On Some Problems Of Topological Entropy

The present thesis carries a study on several classical definitions of topological en-tropy within topological dynamical system. A further analysis on their relationships andproperties has also been touched upon. All together, there are four chapters:Chapter 1 serves as an introduction. It reviews the origin of topological dynamicalsystem, introduces the historical development and the current situation of topologicalentropy, as well as some literatures.The following chapter o?ers basic knowledge we should bear in mind before ouranalysis, including some concepts and marks.Chapter 3 I analyzes the relationships of several classical types of topological entropy,trying to prove their equivalence under the same conditions. They are open covers entropy,Bowen dimension entropy, Bowen set entropy, and Pesin string set entropy. All thosetypes of entropy can equivalent within a compact system (X,f), there, X is a compacttopological space.Chapter 4 proves some main properties of topological entropy, viz. within a compactsystem (X,f), there, X is a compact topological space, then within X, Bowen dimensionentropy of any subset and any images of set are equal. Bowen dimension entropy of theunion of any countable subsets of X is equal to the supermum of Bowen dimension entropyof these countable subsets, and Bowen dimension entropy of any subset of X with respectto the mapping fm is equal to m-times of Bowen dimension entropy of this subset withrespect to mapping f, there, m is any positive integer. If X is a compact metric space,then Pesin string set entropy has the similar properties .