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

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

2011-01-09

arxiv.org/abs/1101.1660v2

Mathematics

Dynamical Systems

10 pages, 1 figure

Scientific paper

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.

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