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

Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves