The absolute continuity of the invariant measure of random iterated function systems with overlaps

We consider iterated function systems on the interval with random perturbation. Let Yε be uniformly distributed in [1−ε, 1+ε] and let fi ∈ C 1+α be contractions with fixpoints ai . We consider the iterated function system {Yε fi + ai (1 − Yε )}, where each of the maps is chosen with probability pi . It is shown that the invariant density is in L2 and its L2 norm does not grow faster than 1/√ε as ε vanishes.
The proof relies on defining a piecewise hyperbolic dynamical system on the cube with
an SRB-measure whose projection is the density of the iterated function system.