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arXiv · 2105.15038

$C^0$-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces

Michael Brandenbursky·Michael Khanevsky
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Abstract

Let $\Sigma$ 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 $C^0$-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 in negative the later question. In particular, we show that on a surface $\Sigma$ 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.

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Michael Brandenbursky, Michael Khanevsky. 2021-05-31. $C^0$-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces. https://arxiv.org/abs/2105.15038

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