# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

### Tom 210 / 2010

Fundamenta Mathematicae 210 (2010), 47-62 MSC: Primary 37C40; Secondary 37H15. DOI: 10.4064/fm210-1-2

#### Streszczenie

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.

#### Autorzy

• Balázs BárányDepartment of Stochastics
Institute of Mathematics
Technical University of Budapest
P.O. Box 91
1521 Budapest, Hungary
e-mail