The absolute continuity of the invariant measure of random iterated function systems with overlaps
(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:
https://lup.lub.lu.se/record/1775560
- author
- Bárány, Balazs and Persson, Tomas LU
- organization
- publishing date
- 2010
- 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}}, }