本文研究拓?fù)鋭恿ο到y(tǒng)的復(fù)雜性理論。對于零熵系統(tǒng),我們研究它們的拓?fù)鋸?fù)雜度、序列熵和熵維數(shù);對于正熵系統(tǒng),我們研究其中的混沌現(xiàn)象;對于無窮熵系統(tǒng),我們研究平均維數(shù)及其相關(guān)的嵌入問題。本文共分五個章節(jié)。第一章是準(zhǔn)備工作,包含了拓?fù)鋭恿ο到y(tǒng)和遍歷論中的一些基本概念和主要結(jié)果,以及在后續(xù)章節(jié)中需要用到的工具和定理。在第二章中,我們研究二維環(huán)面上一類特殊的斜積系統(tǒng),它們都是一維環(huán)面上的某個無理旋轉(zhuǎn)的擴(kuò)充。我們計算了這類系統(tǒng)的拓?fù)鋸?fù)雜度,并利用新的方法證明其極小性,而且給出了這類系統(tǒng)為二階系統(tǒng)的一個等價刻畫。進(jìn)一步,我們構(gòu)造了一個極小distal系統(tǒng),它具有線性的拓?fù)鋸?fù)雜度,但卻不是二步冪零系統(tǒng),從而否定地回答了 Host、Kra和Maass提出的是否每個具有線性拓?fù)鋸?fù)雜度的極小distal系統(tǒng)都是二步冪零系統(tǒng)這一問題。在第三章中,我們研究零熵系統(tǒng)的序列熵和熵維數(shù),建立了動力系統(tǒng)與其誘導(dǎo)系統(tǒng)在這方面的關(guān)系。具體來說,我們證明了對于任意事先給定的正整數(shù)序列,一個拓?fù)鋭恿ο到y(tǒng)沿著該序列的序列熵是零當(dāng)且只當(dāng)它的誘導(dǎo)系統(tǒng)沿著該序列的序列熵也是零。進(jìn)一步地,作為這一結(jié)果的應(yīng)用,我們證明了一個拓?fù)鋭恿ο到y(tǒng)的上... 

【文章來源】：中國科學(xué)技術(shù)大學(xué)安徽省 211工程院校 985工程院校

【文章頁數(shù)】：95 頁

【學(xué)位級別】：博士

【文章目錄】：
摘要
ABSTRACT
緒論
Introduction
1 Preliminaries
    1.1 Topological dynamical systems and measure-preserving systems
        1.1.1 Topological dynamical systems
        1.1.2 A special class: systems of order 2
        1.1.3 Factors of topological dynamical systems
        1.1.4 Equicontinuity
        1.1.5 Invariant measures and measure-preserving systems
        1.1.6 Pointwise good sequences
        1.1.7 Factors of measure-preserving systems
        1.1.8 Conditional expectation and disintegration
    1.2 Sequence entropy
        1.2.1 Topological sequence entropy
        1.2.2 Measure-theoretic sequence entropy
        1.2.3 Relationship between topological and measure-theoretic entrop
        1.2.4 Pinsker σ-algebra and applications
    1.3 Entropy dimension
        1.3.1 Dimension of a sequence of positive integers
        1.3.2 Topological entropy dimension
        1.3.3 Measure-theoretic entropy dimension
    1.4 Mean dimension
    1.5 Li-Yorke chaos
    1.6 Toolbox
        1.6.1 Continued fractions
        1.6.2 Mycielski's theorem and an extension theorem
        1.6.3 Baire spaces
        1.6.4 Linear independence and affine independence
2 Group extensions over irrational rotations on the torus
    2.1 Background
    2.2 Topological complexity
    2.3 Minimality and the maximal equicontinuous factor
    2.4 An example
3 Sequence entropy and entropy dimension
    3.1 Zero topological sequence entropy
    3.2 Topological entropy dimension
    3.3 Zero measure-theoretic sequence entropy
    3.4 Measure-theoretic entropy dimension
4 Mean Li-Yorke chaos along good sequences
    4.1 Characteristic σ-algebras
    4.2 Good sequences for pointwise convergence
    4.3 In positive entropy systems
    4.4 Non-invertible case
5 The embedding problem in dynamical systems
    5.1 Background
    5.2 Rokhlin dimension: an embedding result
    5.3 Takens' embedding theorem
    5.4 The Lindenstrauss-Tsukamoto Conjecture: a remark
Bibliography
Acknowledgements
Publications



本文編號：2959997