Constructive approximations to the invariant densities of higher- dimensional transformations (Q1322593)

Let \(\tau\) be a piecewise \(C^ 2\) transformation of the unit cube which is \(C^ 2\) on each element of a rectangular partition, and suppose that all partial derivatives of \(\tau\) are uniformly greater than one in absolute value. The authors approximate \(\tau\) by a sequence of Jablonski transformations (which are of the same type but with the additional property that restricted to each rectangular partition element they are products of one-dimensional maps) and they show that invariant densities of the approximating transformations approach weakly in \(L^ 1\) the invariant density (or densities) of \(\tau\).