Studies on dendrites and the periodic-recurrent property

Abstract

IIn this paper we evidence the interest of considering three outstanding examples of dendrites with different structures, dendrites F⍵, W and G³. When a dendrite X contains a topological copy of one of them, then it is derived important properties. For example, if X does not contain a topological copy neither F⍵ nor W, then X is a tree. If X does not contain a topological copy of G³ then we obtain that X verifies the Periodic-Recurrent Property (the PR Property) which for dendrites is relevant under the point of view of Topological Dynamics. As an application of the former results, we give a unified proof of the fact that compact intervals of the real line [a; b] (a ≠ b), arcs and trees have also the PR Property.