The PR Property Of Local Dendrite Maps With Countable Endpoints

The PR property is an important topic in one dimensional topological dynamical system.For the PR property of interval mapping,tree mapping and dendrite mapping,an in-depth study have been made by Coven,Hedlund,Mai JieHua,Ye XiangDong and so on.In this thesis,we will talk about the PR property of local dendrite.We introduce the topological properties and mapping properties of local dendrite,define the tracing arc of local dendrite mapping,and by using the related property of tracing arc,we can obtain the relationship between the periodic point and the recurrent point under certain condition.Let X be local dendrite with countable endpoints,f ? Co(X).we denote all simple closed curves on X by O1,O2,…,On,and let O ? O?i=1nOi,Then we have:Theorem 1:Let X be a local dendrite with countable endpoints,f ?Co(X)then R(f)\O=P(f)\O.Theorem 2:Let X be a local dendrite wit.h countable endpoints,f ?Co(X),OiX,if there are only finite branch points on X,then R(f)= P(f)? R(f).Theorem 3:Let X be a,local dendrite with countable endpoints,f ? Co(X),f is a piecewise monotone continuous self-mapping,then R(f)=P(f)U R(f).