Absolutely continuous invariant measures for generic multi-dimensional piecewise affine expanding maps

By a well-known result of Lasota and Yorke, any self-map f of the interval which is piecewise smooth and uniformly expanding, i.e. such that inf \ f'\ > 1, admits absolutely continuous invariant probability measures (or a.c.i.m.'s for short). The generalization of this statement to higher dimension remains an open problem. Currently known results only apply to "sufficiently expanding maps". Here we present a different approach which can deal with almost all piecewise expanding maps. Here, we consider both continuous and discontinuous piecewise affine expanding maps.