## Abstract

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 1+α be contractions with ﬁxpoints ai . We consider the iterated function system {Yε fi + ai (1 − Yε )}, where each of the maps is chosen with probability pi . 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 density of the iterated function system.

The proof relies on deﬁning a piecewise hyperbolic dynamical system on the cube with

an SRB-measure whose projection is the density of the iterated function system.

Original language | English |
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Pages (from-to) | 47-62 |

Journal | Fundamenta Mathematicae |

Volume | 210 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2010 |

## Subject classification (UKÄ)

- Mathematics

## Keywords

- iterated function system
- absolute continuity
- random perturbations