The Research About Some Dynamic Properties Of Alternating System

Discrete dynamical systems is an important branch of dynamical sys-tems,and the use of discrete dynamical systems is commonly accepted by most scholars to describe some practical problems and phenomena.Nev-ertheless,there exist many complicated systems which involve two or more interactions,so the evolution of these systems cannot be described by a single map f generating the dynamical system.To model these systems,more and more scholars are beginning to research the properties of alternating system-s.This thesis mainly study some properties of alternating system(X,[f,g]),such as periodicity,transitivity,topological entropy and so on.The main con-tents are as follows:(1)In this thesis,the definition of n-dimension alternating system(X,F ={f_i}_i~n=1)is given,and the periodic point relationship between n-dimension al-ternating system(X,F ={f_i)_i~n=1)and dynamical system(X,fn(?)fn-1(?)…(?)f1)is found.(2)In this thesis,the two kinds of definitions of topological entropy of alternating system(X,[f,g])are given and proved,and the relationship a-mong two kinds of topological entropy and the topological entropy of f(?)g is found.(3)In this thesis,the transitivity,topological mixing and exactness a-mong alternating system(X,[f,g]),alternating system(X,[g,f]),dynamic system(X,g(?)f)and dynamic system(X,f(?)g)are proved,and some coun-terexamples are given.Such as,there exist continuous mapping f,g which are not surjective satisfy:[g,f]is transitive,but[f,g]is not transitive;there exist continuous mapping f,g satisfy:[f,g]is transitive,but g(?)f is not transitive.