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

Balázs Bárány; Tomas Persson

Fundamenta Mathematicae, Tome 209 (2010), p. 47-62 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider iterated function systems on the interval with random perturbation. Let $Y_{ε}$ be uniformly distributed in [1-ε,1+ ε] and let $f_{i} \in C^{1 + α}$ be contractions with fixpoints $a_{i}$ . We consider the iterated function system $Y_{ε} f_{i} + a_{i} (1 - Y_{ε}) ⁿ_{i = 1}$ , where each of the maps is chosen with probability $p_{i}$ . It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε 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.

Publié le : 2010-01-01

Zbl 1210.37015

EUDML-ID : urn:eudml:doc:282776

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     author = {Bal\'azs B\'ar\'any and Tomas Persson},
     title = {The absolute continuity of the invariant measure of random iterated function systems with overlaps},
     journal = {Fundamenta Mathematicae},
     volume = {209},
     year = {2010},
     pages = {47-62},
     zbl = {1210.37015},
     language = {en},
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Balázs Bárány; Tomas Persson. The absolute continuity of the invariant measure of random iterated function systems with overlaps. Fundamenta Mathematicae, Tome 209 (2010) pp. 47-62. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm210-1-2/