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

Balázs Bárány, Tomas Persson (2010)

Fundamenta Mathematicae

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 C 1 + α be contractions with fixpoints a i . We consider the iterated function system Y ε f i + a i ( 1 - Y ε ) 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...

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