Dynamical properties of monotone dendrite maps

We show that for a monotone dendrite map f:D→D, any ω-limit set is either finite or a minimal Cantor set. We also prove that where P(f), UR(f), R(f) and Λ(f) denote the sets of periodic points, uniformly recurrent points, recurrent points and the union of all ω-limit sets respectively. Moreover, we prove that the following properties are equivalent: (i) R(f)=D, (ii) and (iii) D∖End(D)⊂P(f).