A Finsler Geodesic Flow on $T^2$ With Positive Metric Entropy

TL;DR

This paper constructs a smooth Finsler metric on the 2-torus whose geodesic flow exhibits positive metric entropy, demonstrating complex chaotic behavior close to flat metrics.

Contribution

It provides the first example of a smooth Finsler geodesic flow on the 2-torus with positive metric entropy, using a theorem by Berger and Turaev.

Findings

Existence of Finsler geodesic flow with positive entropy on $T^2$

Flow can be made arbitrarily close to flat metrics in smoothness

Demonstrates complex chaotic dynamics in Finsler geometry

Abstract

We use a theorem of P. Berger and D. Turaev to construct an example of a Finsler geodesic flow on the 2-torus with a transverse section, such that its Poincar\'e return map has positive metric entropy. The Finsler metric generating the flow can be chosen to be arbitrarily $C^\infty$-close to a flat metric.