AREA-PRESERVING IRROTATIONAL DIFFEOMORPHISMS OF THE TORUS WITH SUBLINEAR DIFFUSION

We construct a C-infinity area-preserving diffeomorphism of the two-dimensional torus which is Bernoulli (in particular, ergodic) with respect to Lebesgue measure, homotopic to the identity, and has a lift to the universal covering whose rotation set is {(0, 0)}, which in addition has the property that almost every orbit by the lifted dynamics is unbounded and accumulates in every direction of the circle at infinity.