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

Balázs Bárány¹ ; Tomas Persson²

¹ Department of Stochastics Institute of Mathematics Technical University of Budapest P.O. Box 91 1521 Budapest, Hungary
² Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland

Fundamenta Mathematicae, Tome 210 (2010) no. 1, pp. 47-62

Résumé

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.

Détail

DOI : 10.4064/fm210-1-2

Keywords: consider iterated function systems interval random perturbation varepsilon uniformly distributed varepsilon varepsilon alpha contractions fixpoints consider iterated function system varepsilon varepsilon where each maps chosen probability shown invariant density its norm does grow faster sqrt varepsilon varepsilon vanishes proof relies defining piecewise hyperbolic dynamical system cube srb measure whose projection density iterated function system