Abstract
Let X be a local dendrite and let f : X -> X be a monotone map. Denote by P(f), RR(f), UR(f), R(f) the set of periodic (resp., regularly recurrent, uniformly recurrent, recurrent) points and.(f) the union of all omega-limit sets of f. We show that if P(f) is nonempty, then (i) Lambda(f) = R(f) = UR(f) = RR(f) = <(P(f))over bar>. (ii) R(f) = X if and only if every cut point is a periodic point. If P(f) is empty, then (iii)Lambda(f) = R(f) = UR(f). (iv) R(f) = X if and only if X is a circle and f is topologically conjugate to an irrational rotation of the unit circle S-1. On the other hand, we prove that f has no Li-Yorke pair. Moreover, we show that the family of all omega-limit sets of f is closed with respect to the Hausdorff metric.