The hyperspace {\omega}(f) when f is a transitive dendrite mapping

TL;DR

This paper investigates the topological structure of the hyperspace of omega limit sets for transitive dendrite mappings, revealing conditions under which this hyperspace is totally disconnected or contains arcs.

Contribution

It establishes new topological properties of the omega limit hyperspace for transitive dendrite maps, including conditions for total disconnectedness and the existence of arcs.

Findings

If the space has no isolated points, the omega limit hyperspace has empty interior.

For certain dendrites with transitive maps, the omega limit hyperspace is totally disconnected.

Existence of a transitive map on the universal dendrite with a hyperspace containing an arc.

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 $\omega(f)$, the collection of all omega limits sets $\omega(x,f)$ with $x\in X$. We prove the following: $i)$ If $X$ has no isolated points, then, for every continuous function $f:X\to X$, $int_{2^X}(\omega(f))=\emptyset$. $ii)$ If $X$ is a dendrite for which every arc contains a free arc and $f:X\to X$ is transitive, then the hyperspace $\omega(f)$ 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 $\omega(f)$ contains an arc; hence, $\omega(f)$ is not totally disconnected.