## Abstract

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 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 the negative the latter question. In particular, we show that on a surface Σ 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.

Original language | English |
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Pages (from-to) | 311-324 |

Number of pages | 14 |

Journal | Israel Journal of Mathematics |

Volume | 255 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jun 2023 |

## ASJC Scopus subject areas

- General Mathematics

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