Real Analysis Exchange

On \(\omega\)-Limit Sets of Triangular Induced Maps

Sergiĭ Kolyada and Damoon Robatian

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Abstract

Let \(X\) be a compact topological space and \(T:X\rightarrow X\) a continuous map. Y.N. Dowker, F.G. Friedlander and A.N. Sharkovsky, independently, introduced and studied the notion of \emph{\(T\)-connectedness}. In particular, they showed that any \(\omega\)-limit set of a dynamical system \((X,T)\) is \(T\)-connected. Let \(\big(C(X),F\big)\) be the induced dynamical system of a given system \((X,T)\), where \(C(X)\) is the hyperspace of all compact connected subsets of \(X\) and \(F:C(X)\rightarrow C(X)\) is the induced map of \(T\). In this paper we give a characterization of the induced-map-connected subsets of \(C(I)\), where \(I\) is a compact interval. The characterization is given via the structure of the \(\omega\)-limit sets (located on a fiber) of continuous \emph{triangular induced maps} on \(I\times C(I)\).

Article information

Source
Real Anal. Exchange, Volume 38, Number 2 (2012), 299-316.

Dates
First available in Project Euclid: 27 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.rae/1403894894

Mathematical Reviews number (MathSciNet)
MR3261879

Zentralblatt MATH identifier
1307.37006

Subjects
Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 26A04
Secondary: 26A05

Keywords
omega-limit set triangular map map-connectedness

Citation

Kolyada, Sergiĭ; Robatian, Damoon. On \(\omega\)-Limit Sets of Triangular Induced Maps. Real Anal. Exchange 38 (2012), no. 2, 299--316. https://projecteuclid.org/euclid.rae/1403894894


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