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 $\{x\mapsto d_i + λ_i Y x\}_{i=1}^m$, where $d_i\in \R$ and $λ_i>0$ 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 $χ= E[\log(λY)]$ be the Lyapunov exponent. Assuming that $χ< 0$, we obtain a family of conditional measures $ν_y$ on the line, parametrized by $y = (y_1,y_2,...)$, the sequence of errors. Our main result is that if $h > |χ|$, then $ν_y$ is absolutely continuous with respect to the Lebesgue measure for a.e. $y$. We also prove that if $h < |χ|$, then the measure $ν_y$ is singular and has dimension $h/|χ|$ 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.