Linear parabolic maps on the torus

We investigate Linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of t hese 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 dep endence on the initial conditions. We study the decay of correlations and t he diffusion in the corresponding system on the plane. We also demonstrate how the rationality of the real numbers defining the map influences the dyn amical properties of the system. (C) 1999 Elsevier Science B.V. All rights reserved.