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 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.
