Absolutely continuous invariant measures for most piecewise smooth expanding maps

We introduce transversality conditions for a piecewise smooth map in any dimension with a sufficiently regular discontinuity set. We prove that maps satisfying these conditions have certain nice properties and, among all maps with a common set of discontinuities, form an open and dense subset. This, in conjunction with other works of the author, implies the existence of stochastically stable invariant densities for an open and dense set of piecewise smooth expanding maps.