【作者】 于涛；

【导师】 叶向东；

【作者基本信息】 中国科学技术大学 ， 基础数学， 2017， 博士

【摘要】 本文主要研究族意义下的拓扑与测度敏感性,对不交性问题也有所涉及.具体安排如下:在第一章中,我们简要回顾拓扑动力系统和遍历理论的发展历程和主要的研究内容,并介绍本文研究背景以及主要研究成果.在第二章中,我们简单介绍一些拓扑动力系统和遍历理论的基本定义和性质,以及后文将要用到的一些概念和结论.第三章到第六章是本文的主体部分,详细介绍我们的主要研究成果.在第三章中,我们主要研究拓扑族敏感.特别地,我们引入分块thick敏感,分块IP敏感,强thick敏感和强IP敏感的概念.运用极小流结构定理,我们证明极小系统要么是强thick敏感要么是其极大distal因子的proximal扩充.运用遍历理论的方法和极大无穷步幂零因子的性质,给出了以下的结果(1)极小系统要么是分块IP敏感要么是其极大无穷步幂零因子的几乎一对一扩充;(2)极小系统要么是分块thick敏感要么是其极大等度连续因子的proximal扩充;(3)极小系统要么是强IP敏感要么是其极大distal因子的几乎一对一扩充.这些结果将极小系统在族的意义下的敏感性与系统本身的结构一一对应起来,用敏感的语言给出极小系统结构的另一种刻画.在第四章中,我们主要研究测度族敏感.由于拓扑动力系统(X,T)存在不变测度μ,所以(,Bx,μ,T可以被视为一个保测系统,其中Bx是X的Borel σ-代数.我们引入thick-μ敏感,IP-μ敏感,分块thick-μ敏感和分块IP-μ敏感的概念,并且证明了对极小系统:(1)thick-μ敏感与thick敏感是等价的;(2)分块thick-μ敏感与分块thick敏感是等价的;(3)分块IP-μ敏感与分块IP敏感是等价的.在第五章中,我们引入向量敏感的概念,并且着重研究两种特例:l-敏感和δ-l-敏感.我们证明即使对任意正整数l,系统是l-敏感,系统也不一定是多重敏感的;并且构造一个极小系统是l-敏感但不是(l + 1)-敏感.为了区分δ-l-敏感和δ-(l + 1)-敏感,我们构造一个极小系统(弱混合系统)是δ-l-敏感但不是δ-(l + 1)-敏感.在第六章中,我们研究了群作用下与极小系统都不交的系统的性质.证明了当G是交换群时,如果(X,G)是弱混合系统且distal点稠密,那么(X,G)与极小系统都不交,将董攀登,邵松和叶向东[20]的工作推广到交换群作用.又证明了如果(X,Zd)是传递的且与极小系统都不交,那么(X,Zd)是弱混合M-系统且没有非平凡的的极小因子,将黄文和叶向东[58]的工作推广到Zd作用。更多还原

【Abstract】 In this thesis,we mainly study the properties of sensitivity via Furstenberg families in topological dynamical system and measurable dynamical system,the properties of disjointness are also involved.The thesis is organized as follows:In Chapter 1,we briefly recall the development course and main objectives of the topological dynamical system and ergodic theory.We also introduce the research background and main results of our study.In Chapter 2,we briefly introduce some basic definitions and properties of topo-logical dynamical system and ergodic theory.We also discuss some concepts and propositions of the thesis.In Chapter 3,we will discuss topological sensitivity via Furstenberg families.We introduce the notion of block thick sensitivity,block IP sensitivity,strong thick sensi-tivity and strong IP sensitivity.By using structure theorem for minimal systems,we obtain that a minimal system is either strongly thick sensitive or a proximal extension of its maximal distal factor.By using the methods of ergodic theory and the properties of maximal ∞-step nilfactor,we prove the following results:(1)a minimal system is either block IP sensitive or an almost one-to-one extension of its maximal ∞-step nil-factor;(2)a minimal system is either block thick sensitive or a proximal extension of its maximal equicontinous factor;(3)a minimal system is either strongly IP sensitive or an almost one-to-one extension of its maximal distal factor.These results connect sen-sitivity via Furstenberg families with the structure of minimal systems,giving another way to characterize the structure of minimal systems.In Chapter 4,we will investigate measurable sensitivity via Furstenberg families.For a topological dynamical system(X,T),there exists invariant Borel probability mea-sure μ.Thus(X,Bx,μ,T)can be viewed as a measurable dynamical system,where BX is the Borel σ-algebra of X.We introduce the notion of thick sensitivity for μ,IP sen-sitivity for μ,block thick sensitivity for μ and block IP sensitivity for μ.We show that for minimal system:(1)thick sensitivity for μ is equivalent to thick sensitivity;(2)block thick sensitivity for μ is equivalent to block thick sensitivity;(3)block IP sensitivity for μ is equivalent to block IP sensitivity.In Chapter 5,we introduce the notion of vector sensitivity,and mainly investigate two special cases of vector sensitivity:l-sensitivity and δ-l-sensitivity.We show that even the system is l-sensitive for any positive integer l,the system need not to be multi-sensitive.We also construct a minimal system which is l-sensitive but not(l+1)-sensitive.To distinguish between δ-1-sensitivity and δ-(l + 1)-sensitivity,we construct a minimal system(weakly mixing system)which is δ-l-sensitive but not δ-(l + 1)-sensitive.In Chapter 6,we study the properties of systems which are disjoint from all mini-mal systems under group actions.We show that if(X,G)is weakly mixing with dense distal points and with G being abelian,then(X,G)is disjoint from all minimal systems,extending Dong,Shao and Ye’s work[20]to abelian group actions.We also prove that if(X,Zd)is disjoint from all minimal systems and transitive,then(X,Zd)is a weakly mixing M-system without nontrivial minimal factor,extending Huang and Ye’s work[58]to Zd-actions.更多还原