2025/02/04 by Martínez-Montejano, Jorge M., Méndez, Héctor, Velázquez-Inzunza, Yajaida N.
#Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2502.02778
Let X be a compact metric space. By 2X we denote the hyperspace of all closed and non-empty subsets of X endowed with the Hausdorff metric. Let f:X→ 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∈ X. We prove the following: i) If X has no isolated points, then, for every continuous function f:X→ X, int2X(ω(f))=∅. ii) If X is a dendrite for which every arc contains a free arc and f:X→ X is transitive, then the hyperspace ω(f) is totally disconnected. iii) Let D_∞ be the Wazewski's universal dendrite. Then there exists a transitive continuous function f:D_∞→ D_∞ for which the hyperspace ω(f) contains an arc; hence, ω(f) is not totally disconnected.