## Real Analysis Exchange

### On $\omega$-Limit Sets of Triangular Induced Maps

#### 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

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