The absolute continuity of the invariant measure of random iterated function systems with overlaps

Balazs Bárány, Tomas Persson

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
Pages (from-to)47-62
JournalFundamenta Mathematicae
Volume210
Issue number1
DOIs
Publication statusPublished - 2010

Subject classification (UKÄ)

  • Mathematics

Keywords

  • iterated function system
  • absolute continuity
  • random perturbations

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