Absolute continuity for random iterated function systems with overlaps

Abstract

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in \R$ 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 i.i.d. 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 $h > |\chi|$, then $\nu_y$ is absolutely continuous with respect to the Lebesgue measure for a.e. $y$. We also prove that if $h < |\chi|$, then the measure $\nu_y$ is singular and has dimension $h/|\chi|$ for a.e. $y$. These results are applied to a randomly perturbed IFS suggested by Y. Sinai, and to a class of random sets considered by R. Arratia, motivated by probabilistic number theory.