  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.
