### 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}. 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 independent and identically distributed errors y_{1}, y_{2},..., distributed as Y, independent of everything else. Let h be the entropy of the process, and let χ =double struck E sign[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 = (y_{1}, y_{2}, . . . ), 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 almost every y. We also prove that if h <|χ|, then the measure νy is singular and has dimension h/|χ| almost every y. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia. motivated by probabilistic number theory.

Original language | English |
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Pages (from-to) | 739-756 |

Number of pages | 18 |

Journal | Journal of the London Mathematical Society |

Volume | 74 |

Issue number | 3 |

DOIs | |

Publication status | Published - Dec 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the London Mathematical Society*,

*74*(3), 739-756. https://doi.org/10.1112/S0024610706023258

**Absolute continuity for random iterated function systems with overlaps.** / Peres, Yuval; Simon, K.; Solomyak, Boris.

Research output: Contribution to journal › Article

*Journal of the London Mathematical Society*, vol. 74, no. 3, pp. 739-756. https://doi.org/10.1112/S0024610706023258

}

TY - JOUR

T1 - Absolute continuity for random iterated function systems with overlaps

AU - Peres, Yuval

AU - Simon, K.

AU - Solomyak, Boris

PY - 2006/12

Y1 - 2006/12

N2 - We consider linear iterated function systems with a random multiplicative error on the real line. Our system is {x → di + λiYx. 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 independent and identically distributed errors y1, y2,..., distributed as Y, independent of everything else. Let h be the entropy of the process, and let χ =double struck E sign[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 almost every y. We also prove that if h <|χ|, then the measure νy is singular and has dimension h/|χ| almost every y. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia. motivated by probabilistic number theory.

AB - We consider linear iterated function systems with a random multiplicative error on the real line. Our system is {x → di + λiYx. 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 independent and identically distributed errors y1, y2,..., distributed as Y, independent of everything else. Let h be the entropy of the process, and let χ =double struck E sign[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 almost every y. We also prove that if h <|χ|, then the measure νy is singular and has dimension h/|χ| almost every y. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia. motivated by probabilistic number theory.

UR - http://www.scopus.com/inward/record.url?scp=33846080538&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846080538&partnerID=8YFLogxK

U2 - 10.1112/S0024610706023258

DO - 10.1112/S0024610706023258

M3 - Article

AN - SCOPUS:33846080538

VL - 74

SP - 739

EP - 756

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 3

ER -