Characterization of geodesic flows on T^2 with and without positive topological entropy

TL;DR

This paper analyzes the geodesic flow on the 2-torus, showing that zero topological entropy implies integrable-like behavior with constants of motion, while positive entropy indicates chaotic dynamics.

Contribution

It provides a characterization of geodesic flows on T^2, establishing conditions for zero entropy and describing the flow's properties in both cases.

Findings

Zero topological entropy implies integrable-like properties.

Existence of a continuous constant of motion measuring geodesic direction.

Necessary and sufficient conditions for zero entropy based on geodesic intersections.

Abstract

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering $\Br^2$. Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on $T^2$.