Topological Transitivity,Topological Ergodicity And Chain Ergodicity

Topological mixing,topological transitivity are two important study objects of dynamical system.In 1993,Akin E. gave the definition of topological ergodicity,and pointed out that topological mixing implied topological ergodicity,topological ergodicity implied topological transitivity. Henceforth,people began to study different properties of topological ergodicity, and extend to all aspects.Yang Runsheng have studied the properties of topological ergodicity in [2]and [3] and its relationship with chaos,he pointed out that topological ergodicity had better properties than topological transitivity and got a series of results.This thesis goes on to study the properties of topological ergodicity and topological double ergodicity following ancestor's work.Chain mixing and chain transitivity are also important study objects of dynamical system,with the enlightenment of the definition of topological ergodicity,the defination of chain ergodicity is given in this thesis,then we discuss its relationship with topological ergodicity as well as topological transitivity etc.Just like the definition of topological ergodicity, chain ergodicity is the definition between chain mixing and chain transitivity used to describe the properties of dynamical system.Imitating the method used to study topological ergodicity,this thesis studies the properties of chain ergodicity on inverse limit space,and give the sufficient and necessary condition when a continuous map on compact space is chain ergodicity and sensitive.At last,this article introduces the definition that the superspace mapping f is induced from the napping f of metric space and depicts an equal portriat of the mapping f of the topological ergodicity.