An entropic characterization of the flat metrics on the two torus

TL;DR

This paper characterizes flat metrics on the two torus by showing that their geodesic flow uniquely minimizes polynomial entropy among all torus geodesic flows.

Contribution

It provides a novel entropic characterization that uniquely identifies flat metrics on the two torus, linking geometric properties with dynamical entropy.

Findings

Flat metrics minimize polynomial entropy among all geodesic flows on the torus.

The property characterizes flat metrics uniquely on the two torus.

The result connects geometric flatness with dynamical entropy minimization.

Abstract

The geodesic flow of the flat metric on a torus is minimizing the polynomial entropy among all geodesic flows on this torus. We prove here that this properties characterises the flat metric on the two torus.