The absolute continuity of the invariant measure of random iterated function systems with overlaps
Volume 210 / 2010
Fundamenta Mathematicae 210 (2010), 47-62
MSC: Primary 37C40; Secondary 37H15.
DOI: 10.4064/fm210-1-2
Abstract
We consider iterated function systems on the interval with random perturbation. Let $Y_\varepsilon$ be uniformly distributed in $[1- \varepsilon, 1 + \varepsilon]$ and let $f_i \in C^{1+\alpha}$ be contractions with fixpoints $a_i$. We consider the iterated function system $\{ Y_\varepsilon f_i + a_i (1 - Y_\varepsilon) \}_{i=1}^n$, where each of the maps is chosen with probability $p_i$. It is shown that the invariant density is in $L^2$ and its $L^2$ norm does not grow faster than $1/\sqrt{\varepsilon}$ as $\varepsilon$ vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection is the density of the iterated function system.
