I wonder if there is some idealized definition of "stirring" being used here. It seems that actual stirring could conceivably result in any possible rearrangeme

I wonder if there is some idealized definition of "stirring" being used here. It seems that actual stirring could conceivably result in any possible rearrangement of particles. As a simple example, which is illustrative even though it seems unlikely as a result of stirring: leave the (x, y) coordinates of each particle the same, while changing the z coordinate as z -> z+k mod h. The result is effectively a discontinuous map (violating the theorem), even though the path of each individual particle must be continuous for physical reasons. Two different notions of continuity here.