The depths of the centres and the attracting centres of a class of dendrite maps

Let D 1 be the set of dendrites with the number of the accumulation points of branch points being finite, D ∈ D 1 and f be a continuous map from D to D. Denote by R ( f ) , Ω ( f ) and ω ( x , f ) the set of recurrent points of f, the set of non-wandering points of f and the set of ω-limit points of x under f, respectively. Write ω ( f ) = ∪ x ∈ D ω ( x , f ) and ω n + 1 ( f ) = ∪ x ∈ ω n ( f ) ω ( x , f ) and Ω n + 1 ( f ) = Ω ( f | Ω n ( f ) ) for any n ∈ N . In this paper, we show that Ω 4 ( f ) = R ( f ) ‾ and the depth of f is at most 4, and ω 4 ( f ) = ω 3 ( f ) . Furthermore, we show that there exist dendrites D 1 , D 2 ∈ D 1 and f 1 ∈ C 0 ( D 1 ) and f 2 ∈ C 0 ( D 2 ) such that Ω 3 ( f 1 ) ≠ R ( f 1 ) ‾ and ω 3 ( f 2 ) ≠ ω 2 ( f 2 ) .