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 "sufficientl y 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.