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

Balázs Bárány; Tomas Persson

Fundamenta Mathematicae (2010)

  • Volume: 210, Issue: 1, page 47-62
  • ISSN: 0016-2736

Abstract

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We consider iterated function systems on the interval with random perturbation. Let Y ε be uniformly distributed in [1-ε,1+ ε] and let f i C 1 + α be contractions with fixpoints a i . We consider the iterated function system Y ε f i + a i ( 1 - Y ε ) i = 1 , where each of the maps is chosen with probability p i . 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.

How to cite

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Balázs Bárány, and Tomas Persson. "The absolute continuity of the invariant measure of random iterated function systems with overlaps." Fundamenta Mathematicae 210.1 (2010): 47-62. <http://eudml.org/doc/282776>.

@article{BalázsBárány2010,
abstract = {We consider iterated function systems on the interval with random perturbation. Let $Y_ε$ be uniformly distributed in [1-ε,1+ ε] and let $f_i ∈ C^\{1+α\}$ be contractions with fixpoints $a_i$. We consider the iterated function system $\{Y_\{ε\}f_\{i\} + a_\{i\}(1-Y_\{ε\})\}ⁿ_\{i=1\}$, where each of the maps is chosen with probability $p_i$. 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.},
author = {Balázs Bárány, Tomas Persson},
journal = {Fundamenta Mathematicae},
keywords = {random iterated function systems},
language = {eng},
number = {1},
pages = {47-62},
title = {The absolute continuity of the invariant measure of random iterated function systems with overlaps},
url = {http://eudml.org/doc/282776},
volume = {210},
year = {2010},
}

TY - JOUR
AU - Balázs Bárány
AU - Tomas Persson
TI - The absolute continuity of the invariant measure of random iterated function systems with overlaps
JO - Fundamenta Mathematicae
PY - 2010
VL - 210
IS - 1
SP - 47
EP - 62
AB - We consider iterated function systems on the interval with random perturbation. Let $Y_ε$ be uniformly distributed in [1-ε,1+ ε] and let $f_i ∈ C^{1+α}$ be contractions with fixpoints $a_i$. We consider the iterated function system ${Y_{ε}f_{i} + a_{i}(1-Y_{ε})}ⁿ_{i=1}$, where each of the maps is chosen with probability $p_i$. 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.
LA - eng
KW - random iterated function systems
UR - http://eudml.org/doc/282776
ER -

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