On the entropy norm on $Ham(S^2)$

TL;DR

This paper proves that for every positive integer m, the integer lattice Z^m can be embedded in the Hamiltonian diffeomorphism group of the 2-sphere with the entropy metric, including the autonomous metric.

Contribution

It establishes the existence of bi-Lipschitz embeddings of Z^m into Ham(S^2) with the entropy and autonomous metrics, revealing the metric's rich geometric structure.

Findings

Z^m embeds bi-Lipschitz into Ham(S^2) with the entropy metric

The same embedding result holds for the autonomous metric

Demonstrates the complexity of the entropy metric on Ham(S^2)

Abstract

In this note we prove that for each positive integer $m$ there exists a bi-Lipschitz embedding $Z^m\to Ham(S^2)$, where $Ham(S^2)$ is equipped with the entropy metric. In particular, the same result holds when the entropy metric is substituted with the autonomous metric.