C0-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces

Michael Brandenbursky, Michael Khanevsky

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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 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 in negative the later 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 GB
JournalIsrael Journal of Mathematics
StateAccepted/In press - 19 Jan 2022

Keywords

  • Mathematics - Dynamical Systems
  • Mathematics - Symplectic Geometry

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