05/06 by Damasceno, J. G., Miranda, J. A. G., Perona, L. G.
#37B40 #37J50 #37J99 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.03108

Abstract

In this work we study the dynamical behavior Tonelli Lagrangian systems defined on the tangent bundle of the torus \mathbbT²=ℝ² / ℤ². We prove that the Lagrangian flow restricted to a high energy level E_L^-1(c) (i.e c> c₀(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.

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