  We consider transport properties of the chaotic (strange) attractor along
unfolded trajectories of the dissipative standard map. It is shown that the
diffusion process is normal except of the cases when a control parameter is
close to some special values that correspond to the ballistic mode dynamics.
Diffusion near the related crisises is anomalous and non-uniform in time: there
are large time intervals during which the transport is normal or ballistic, or
even superballistic. The anomalous superdiffusion seems to be caused by
stickiness of trajectories to a non-chaotic and nowhere dense invariant Cantor
set that plays a similar role as cantori in Hamiltonian chaos. We provide a
numerical example of such a sticky set. Distribution function on the sticky set
almost coincides with the distribution function (SRB measure) of the chaotic
attractor.
