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Studies on dendrites and the periodic-recurrent property. (English) Zbl 1229.37035

Summary: We evidence the interest of considering three outstanding examples of dendrites with different structures, namely the dendrites \(F_\omega\), \(W\) and \(G^3\). When a dendrite \(X\) contains a topological copy of one of them, then it has important properties. For example, if \(X\) does not contain a topological copy of neither \(F_\omega\) nor \(W\), then \(X\) is a tree. If \(X\) does not contain a topological copy of \(G^3\) then \(X\) verifies the periodic-recurrent property (PR property) which is relevant for dendrites 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 \([a,b]\) \((a\neq b)\), arcs and trees also have the PR property.

MSC:

37E25 Dynamical systems involving maps of trees and graphs
37B45 Continua theory in dynamics