A gap in the Hofer metric between integrable and autonomous Hamiltonian diffeomorphisms on surfaces

Michael Khanevsky

Research output: Working paperPreprint


Let $\Sigma$ be a compact 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 natural generalization of this question is to ask to what extent one family of `simple' Hamiltonian diffeomorphisms of $\Sigma$ can be approximated by the other. In this paper we show that the set of autonomous Hamiltonian diffeomorphisms is not Hofer-dense in the set of integrable Hamiltonians. We construct explicit examples of integrable diffeomorphisms which cannot be Hofer-approximated by autonomous ones.
Original languageUndefined/Unknown
StatePublished - 6 May 2022


  • math.SG
  • math.DS

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