Topological Space, Continuous Mapping Of Topological Entropy

Topological entropy, introduced by Adler. Konheim and M(?)Andrew for compact dynamical systems in 1965. is a numerical measure that determines the dynamical complexity of a topological dynamical system (a space and a continuous transformation on the space). As such, the computation or estimation (even an appropriate defintion) of a topological entropy for a general dynamical system becomes important and difficult, and therefore is widely considered as a forever research topic in dynamics. In 1971. Bowen generalized the concept of topological entropy to continuous mappings defined on non-compact metric spaces, which however is metric-dependent as pointed out by Walters. Consequently, a significant work is to seek a metric-independent entropy. This thesis addresses such a gap in the literature of dynamics by generalizing the concept of topological entropy to continuous mappings defined on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required).The contents of the thesis are outlined below:In Chapter one, we review the history, current development and some applications, and explore the relations between the existing definitions of topological entropy. In particular, a finding from this literature survey is that topological entropy is an analogous invariant under conjuga(?)y of topological dynamical systems and can be obtained by maximizing the metric entropy over a suitable class of measures defined on a dynamical system, implying that topological entropy and measure-theoretic entropy (Kolmogorov, 1950's) are closely related: topological entropy bounds measure-theoretic entropy (Goodwyn, 1969 and 1971). The purpose and main results are also summarized in this chapter.Chapter two proposes our definition of topological entropy for continuous mappings defined on general topological spaces. The idea of this definition is the consideration of all non-empty invariant compact subsets of a mapping. We then investigate basic properties of this new entropy, explore its relations to the existing definitions and provide an example in an effort to demonstrate the appropriateness of this new definition.In Chapter three, we explore further fundamental properties of the defined new entropy and present the main results, including that topological entropy of a subsystem is less than or equal to that of the original system, and topologically conjugated systems have an identical entropy. It should be emphasized that these two properties for arbitary dynamical systems arc among those most significant proeprties for compact dynamical systems.Chapter four considers the entropies of locally compact metric spaces, which is practically useful for the study of dynamical systems defined on manifolds. In addition, entropies of hyperspace dynamical systems of locally compact underlying spaces are also discussed in this chapter: the entropy of the induced hyperspace system is larger than or equal to that of the original system.Chapter five summarizes the results and innovation of this thesis, and presents our view of perspectives for the future investigation of topological entropy.