Stone - Cech Tychonoff Topological Dynamical System And The Corresponding Relations Between The Expansion Of Power System

The study of non-compact topological dynamics is one of the important fields in the studies of dynamics. But the work on it is little by now. And there are many differences between non-compact topological dynamics and compact topological dynamics, so the study of non-compact topological dynamics is necessary.In this paper, to begin with, we conclude the existing methods and analyze Stone-Cech theorem and other results on non-compact topological dynamics.What’s more, we obtain that each Tychonoff dynamic (X,f) has the unique Stone-Cech compactification (βX,βf) in the sense of topological conjugate, which is the extension of original dynamic.And, we discuss some relations between the original Tychonoff dynamic and its Stone-Cech compactification with respect to some properties of topological dynamics.Finally, taking almost periodic points and minimal sets for example, we use those relations to give a method of generalizing the result in compact dynamics to non-compact dynamics.