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

Let Σ be a surface equipped with an area form. There is a long-standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the C⁰-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 C⁰-closure of the set of autonomous diffeomorphisms?” In this paper we answer in the negative the latter question. In particular, we show that on a surface Σ the set of autonomous Hamiltonian diffeomorphisms is not C⁰-dense in the set of entropy-zero Hamiltonians. We explicitly construct examples of such Hamiltonians which cannot be approximated by autonomous diffeomorphisms.