A gap in the Hofer metric between integrable and autonomous Hamiltonian diffeomorphisms on surfaces

TL;DR

This paper investigates the Hofer metric's properties on surfaces, demonstrating that autonomous Hamiltonian diffeomorphisms are not dense among integrable Hamiltonians, with explicit examples showing the limitations of approximation.

Contribution

It proves that autonomous Hamiltonian diffeomorphisms are not Hofer-dense in the set of integrable Hamiltonians on surfaces, providing explicit counterexamples.

Findings

Autonomous diffeomorphisms are not Hofer-dense among integrable ones.

Explicit examples of integrable diffeomorphisms cannot be approximated by autonomous ones.

Addresses a gap related to the Hofer metric and Hamiltonian diffeomorphisms on surfaces.

Abstract

Let $\Sigma$ be a compact surface equipped with an area form. There is an long standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure of the set of integrable diffeomorphisms. A natural generalization of this question is to ask to what extent one family of `simple' Hamiltonian diffeomorphisms of $\Sigma$ can be approximated by the other. In this paper we show that the set of autonomous Hamiltonian diffeomorphisms is not Hofer-dense in the set of integrable Hamiltonians. We construct explicit examples of integrable diffeomorphisms which cannot be Hofer-approximated by autonomous ones.