# The absolute continuity of the invariant measure of random iterated function systems with overlaps - Mathematics > Dynamical Systems

Abstract: We consider iterated function systems on the interval with randomperturbation. Let $Y \epsilon$ be uniformly distributed in $1- \epsilon, 1 +\epsilon$ and let $f i \in C^{1+\alpha}$ be contractions with fixpoints $a i$.We consider the iterated function system $\{Y \epsilon f i + a i 1 -Y \epsilon \} {i=1}^n$, were each of the maps are chosen with probability$p i$. It is shown that the invariant density is in $L^2$ and the $L^2$-normdoes not grow faster than $1-\sqrt{\epsilon}$, as $\epsilon$ vanishes.