  We investigate linear parabolic maps on the torus. In a generic case these
maps are non-invertible and discontinuous. Although the metric entropy of these
systems is equal to zero, their dynamics is non-trivial due to folding of the
image of the unit square into the torus. We study the structure of the maximal
invariant set, and in a generic case we prove the sensitive dependence on the
initial conditions. We study the decay of correlations and the diffusion in the
corresponding system on the plane. We also demonstrate how the rationality of
the real numbers defining the map influences the dynamical properties of the
system.
