Special $$\alpha $$ α -Limit Points and $$\gamma $$ γ -Limit Points of a Dendrite Map

Abstract

Let (X, d) be a compact metric space and f be a continuous map from X to X. Denote by R(f), \({ SA}(f)\) and \(\Gamma (f)\) the set of recurrent points, the set of special \(\alpha \)-limit points and the set of \(\gamma \)-limit points of f, respectively. It is well known that for an interval map f, the following three statements hold: (1) \(R(f)\subset { SA}(f)\cap \Gamma (f)\); (2) \({ SA}(f)= \Gamma (f)\); (3) \({ SA}(f)\cup \Gamma (f)\subset \overline{R(f)}\). The aim of this paper is to show that the above statement (1) holds for maps of dendrites with the number of endpoints being \(\aleph _{\mathbf{0}}\) (the cardinal number of the set of positive integers) and the above statements (2) and (3) do not hold for maps of dendrites with the number of endpoints being \(\aleph _{\mathbf{0}}\). Besides, we also study unilateral \(\gamma \)-limit points for maps of dendrites with the number of branch points being finite.