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

TL;DR

The paper investigates when the hyperspace of ω-limit sets $ω(f)$, viewed as a subspace of $2^{X}$ with the Hausdorff metric, has nontrivial interior or a connected structure. It proves that for any $f$ on a space with no isolated points, the interior of $ω(f)$ is empty, and that for a dendrite $D$ where every arc contains a free arc, a transitive $f$ yields $ω(f)$ which is totally disconnected. It then exhibits sharpness by constructing a transitive map on the universal dendrite $D_ty$ for which $ω(f)$ contains an arc, showing that total disconnectedness can fail in non-graph continua. The results delineate when ω-limit hyperspaces must be totally disconnected and provide a concrete counterexample on the universal dendrite, advancing understanding of ω-limit structures in dendritic dynamics.

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.

Theorems & Definitions (17)

• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• Theorem 3.2
• Claim 1
• Claim 2
• Claim 3