Hofer's distance between eggbeaters and autonomous Hamiltonian diffeomorphisms on surfaces

TL;DR

This paper constructs eggbeater Hamiltonian diffeomorphisms on surfaces that are arbitrarily far from autonomous Hamiltonians in the Hofer metric, providing a new result for genus one surfaces with a simple construction.

Contribution

It introduces a simple construction of eggbeater Hamiltonian diffeomorphisms on genus one surfaces that are far from autonomous Hamiltonians in the Hofer metric.

Findings

Eggbeater Hamiltonian diffeomorphisms can be arbitrarily far in Hofer metric from autonomous ones.

The construction is simpler than previous methods for genus ≥ 2.

The case for genus 1 surfaces is newly established.

Abstract

Let $\Sigma$ be a compact surface of genus $g \geq 1$ equipped with an area form. We construct eggbeater Hamiltonian diffeomorphisms which lie arbitrarily far in the Hofer metric from the set of autonomous Hamiltonians. This result is already known for $g \geq 2$ (our argument provides an alternative, very simple construction compared to previous publications) while the case $g = 1$ is new.