Invariant sets for discontinuous parabolic area - preservingtorus

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in both components. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For certain cases (where the entries in the matrix are rational) we show that the maximal invariant set has positive Lebesgue measure and give bounds on the measure. For certain examples we nd expressions for the measure of the invariant set.