| ABSOLUTE CONTINUITY FOR RANDOM ITERATED FUNCTION SYSTEMS WITH OVERLAPS (2009) | |||||||||||||||
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| Abstract. We consider linear iterated function systems with a random mul-tiplicative error on the real line. Our system is {x ↦ → di + λiY x} m i=1, where di ∈ R and λi> 0 are fixed and Y> 0 is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of i.i.d. errors y1, y2,..., dis-tributed as Y, independent of everything else. Let h be the entropy of the process, and let χ = E [log(λY)] be the Lyapunov exponent. Assuming that χ < 0, we obtain a family of conditional measures νy on the line, parametrized by y = (y1, y2,...), the sequence of errors. Our main result is that if h> |χ|, then νy is absolutely continuous with respect to the Lebesgue measure for a.e. y. We also prove that if h < |χ|, then the measure νy is singular and has di-mension h/|χ | for a.e. y. These results are applied to a randomly perturbed IFS suggested by Y. Sinai, and to a class of random sets considered by R. Arratia, motivated by probabilistic number theory. 1. | |||||||||||||||
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