Absolute continuity for random iterated function systems with overlaps

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is {x -> d(i) + lambda(i)Yx}(i=1)(m), where d(i) is an element of R and lambda(i) > 0 are fixed and Y > 0 is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors y(1), y(2),., distributed as Y, independent of everything else. Let h be the entropy of the process, and let chi = E[log(lambda Y)] be the Lyapunov exponent. Assuming that chi < 0, we obtain a family of conditional measures nu(y) on the line, parametrized by y = (y(1), y(2),...), the sequence of errors. Our main result is that if h > vertical bar chi vertical bar, then nu(y) is absolutely continuous with respect to the Lebesgue measure for almost every y. We also prove that if h < vertical bar chi vertical bar, then the measure nu(y) is singular and has dimension h/vertical bar chi vertical bar for almost every y. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia, motivated by probabilistic number theory.