Absolute continuity for random iterated function systems with overlaps

2005/02/09 by Yuval Peres, Peres, Yuval, Károly Simon +3 · 1 citation
Mathematics · #28A80 #37C45 #60D05 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR) #math.DS #math.PR #msc:28A80 #msc:37C45 #msc:60D05

paper · pdf · doi:10.48550/arxiv.math/0502200

22 pages

arxiv created 2005/02/09 · arxiv updated 2009/12/01

Abstract

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is \x↦ d_i + λ_i Y x\_i=1^m, where d_i∈ \R and λ_i>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,..., distributed 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 ν_y on the line, parametrized by y = (y₁,y₂,...), the sequence of errors. Our main result is that if h > |χ|, then ν_y is absolutely continuous with respect to the Lebesgue measure for a.e. y. We also prove that if h < |χ|, then the measure ν_y is singular and has dimension 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.

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