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

TL;DR

This paper studies the invariant measure of a random iterated function system with overlaps and shows that its density remains in L^2 with a norm bounded by 1/√ε as the perturbation parameter ε approaches zero.

Contribution

It demonstrates that the invariant density of the system is in L^2 and provides bounds on its L^2-norm relative to the perturbation size, extending understanding of overlaps in random IFS.

Findings

Invariant density is in L^2.

L^2-norm of the density is bounded by 1/√ε.

Results hold as ε approaches zero.

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.