Entropy collapse versus entropy rigidity for Reeb and Finsler flows

TL;DR

This paper compares the behavior of Reeb flows and Finsler geodesic flows on closed manifolds, showing that Reeb flows can have arbitrarily small entropy while Finsler flows have a positive lower bound.

Contribution

It demonstrates a fundamental difference in entropy properties between Reeb flows and Finsler geodesic flows on closed manifolds.

Findings

Reeb flows can have arbitrarily small topological entropy.

Finsler geodesic flows have a positive lower bound for entropy on many manifolds.

The results highlight contrasting entropy rigidity and collapse phenomena.

Abstract

On every closed contact manifold there exist contact forms with volume one whose Reeb flows have arbitrarily small topological entropy. In contrast, for many closed manifolds there is a uniform positive lower bound for the topological entropy of (not necessarily reversible) normalized Finsler geodesic flows.