Minimal geodesic foliation on T^2 in case of vanishing topological entropy

Abstract

On a Riemannian 2-torus $(T^2,g)$ we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all $r\in \mathbb{R}\cup \{\infty \}$ the universal cover $\Br^2$ is foliated by minimal geodesics of rotation number $r$. For irrational $r\in \mathbb{R}$ all geodesics are minimal, for rational $r\in \mathbb{R}\cup \{\infty \}$ all geodesics stay in strips between neighboring minimal axes. In such a strip the minimal geodesics are asymptotic to the neighboring minimal axes and generate two foliations.

10 pages, 1 figure

Cited by in corpus (2)

• Minimal geodesics and integrable behavior in geodesic flows
• Tonelli Lagrangian systems on the 2-torus and topological entropy