C0-Gap Between Entropy-Zero Hamiltonians and Autonomous Diffeomorphisms of Surfaces

Michael Brandenbursky, Michael Khanevsky

Research output: Contribution to journalArticlepeer-review

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 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 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.
Original languageEnglish
Pages (from-to)311-324
Number of pages14
JournalIsrael Journal of Mathematics
Volume255
Issue number1
Early online date5 Dec 2022
DOIs
StatePublished - Jun 2023

ASJC Scopus subject areas

  • General Mathematics

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