Fractal measures with uniform marginals

We provide several constructions of self-affine probability measures on the unit square with uniform marginals. These constructions include and extend constructions of previous authors and are parameterized in a natural way. In addition, for each different construction we determine the dimension of the parameter space and thus the level of flexibility (for instance, for approximation purposes) each construction allows. Finally, we give some simple approximation results showing how to approximate any measure with uniform marginals on the unit square with a fractal measure resulting from one of our constructions.