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 UR ( f ) = R ( f ) = Λ ( f ) = P ( f ) ¯ 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) R ( f ) ¯ = D and (iii) D ∖ End ( D ) ⊂ P ( f ) .