Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus

Abstract

We study the Euler–Lagrange flow of a Tonelli Lagrangian on the 2-torus$\mathbb{T}^{2}$at a fixed energy level${\mathcal{E}}\subset T\mathbb{T}^{2}$strictly above Mañé’s strict critical value. We prove that, if for some rational direction${\it\zeta}\in S^{1}$there is no invariant graph${\mathcal{T}}\subset {\mathcal{E}}$over$\mathbb{T}^{2}$for the Euler–Lagrange flow with the property that all orbits on${\mathcal{T}}$have an asymptotic direction equal to${\it\zeta}$, then there are chaotic dynamics in${\mathcal{E}}$. This implies that, if the topological entropy of the Euler–Lagrange flow in${\mathcal{E}}$vanishes, then in${\mathcal{E}}$there are invariant graphs for all asymptotic directions${\it\zeta}\in S^{1}$and integrable-like behavior on a large scale.