Invariant sets for discontinuous parabolic area-preserving torus maps

We analyse a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and t aking module one in each component. We show that within this two-parameter family of maps, the set of non-invertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal inva riant set has positive Lebesgue measure and we give bounds on the measure. For several examples we find expressions for the measure of the invariant s et but we leave open the question as to whether there are parameters for wh ich this measure is zero.