## Real Analysis Exchange

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

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

Sergiĭ Kolyada and Damoon Robatian

#### 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