How to construct asymptotically stable iterated function systems
type: Item
instance of: scholarly article
title: How to construct asymptotically stable iterated function systems
zbMATH Open document ID: 1157.60069
author: Joanna Jaroszewska
published in: Statistics \& Probability Letters
publication date: 2008-09-17T00:00:00Z
review text: The paper is devoted to the following problem: given a countable or finite family \(\{S_i, \, i\in I\}\) of continuous transformations \(S_i:X\to X\), is there exists a family \(\{p_i, \, i\in I\}\), such that \(\sum_{i\in I} p_i=1\), and the iterated function system \(\{(S_i, p_i), \, i\in I\}\) is asymptotically stable. The answer to this problem was given in two following cases. 1. If for every \(i\in I\) a transformation \(S_i\) is Lipschitzian on every bounded subset of \(X\), and for at least one element of \(I\) (say, \(j\in I\)), \(S_j\) is a generalized contraction with a fixed point \(c\in X\). Then there exists a family \(\{p_i, \, i\in I\}\) of continuous functions such that \(p_i\in (0,1)\) for all \(x\neq c\), and the family \(\{(S_i, p_i), \, i\in I\}\) is asymptotically stable iterated function system. 2. Assume that for every \(i\in I\) a transformation \(S_i\) is Lipschitzian with the constant 1, and for at least \(j\in I\) \(S_j\) is a generalized contraction. Then if \(\{p_i, \, i\in I\}\) is an arbitrary sequence of numbers with \(p_0>0\), \(\sum_{i\in I}p_i=1\), then the iterated function system \(\{(S_i, p_i), \, i\in I\}\) is asymptotically stable. See also \textit{A. Lasota} and \textit{J. Myjak} [Bull. Pol. Acad. Sci., Math. 46, No. 2, 179--196 (1998; Zbl 0921.28007)], \textit{A. Lasota} and \textit{M. Mackey} [Chaos, fractals, and noise: Stochastic aspects of dynamics. Applied Mathematical Sciences. 97. New York, NY: Springer-Verlag (1994; Zbl 0784.58005)], \textit{T. Szarek} [Ann. Pol. Math. 67, No. 3, 247--257 (1997; Zbl 0903.60052)] for related results.
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 5344885
zbMATH Keywords: iterated function system
zbMATH Keywords: Markov operator
zbMATH Keywords: asymptotical stability
zbMATH Keywords: invariant distribution
reviewed by: V. Knopova
MaRDI profile type: Publication
full work available at URL: j.spl.2008.01.018
OpenAlex ID: W2021889948
cites work: Q4127104
cites work: Q3941926
cites work: Convergence de la répartition empirique vers la répartition théorique
cites work: Q3731958
cites work: Q4807039
cites work: Chaos, fractals, and noise: Stochastic aspects of dynamics.
cites work: Q4399192
cites work: Q4297801
cites work: A note on a theorem of Karlin
cites work: Markov operators acting on Polish spaces
cites work: Invariant measures for nonexpansive Markov operators on Polish spaces
cites work: Fortet-Mourier norms associated with some iterated function systems
DOI: 10.1016/J.SPL.2008.01.018
description: scientific article; zbMATH DE number 5344885