Flat metrics are local strict minimizers for the polynomial entropy

TL;DR

This paper demonstrates that flat metrics on the torus are local strict minimizers of polynomial entropy within a class of geodesic flows that are Bott integrable and dynamically coherent, extending previous results.

Contribution

It establishes the local minimality of flat metrics for polynomial entropy among a specific class of integrable geodesic flows, using a graph property for invariant tori.

Findings

Flat metrics are local strict minima for polynomial entropy.

Invariant Lagrangian tori satisfy a graph property in near-integrable systems.

The result extends previous global minimality to a local setting.

Abstract

As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for the polynomial entropy. To this aim, we prove a graph property for invariant Lagrangian tori in near-integrable systems.