Let D be a dendrite and f be a continuous map on D.Denote by R(f),Ω(f)and ω(x,f)the set of recurrent points,the set of non-wandering points and the set of ω-limit points of x under f,respectively.Write Ωk+1(f)= Ω(f|Ωk(f))and ωk+1(f)= Ux∈ωk(f)ω(x,f)for any positive integer k,where Ω1(f)= Ω(f)and ω(f)= Ux∈D ω(x,f).ωm(f)is called the attracting centre of f ifωm+1(f)= ωm(f).In this paper,we show that if the rank of D is n-1,then we have the following results:(1)ωn+2(f)= ωn+1(f)and the attracting centre of f is ωn+1(f);(2)Ωn+2(f)=(R(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,then Ωn+1(f)=(R(f))and the depth of f is at most n+1.Besides,we show that there exist a dendrite D1 whose rank is n-1 and the set of(n-1)-order accumulation points of Br(D1)is a singleton,and a continuous map g on D1 such that ωn+1(g)≠ ωn(g)and Ωn(f)≠(R(f)).

Guang Wang SU;Cai Hong HAN;Tai Xiang SUN;Lue LI

Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing,Guangxi University of Finance and Economics,Nanning 530003,P.R.China;College of Information and Statistics,Guangxi University of Finance and Economics,Nanning 530003,P.R.China

数学学报（英文版）

[1]Guang Wang SU;Cai Hong HAN;Tai Xiang SUN;Lue LI-.The Depths and the Attracting Centres for Continuous Maps on a Dendrite Whose Rank is Finite)[J].数学学报（英文版）,2022(09):1643-1652

Attracting,Centres,Ux

Depths,Continuous,Maps,Dendrite,Whose,Rank,Finite,Let,be,dendrite,continuous,map,Denote,by,set,recurrent,points,wandering,limit,under,respectively,Write,k+1,any,positive,integer,where,called,attracting,centre,if,m+1,In,this,paper,we,show,that,rank,then,have,following,results,n+2,n+1,depth,most,Further,order,accumulation,Br,branch,singleton,Besides,there,exist,D1,whose,such

0.455806