## Real Analysis Exchange

### Chaotic maps in hyperspace

Hermann Haase

#### 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

#### Citation

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