导　　师： 马东魁

学科专业： 070104

授予学位： 硕士

作　　者： ;

机构地区： 华南理工大学

摘　　要： 本文讨论了拓扑动力系统中拓扑熵的几种经典定义,对它们之间的关系和性质进行了研究,分为以下四章: 第一章是绪论,回顾了拓扑动力系统的起源和主要研究内容,介绍了拓扑熵的发展历史、发展现状及其相关的一些文献. 第二章是预备知识,介绍本文相关的一些基本概念、记号及相关的基础知识.在第三章中研究了几种经典拓扑熵之间的关系,证明了在条件一致的情况下这几种经典拓扑熵是等价的.即对紧致系统/(X,f/),这里X是紧致度量空间,则开覆盖熵、Bowen集熵、Bowen维数熵、Pesin串集熵相互等价. 在第四章中证明了拓扑熵的一些重要性质.即对紧致系统/(X,f/),这里X是紧致拓扑空间,则X中任意子集的Bowen维数熵和集合的像的Bowen维数熵相等; X中任意可列个集合的并的Bowen维数熵等于这可列个集合的Bowen维数熵的上确界. X中任意子集关于映射fm的Bowen维数熵等于该集合关于映射f的Bowen维数熵的m倍,这里m是任意正整数.若X是紧致度量空间,则对于Pesin串集熵也有类似的性质. 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 .

关 键 词： 拓扑动力系统 拓扑熵 开覆盖熵 维数熵 集熵 串集熵

分 类 号： [O189.11]

领　　域： [理学] [理学]