The hyperspace ω(f) when f is a transitive dendrite mapping

Abstract

Let $X$ be a compact metric space. By $2^X$ we denote the hyperspace of all closed and non-empty subsets of $X$ endowed with the Hausdorff metric. Let $f:X\to X$ be a continuous function. In this paper we study some topological properties of the hyperspace $ω(f)$, the collection of all omega limits sets $ω(x,f)$ with $x\in X$. We prove the following: $i)$ If $X$ has no isolated points, then, for every continuous function $f:X\to X$, $int_{2^X}(ω(f))=\emptyset$. $ii)$ If $X$ is a dendrite for which every arc contains a free arc and $f:X\to X$ is transitive, then the hyperspace $ω(f)$ is totally disconnected. $iii)$ Let $D_\infty$ be the Wazewski's universal dendrite. Then there exists a transitive continuous function $f:D_\infty\to D_\infty$ for which the hyperspace $ω(f)$ contains an arc; hence, $ω(f)$ is not totally disconnected.