  It is an open question in smooth ergodic theory whether there exists a
Hamiltonian disk map with zero topological entropy and (strong) mixing
dynamics. Weak mixing has been known since Anosov and Katok first constructed
examples in 1970. Currently all known examples with weak mixing are irrational
pseudo-rotations with Liouvillean rotation number on the boundary. Our main
result however implies that for a dense subset of Liouville numbers (strong)
mixing cannot occur. Our approach involves approximating the flow of a
suspension of the given disk map by pseudoholomorphic curves. Ellipticity of
the Cauchy-Riemann equation allows quantitative L^2-estimates to be converted
into C^0-estimates between the pseudoholomorphic curves and the trajectories of
the flow on growing time scales. Arithmetic properties of the rotation number
enter through these estimates.
