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

Abstract

Let D1 be the set of dendrites with the number of the accumulation points of branch points being finite, D∈D1 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∈Dω(x,f) and ωn+1(f)=∪x∈ωn(f)ω(x,f) and Ωn+1(f)=Ω(f|Ωn(f)) for any n∈N. In this paper, we show that Ω4(f)=R(f)‾ and the depth of f is at most 4, and ω4(f)=ω3(f). Furthermore, we show that there exist dendrites D1,D2∈D1 and f1∈C0(D1) and f2∈C0(D2) such that Ω3(f1)≠R(f1)‾ and ω3(f2)≠ω2(f2).