## Abstract

Let Σ 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

-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 Σ the set of autonomous Hamiltonian diffeomorphisms is not

Hamiltonian diffeomorphism of a surface lies in the

*C*-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^{0}*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 Σ the set of autonomous Hamiltonian diffeomorphisms is not

*C*-dense in the set of entropy-zero Hamiltonians. We explicitly construct examples of such Hamiltonians which cannot be approximated by autonomous diffeomorphisms.^{0}Original language | English |
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Journal | arXiv preprint |

State | Published - 1 May 2021 |

## Keywords

- Mathematics - Dynamical Systems
- Mathematics - Symplectic Geometry