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

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 -