  We construct a $C^\infty$ 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.
