$C^0$-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces

TL;DR

This paper demonstrates that on surfaces, entropy-zero Hamiltonian diffeomorphisms cannot always be approximated by autonomous diffeomorphisms in the $C^0$ topology, providing explicit counterexamples.

Contribution

It proves that the set of autonomous Hamiltonian diffeomorphisms is not $C^0$-dense among entropy-zero Hamiltonians on surfaces, answering a long-standing open question.

Findings

Autonomous Hamiltonian diffeomorphisms are not $C^0$-dense among entropy-zero Hamiltonians.

Explicit examples of entropy-zero Hamiltonians not approximable by autonomous diffeomorphisms.

Negative answer to the question about $C^0$-closure of autonomous diffeomorphisms.

Abstract

Let $\Sigma$ be a 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 slightly weaker version of this question asks: ``Does every entropy-zero Hamiltonian diffeomorphism of a surface lie in the $C^0$-closure of the set of autonomous diffeomorphisms?'' In this paper we answer in negative the later question. In particular, we show that on a surface $\Sigma$ the set of autonomous Hamiltonian diffeomorphisms is not $C^0$-dense in the set of entropy-zero Hamiltonians. We explicitly construct examples of such Hamiltonians which cannot be approximated by autonomous diffeomorphisms.