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

Summary, in English

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.

• Författare: Balazs Bárány; Tomas Persson
• Avdelning/ar: Matematik LTH; Dynamical systems
• Publiceringsår: 2010
• Språk: Engelska
• Sidor: 47-62
• Publikation/Tidskrift/Serie: Fundamenta Mathematicae
• Volym: 210
• Issue: 1
• Länkar: https://arxiv.org/abs/0903.2166; http://dx.doi.org/10.4064/fm210-1-2
• Dokumenttyp: Artikel i tidskrift
• Förlag: Institute of Mathematics, Polish Academy of Sciences
• Ämne: Mathematics
• Nyckelord: iterated function system; absolute continuity; random perturbations
• Status: Published
• Forskningsgrupp: Analysis and Dynamics; Dynamical systems
• ISBN/ISSN/Övrigt: ISSN: 0016-2736