Tonelli Lagrangian systems on the 2-torus and topological entropy

TL;DR

This paper investigates the structure of Tonelli Lagrangian systems on the 2-torus, revealing conditions for integrability and examples of complex dynamics with zero topological entropy.

Contribution

It characterizes invariant tori for all rotation vectors when topological entropy vanishes and constructs examples of Finsler geodesic flows with ergodic components despite zero entropy.

Findings

Invariant tori exist for all rotation vectors with zero topological entropy.

Examples of reversible Finsler flows with ergodic components despite zero entropy.

Analysis of global minimizers in Tonelli systems on the 2-torus.

Abstract

We study Tonelli Lagrangian systems on the 2-torus in energy levels above Ma\~n\'e's strict critical value and analyize the structure of global minimizers in the spirit of Morse, Hedlund and Bangert. In the case where the topological entropy of the Euler-Lagrange flow on the fixed energy level vanishes, we show that there are invariant tori for all rotation vectors indicating integrable-like behavior on a large scale. On the other hand, using a construction of Katok, we give examples of reversible Finsler geodesic flows with vanishing topological entropy, but having ergodic components of positive measure in the unit tangent bundle.