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

2022/05/06 by Khanevsky, Michael
#Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2205.03492

Abstract

Let Σ 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 C⁰-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 Σ 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.

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