Pointwise Chain Recurrent Maps Of A Class Of Dendrites

A dendrite is a compact,connected and locally connected one-dimensional topological space.In recent years,dynamics of continuous self-maps of den-drite have been studied by many authors,and obtained a lot of meaningful results.In this thesis,we mainly study the pointwise chain recurrent of a class of continuous self-maps on dendrites with an unique branching point.More specifically,we let Dn={r·ei?:?-?/4n,0<r?1/2n},D-n={r·ei?:?=?-?/4n,0<r?1/2n},for any n>1,and denote D*= D*+ UD*-U {o},where D+*= Un=1?n and D*-Un=1 ? D-n.In the thesis,we study the pointwise chain recurrence of the dendrite maps satisfying the unique branching point is not a fixed point.Concretely,if f:D*? D*is a pointwise chain recurrent map on dendrite D*,and o is the unique branching point on D*,then the following conclusion hold:If f(o)?Di,then there exists a positive integerno such that f(Dn)(?)Di whenever|n|>no,and there are points o1,z ? Di such that f(o1)=o,f(z)=z,and one of the following properties hold:(1)If f-1(Z)= {z},then either f2 is the identity map or f2 is turbulent.(2)If f-1(z)? {z},there exists j ? 2n0 such that either fj is the identity map or fj is turbulent.Otherwise,we also mainly study the pointwise chain recurrence of the dendrite maps satisfying the unique branching point is a fixed point.Con-cretely,if f:D*? D*is a pointwise chain recurrent map on dendrite D*,and o is the unique branching point on D*with f(o)= 0,then the following conclusion hold:(1)If f-1{0)= {o},we set Ai = {j:3m ? Z s.t.fm(Dj):Di},Bi = {j:3m E Z s.t.fm(Di)Dj} for any i ?N,then one of the following properties hold:(i)If Ai and Bi are infinite sets,then lim fn(Di)=n??lim f-n(Di)= {o};(ii)if Ai is an infinite set,Bi is a finite set and lim f-n(Di)n??={o},then there exists j0 ? Bi and a positive integer 0<k?<|Bi| such that fk(Bj0)= Bj0,and either fk|Bj0 is the identity map or it is turbulent;(iii)if Ai is a finite set,then Bi is a finite set,fk(Bi)= Bi and either fk|Bi is the identity map or it is turbulent,where k = |Bi|.(2)If f-1(o)? {O},then one of the following properties hold:(i)if there exists a positive integer N such that Dn(?)[P(f)for all |n|>N,then there exists a positive integer 0<k ? 2N such that either fk is turbulent or fk is the identity map;(ii)if there exists|ni| ? ? such that Dni(?)P(f),then there exists a positive integer k such that fk is turbulent or lim f-n(x)= {o} for any x ? Dni.