## 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

In this paper we answer the negative the latter question. In particular, we show that on a surface Σ the set of autonomous Hamiltonian diffeomorphisms is not

*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 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 |
---|---|

Pages (from-to) | 311-324 |

Number of pages | 14 |

Journal | Israel Journal of Mathematics |

Volume | 255 |

Issue number | 1 |

Early online date | 5 Dec 2022 |

DOIs | |

State | Published - Jun 2023 |

## ASJC Scopus subject areas

- General Mathematics

## Fingerprint

Dive into the research topics of '*C*

^{0}-Gap Between Entropy-Zero Hamiltonians and Autonomous Diffeomorphisms of Surfaces'. Together they form a unique fingerprint.