Characterizing Dendrites with the Recurrent=Periodic Point Property
by
Alejandro Illanes
Universidad Nacional Autonoma de Mexico

A continuum is a compact connected metric space. A dendrite is a locally connected continuum without simple closed curves. Given a map f from a continuum X into itself, let P(f) be the set of periodic points of f and let R(f) be the set of recurrent points of f. A continuum X has the Periodic = Recurrent Point Property (P=R Property) if for every map f from X into itself Cl(R(f))=Cl(P(f)).

Coven and Hedlund (1980) showed that the interval [0, 1] has the P=R Property. This result was extended to trees by Ye (1993). In 1995, Kato showed that this result is not true for dendrites. He showed that the Gehman dendrite does not have the P=R Property.

In this paper we prove that a dendrite X does not have the P=R Property if and only if X contains a topological copy of the Gehman dendrite.

Date received: June 30, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-50.