Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

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

Bárány, Balazs and Persson, Tomas LU orcid (2010) In Fundamenta Mathematicae 210(1). p.47-62
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.
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
iterated function system, absolute continuity, random perturbations
in
Fundamenta Mathematicae
volume
210
issue
1
pages
47 - 62
publisher
Institute of Mathematics, Polish Academy of Sciences
external identifiers
  • scopus:84855866156
ISSN
0016-2736
DOI
10.4064/fm210-1-2
language
English
LU publication?
yes
id
e893934f-e545-4349-abb6-72ed64e4ea90 (old id 1775560)
alternative location
https://arxiv.org/abs/0903.2166
date added to LUP
2016-04-01 10:34:58
date last changed
2022-01-26 00:39:27
@article{e893934f-e545-4349-abb6-72ed64e4ea90,
  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.<br/><br>
The proof relies on defining a piecewise hyperbolic dynamical system on the cube with<br/><br>
an SRB-measure whose projection is the density of the iterated function system.}},
  author       = {{Bárány, Balazs and Persson, Tomas}},
  issn         = {{0016-2736}},
  keywords     = {{iterated function system; absolute continuity; random perturbations}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{47--62}},
  publisher    = {{Institute of Mathematics, Polish Academy of Sciences}},
  series       = {{Fundamenta Mathematicae}},
  title        = {{The absolute continuity of the invariant measure of random iterated function systems with overlaps}},
  url          = {{http://dx.doi.org/10.4064/fm210-1-2}},
  doi          = {{10.4064/fm210-1-2}},
  volume       = {{210}},
  year         = {{2010}},
}