Topological Entropy Of Proper Map

By using the Caratheodory-structure(C-structure), the topological entropy on the whole topological space introduced for a proper map, is generalized to the cases of arbi-trary subset. The lower and upper capacities topological entropy are the extension of the Patrao entropy and Adler-Konheim-McAndrew entropy. The topological entropy is the extension of Pesin-Pitskel entropy and Bowen dimensional entropy. Some of the prop-erties of the new topological entropy and. the relationships between the new topological entropy and some classical topological entropies are provided. Two examples are given, one of them is that, for f:R→R,f(x)=x2, we give the topological entropy on any subset of R. This paper is organized as follows.In Chapter 1, we introduce the development history and the investigation status of topological entropy.Some general conceptions, knowledge and notations are given in preliminaries in Chapter 2.In Chapter 3, we define a C-structure on topological dynamical system (X, f), where X is topological space. Then by applying the C-structure, we give definitions of topolog-ical entropy, upper (lower) capacity topological entropy of the proper map on any subset of X and some basic properties of the new entropy. Finally, we show some examples.In Chapter 4, we give the relations between the new topological entropy and sev-eral classical topological entropies, i.e. upper(lower) capacity topological entropy is the generalized form of Alder entropy and Patrao entropy. At the same time, we also prove that upper (lower) capacity topological entropy is not larger than the topological entropy which is defined by separated sets of Bowen. Upper(Lower) capacity topological entropy equals infimum of the topological entropy defined by separated sets of Bowen when X is the local compact separable metric space. In our proof, we extend Lebesgue Covering Lemma from the compact metric space to general metric space.