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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
2021-09-22 01:17:58
@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},
  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},
}