The absolute continuity of the invariant measure of random iterated function systems with overlaps

Abstract

We consider iterated function systems on the interval with random perturbation. 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 $L^2$-norm does not grow faster than $1/\sqrt{\epsilon }$, as $\epsilon$ vanishes.

14 pages

References in corpus (1)

• On the fine structure of stationary measures in systems which contract-on-average

Cited by in corpus (1)

• Random iterated function systems with smooth invariant densities