The Depths and the Attracting Centres for Continuous Maps on a Dendrite Whose Rank is Finite

Let D be a dendrite and f be a continuous map on D.Denote by R(f),Ω(f)andω(x,f)the set of recurrent points,the set of non-wandering points and the set ofω-limit points of x under f,respectively.WriteΩ_(k+1)(f)=Ω(f|_(()Ω_(k)(f)))andω^(k+1)(f)=∪_(()x∈ω~k(f))ω(x,f)for any positive integer k,whereΩ_(1)(f)=Ω(f)andω(f)=∪_(x∈D)ω(x,f).ω~m(f)is called the attracting centre of f ifω^(m+1)(f)=ω~m(f).In this paper,we show that if the rank of D is n-1,then we have the following results:(1)ω^(n+2)(f)=ω^(n+1)(f)and the attracting centre of f isω^(n+1)(f);(2)Ω_(n+2)(f)=■and the depth of f is at most n+2.Further,if the set of(n-1)-order accumulation points of Br(D)(the set of branch points of D)is a singleton,thenΩ_(n+1)(f)=■and the depth of f is at most n+1.Besides,we show that there exist a dendrite D_(1)whose rank is n-1 and the set of(n-1)-order accumulation points of Br(D_(1))is a singleton,and a continuous map g on D_(1)such thatω^(n+1)(g)≠ω~n(g)andΩn(f)≠■.