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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
