非交换换面上的Sarnak问题以及若干拓扑模型的研究

发布时间：2018-04-05 19:10

本文选题：Sarnak猜测　切入点：幂零序列　出处：《中国科学技术大学》2016年博士论文

【摘要】：本论文包括两个主题：第一部分是关于非交换环面自同构等系统上的Samak猜测的研究：第二部分涉及遍历系统的拓扑模型以及相关的应用等。本文具体安排如下：在引言中,我们详细介绍了Sarnak猜测和遍历系统的拓扑模型理论的背景和最近的一些进展。与此同时也简要介绍了我们的相关工作。在第一章中,我们介绍论文中所涉及的关于拓扑动力系统和遍历理论、C*代数等的一些基本概念和结果。特别的,我们介绍了Sarnak猜测在C*代数中的等价命题及其证明概要。在第二章中,我们证明了环面、幂零流形和紧交换群上的仿射为零熵的当且仅当空间中点的轨道闭包为幂零系统的逆极限。尤其,如果这些系统是零熵的,那么其上生成的序列为几乎幂零序列,由此给出Sarnak猜测对于这些系统成立的新证明。在第三章中,我们通过研究Voiculescu-Brown熵为零的非交换环面自同构,证明了对于非交换环面自同构Sarnak猜测的C*代数版本是成立的。具体的讲,我们证明了任何由零Voiculescu-Brown熵非交换环面自同构生成的序列为几乎幂零序列,从而根据Green-Tao的结果它与Mobius函数正交,由此证明了对于非交换环面自同构Samak猜测的C*代数版本成立。为了得到上述结果,我们将问题转化为Hilbert空间上关于酉算子的一个一般性结果。从第四章开始我们研究遍历系统的拓扑模型。动力系统中一个经典结果指出：任何遍历系统间的扩充都有拓扑实现。我们在第四章中加强了这个结果,证明了任何遍历系统的扩充都有拓扑弱混合或者有限对一的拓扑实现。在第五章中,我们研究了遍历系统非极小的拓扑模型。我们证明了对于任何遍历系统,它具有两个非极小拓扑弱混合且全支撑的拓扑模型：其中一个的极小点稠密,另外一个有且仅有一个极小点。作为应用,我们构造了诸如强混合但是渐近的Kolmogorov系统等例子回答了拓扑动力系统中的一些问题。在证明中,我们加强了Weiss的一个结果,解释了如何从一个柱高互素的Kakutani-Rokhlin塔得到新的更高的柱高互素的Kakutani-Rokhlin塔。这个引理可以简化Weiss关于Jewett-Krieger定理和他加倍极小模型定理的证明,以及应用到类似的问题中去。
[Abstract]:This thesis includes two topics: the first part is about the Samak conjecture on systems such as noncommutative torus automorphism, and the second part deals with the topological model of ergodic system and its related applications.In the introduction, we introduce the background of topological model theory of Sarnak conjecture and ergodic system and some recent developments.At the same time, we also briefly introduced our related work.In the first chapter, we introduce some basic concepts and results about topological dynamical system and ergodic theory C * algebra.In particular, we introduce the equivalent propositions of Sarnak conjectures in C * algebras and their propositions.In chapter 2, we prove that affine on torus, nilpotent manifold and compact commutative group is zero entropy if and only if the orbital closure of the point in a space is the inverse limit of a nilpotent system.In particular, if these systems are of zero entropy, the sequences generated on them are almost nilpotent sequences, so a new proof of Sarnak's conjecture for these systems is given.In chapter 3, by studying the noncommutative torus automorphism with zero Voiculescu-Brown entropy, we prove that the C * algebraic version of Sarnak conjecture for noncommutative torus automorphism is true.In particular, we prove that any sequence generated by zero Voiculescu-Brown entropy noncommutative torus automorphism is almost nilpotent sequence, which is orthogonal to Mobius function according to the result of Green-Tao.It is proved that the C * algebraic version of Samak conjecture for noncommutative torus automorphism is true.In order to obtain the above results, we transform the problem into a general result on unitary operators on Hilbert spaces.In chapter 4, we study the topological model of ergodic system.A classical result of dynamical systems shows that any extension between ergodic systems has topological implementation.In chapter 4, we strengthen this result and prove that any extension of ergodic systems has topology weakly mixed or finitely one-to-one topological implementations.In chapter 5, we study the nonminimal topological model of ergodic systems.We prove that for any ergodic system, it has two nonminimal topological weakly mixed and fully supported topological models: one has dense minimal points and the other has and has only one minimal point.As applications, examples such as strongly mixed but asymptotically Kolmogorov systems are constructed to answer some problems in topological dynamical systems.In the proof, we strengthen one of the results of Weiss and explain how to obtain a new and higher column high reciprocal Kakutani-Rokhlin tower from a column high reciprocity Kakutani-Rokhlin tower.This Lemma can simplify the proof of Weiss's theorem on Jewett-Krieger 's theorem and his theorem of double minimal model and its application to similar problems.
【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】：2016
【分类号】：O19

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