【作者】 庞琳娜；

【导师】 曾凡平；

【作者基本信息】 广西大学 ， 基础数学， 2007， 硕士

【摘要】 本文主要研究了华沙圈上连续映射的混合性质及树映射的稠密混沌．在第一章，我们简要介绍拓扑动力系统的历史背景和本文的写作背景．在第二章，我们主要研究华沙圈W上连续映射的混合性质．对于连续映射f∶W→W而言，我们证明了以下结论：（1）f是拓扑传递的当且仅当f是Devaney混沌的；（2）f是拓扑传递的当且仅当f是混合的；（3）f是拓扑传递的则f含有马蹄；（4）f传递蕴含对所有的整数n，f含有n周期的周期点．在第三章，我们主要研究了树T映射的混沌．令f∶T→T是连续映射．证明了下列性质是等价的：（1）f是通用混沌，（2）对某个δ＞0，f是通用δ-混沌，（3）对某个δ＞0，f是稠密δ-混沌，（4）或者存在唯一的传递的非退化的连通闭集，或者存在κ（κ≥2）个有公共端点的传递的非退化的连通闭集；且如果J是非退化的连通集合，则f（J）是非退化的，且存在传递的连通集合I₀和整数n使得fⁿ（J）∩Int（I₀）≠（?）．更多还原

【Abstract】 In this paper we mainly study the mixing properties on maps of warsaw circle and dense chaos of tree maps.In Chapter One, we introduce simply the development about topological dynamical system and the background of this paper.In Chapter Two, we study mainly the mixing properties on maps of Warsaw circle W. For a coutinuous map f: W→W, we prove the following results:（1） f is topological transitive if and only if f is chaotic in the sense Devaney; （2） f is topological transitive if and only if f is mixing; （3） f is topological transitive implies f has a horseshoe; （4） f is topological transitive implies f have periodic points of n period for each integer n.In Chapter Three, we study mainly dense chaos of tree maps. Let T be a tree and f : T→T be a continuous map, we show the following four conditions are equivalent:（1） f is generically chaotic, （2） f is genericallyδ—chaotic for someδ> 0, （3） f is denselyδ- chaotic for someδ> 0, （4） either there exists a unique transitive closed non degenerate connected set or there exist k（k≥2） transitive closed non degenerate connected components having a common endpoint; moreover, if J is a non degenerate connected set then f（J） is non degenerate, and there exist a transitive connected setⅠ_o and an integer n such that fⁿ（J） n Int（Ⅰ₀）≠φ.更多还原