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

Michael Brandenbursky, Michael Khanevsky

Research output: Contribution to journalArticlepeer-review

4 Downloads (Pure)

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
JournalarXiv preprint
StatePublished - 1 May 2021

Keywords

  • Mathematics - Dynamical Systems
  • Mathematics - Symplectic Geometry

Fingerprint

Dive into the research topics of '<i>C<sup>0</sup></i>-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces'. Together they form a unique fingerprint.

Cite this