The Absolute Continuity of the Invariant Measure of Random Iterated Function Systems with Overlaps
نویسنده
چکیده
We consider iterated function systems on the interval with random perturbation. Let Yε be uniformly distributed in [1 − ε, 1 + ε] and let fi ∈ C be contractions with fixpoints ai. We consider the iterated function system {Yεfi + ai(1− Yε)}i=1, were each of the maps are chosen with probability pi. It is shown that the invariant density is in L and the L-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 with the property that its projection is the density of the iterated function system.
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