Real Analysis Exchange

Chaotic maps in hyperspace

Hermann Haase

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Abstract

The dynamical system \((\mathcal{F}(X),T)\) which arises from an iterated function system \((X;w_1,\ldots ,w_m)\), where \(X\) is a compact metric space identified with the attractor of the system and the \(w_i\)’s are contractive invertible maps, is chaotic provided that the iterated function system satisfies the open set condition. The map \(T\) on the hyperspace \(\mathcal{F} (X)\) of the closed subsets of \(X\) is defined for a closed subset \(E\) as \begin{equation*} T(E)=w_1^{-1}(E)\cup \ldots \cup w_m^{-1}(E). \end{equation*} This extends results about the shift dynamical system for the non-overlapping case \cite{ba}.

Article information

Source
Real Anal. Exchange, Volume 21, Number 2 (1995), 689-695.

Dates
First available in Project Euclid: 14 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1339694096

Mathematical Reviews number (MathSciNet)
MR1407280

Zentralblatt MATH identifier
0879.54044

Subjects
Primary: 54B20: Hyperspaces 54H20: Topological dynamics [See also 28Dxx, 37Bxx]
Secondary: 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]

Keywords
IFS hyperspace set-valued chaotic map

Citation

Haase, Hermann. Chaotic maps in hyperspace. Real Anal. Exchange 21 (1995), no. 2, 689--695. https://projecteuclid.org/euclid.rae/1339694096


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