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 + λ i Y x } i = 1 m , where di∈ ℝ and λi > 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 y1,y2,…, distributed as Y, independent of everything else. Let h be the entropy of the process, and let χ = 피[log(λ Y)] be the Lyapunov exponent. Assuming that χ < 0, we obtain a family of conditional measures νy on the line, parametrized by y = (y1,y2,…), the sequence of errors. Our main result is that if h > |χ|, then νy is absolutely continuous with respect to the Lebesgue measure for almost every y. We also prove that if h < |χ|, then the measure νy is singular and has dimension h/|χ| 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.