The Research Of The Topological Complexity Of Tree Maps

T is called a tree,which is one-dimensional compact connected man-ifold with no circles.Any subset of T is called a subtree of T,if it is a tree itself.Take any x?T,let V(x)denote the number of the connected branch of T-{x}.If V(x)?3,then x is called a branch point of T;If V(x)= 1,then x is called a endpoint of T.In recent years,many experts and scholars have studied dynamics generated by tree maps,for example,the characteristics of the ?-limit set,the topological transitiveness and the topological mixing,the set of chain equivalent points and the turbulence,the attractive center and the topological entropy,and so on.This thesis mainly researches the topological complexity of tree maps,which involves the positive topological entropy and pattern entropy of tree maps.In the first place,we have discussed some equivalent conditions for tree maps with positive topological entropy.Specifically,let f:T ?T be a con-tinuous map,the following conditions have been proved to be equivalent:(1)f is strongly mixing;(2)topological complexity function C((?))>n for every open cover(?)which consists of two non dense open sets,where C((?))=(?)N(Vi=1n f-i((?))),n is the endpoint of T;(3)f is uniformly positive entropy;(4)f is topological K system.Then,we have discussed the pattern entropy of tree maps.Specifically,let f:T ? T be a continuous map,we have proved that if the topological entropy of f is zero,then the pattern entropy of each open cover(?)of T is of polynomial order.