动力系统敏感性和不交性中若干问题的研究

发布时间：2018-03-19 19:20

本文选题：敏感　切入点：测度敏感　出处：《中国科学技术大学》2017年博士论文　论文类型：学位论文

【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。
[Abstract]:In the first chapter, we briefly review the development of topological dynamical systems and ergodic theory and the main research contents. In the second chapter, we briefly introduce some basic definitions and properties of topological dynamical system and ergodic theory. Chapter 3 to 6th is the main part of this paper, which introduces our main research achievements in detail. In Chapter 3, we mainly study the sensitivity of topological families. We introduce the concepts of block thick sensitivity, block IP sensitivity, strong thick sensitivity and strong IP sensitivity. We prove that the minimal system is either strongly thick sensitive or a proximal extension of its maximal distal factor. The following results are given: 1) the minimal system is either block IP sensitive or almost one-to-one expansion of its maximal infinite step nilpotent factor) minimal system is either block thick sensitive or proximal extension of its maximal equicontinuous factor. The minimal system is either strongly IP sensitive or an almost one-to-one extension of its maximal distal factor. These results correspond the sensitivity of the minimal system in the sense of a family to the structure of the system itself. Another description of the structure of minimal system is given in sensitive language. In Chapter 4th, we mainly study the family of measure sensitivities. Because of the existence of invariant measure 渭 in topological dynamical system, BX, 渭 T can be regarded as a preserving system. Where BX is the Borel 蟽 -algebra of X. We introduce the concepts thick- 渭 sensitive IP- 渭 sensitivity, block thick- 渭 sensitivity and block IP- 渭 sensitivity. It is proved that the block thick- 渭 sensitivity and the block thick sensitivity are equivalent to that of the minimal system: 1 / 1 thick- 渭 sensitive and the thick sensitivity are equivalent. In Chapter 5th, the concept of vector sensitivity is introduced, which is equivalent to the block IP- 渭 sensitivity and the block IP sensitivity. We also study two special cases: l-sensitivity and 未-l-sensitivity. We prove that even for any positive integer l, the system is l-sensitive, and the system is not necessarily multi-sensitive; In order to distinguish 未 -l-sensitivity from 未 -l-sensitive, we construct a minimal system (weakly mixed system) which is 未 -l-sensitive but not 未 -l1sensitive. In Chapter 6th, We study the properties of systems which do not intersect with minimal systems under group action. It is proved that when G is a commutative group, if XG is a weakly mixed system and the distal points are dense, then the XG) does not intersect with the minimal system and climb Dong. The work of Shao Song and Ye Xiangdong [20] is extended to the interaction of exchange groups. It is also proved that if XG Zd) is transitive and does not intersect with the minimal system, then XG Zd) is a weakly mixed M- system and has no nontrivial minimal factor. The work of Huang Wen and Ye Xiangdong [58] will be extended to the role of Zd.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O189.1;O19

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