Abstract
Given a local dendrite X with set End(X) of endpoints of X countable and f : X -> X be a continuous map, we let P(f), AP(f), R(f), Omega(f), EP(f), EAP(f) the sets of periodic points, almost periodic points, recurrent points, nonwandering points, eventually periodic points and eventually almost periodic points of f, respectively. We show that R(f) subset of <(EAP(f))over bar>. On the other hand, we show that Omega(f) subset of <(EAP(f))over bar>, whenever P(f)= phi. As a consequence, we prove that if the set B(X) of branch points is finite, then <(R(f))over bar>R(f) over bar = <(AP(f))over bar>. We give a counter-example showing that the above results do not hold whenever B(X) is infinite or End(X) is uncountable.