Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

TL;DR

This paper demonstrates that smooth, area-preserving diffeomorphisms of the 2-disk with at most one periodic point can be uniformly approximated by periodic smooth diffeomorphisms, using pseudoholomorphic curve methods.

Contribution

It introduces a novel approach employing pseudoholomorphic curves to approximate irrational pseudo-rotations by periodic systems.

Findings

Any smooth irrational pseudo-rotation can be approximated by integrable systems.

Addresses a long-standing question of A. Katok on zero entropy Hamiltonian systems.

Uses symplectic geometry techniques in dynamical systems approximation.

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.