Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

Pages 1033-1086 from Volume 181 (2015), Issue 3 by Barney Bramham

Abstract

We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed $2$-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular, every smooth irrational pseudo-rotation can be $C^0$-approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.

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Keywords
, , , ,
Mathematical Subject Classification
Primary: ,
DOI
https://doi.org/10.4007/annals.2015.181.3.4
MR
3296820
zbMATH
06446408
Milestones
Received: 30 April 2012
Revised: 9 April 2014
Accepted: 12 May 2014
Published online: 1 May 2015

Authors

Barney Bramham

Ruhr-Universität Bochum, Bochum, Germany