Chaotic maps in hyperspaces
type: Item
instance of: scholarly article
title: Chaotic maps in hyperspaces
zbMATH Open document ID: 0879.54044
author: Hermann Haase
published in: Real Analysis Exchange
publication date: 1997-12-10T00:00:00Z
review text: Suppose \(X\) to be a compact metric space, \(w_1, w_2, \dots , w_m\) invertible contractive selfmaps. The iterated function system \((X;w_1,w_2,\dots ,w_m)\) satisfies the open set condition, if \(X\) is an attractor of the system \((X,w_1,w_2,\dots ,w_m)\) and there is a nonvoid open \(O\subseteq X\) such that \(w_i[O]\cap w_j[O]=\emptyset\) for \(i\neq j\) and \(\bigcup_{1\leq i\leq m} w_i[O]\subseteq O\). The author considers an associated dynamical system \((\mathcal F(X),T)\), where \(\mathcal F(X)\) is the set of all nonempty compact subsets of \(X\) with Hausdorff metric, and the mapping \(T\) is a slight modification of the Barnsley operator: \(T(E)=\bigcup_{1\leq i \leq m} w_i^{-1}[E]\). The paper is devoted to the proof of the following Theorem 1: \((\mathcal F(X),T)\) is a chaotic dynamical system provided that for the initial IFS the open set condition is satisfied. This generalizes the result from [\textit{M. F. Barnsley}, Fractals everywhere (1988; Zbl 0691.58001)].
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 1095860
zbMATH Keywords: iterated function system
zbMATH Keywords: hyperspace
zbMATH Keywords: set-valued chaotic map
zbMATH Keywords: IFS
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1095860