Topological Limits and ω-Limit Sets of the Dendrite Maps

Abstract

Let (X, d) be a metric space and f be a continuous map from X to X. Denote by ω(f) and P(f) the ω-limit set and the set of periodic points of f, respectively. It is well known that for an interval map f, the following statements hold: (1) If P(f) = {x: f(x) = x}, then for any nonempty connected subset A of [0, 1], the topological limit of trajectory of A under f exists. (2) If x ∈ ω(f) - P(f), then the orbit O(x, f) of x under f is an infinite set. The aim of this note is to show that the above statements do not hold for dendrite maps.