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 \(I\times C(I)\).