設(shè)（X,d）是緊致度量空間,T:X→為連續(xù)映射,則稱（X,d,T）為拓?fù)鋭?dòng)力系統(tǒng)。動(dòng)力系統(tǒng)主要研究連續(xù)映射軌道漸近性質(zhì),通常利用拓?fù)潇�、拓�(fù)鋲�、混沌和Lyapunov指數(shù)等來(lái)刻畫(huà)這種軌道性質(zhì)。動(dòng)力系統(tǒng)軌道的回復(fù)性是動(dòng)力系統(tǒng)研究的重要課題,它與數(shù)論,分形幾何,微分方程等學(xué)科有著深刻的關(guān)聯(lián)。我們把重點(diǎn)放在動(dòng)力系統(tǒng)中度量丟番圖逼近問(wèn)題相關(guān)回復(fù)性質(zhì)的量化研究,也就是利用拓?fù)潇?拓?fù)鋲?Hausdorff維數(shù)等來(lái)對(duì)動(dòng)力系統(tǒng)中的回復(fù)行為,收縮靶問(wèn)題進(jìn)行量化的研究。本文主要利用動(dòng)力系統(tǒng)中的軌道跟蹤性質(zhì)來(lái)構(gòu)造Moran分形集來(lái)刻畫(huà)動(dòng)力丟番圖集合占系統(tǒng)的比重。在重分形分析的觀點(diǎn)來(lái)看,是研究一類(lèi)具有一定的回復(fù)、收縮行為的度量丟番圖逼近水平集。第一章主要是推廣了 Bosher-nitzan的關(guān)于定量回復(fù)的結(jié)果到半群作用的動(dòng)力系統(tǒng);第二章,研究滿足非一致結(jié)構(gòu)的子位移系統(tǒng)中的回復(fù)行為,給出了關(guān)于回復(fù)的動(dòng)力丟番圖的水平集的Hausdorff維數(shù)的估計(jì);第三章,研究非一致系統(tǒng)的的飽和集并建立了相應(yīng)的條件變分原理;第四章,定義了一類(lèi)新的水平集刻畫(huà)拓?fù)浠旌嫌邢扌偷陌l(fā)散點(diǎn)集中的回復(fù)性;第五章,對(duì)于具有specificat... 

【文章來(lái)源】：南京師范大學(xué)江蘇省 211工程院校

【文章頁(yè)數(shù)】：123 頁(yè)

【學(xué)位級(jí)別】：博士

【文章目錄】：
摘要
Abstract
Preface
    0.1 Metric Diophantine approximation
    0.2 Nondense orbit set
    0.3 Multifractal analysis
Chapter 1 Quantitative recurrence properties for free semigroup actions
    1.1 Preliminaries and main results
    1.2 Proof of Theorem 1.1.4 and 1.1.5
Chapter 2 Quantitative recurrence properties for systems with non-uniformstructure
    2.1 Preliminaries and main results
        2.1.1 Topological pressure
    2.2 Proof of Theorem 2.1.1
        2.2.1 Proof of upper bound
        2.2.2 Proof of lower bound
    2.3 Proof of Theorem 2.1.2
        2.3.1 Proof of upper bound
        2.3.2 Proof of lower bound
    2.4 Applications
Chapter 3 Topological pressure of generic points sets with non-unifromstructure
    3.1 Preliminaries and main results
    3.2 Proof of Theorem 3.1.2
        3.2.1 Choose the sequence {n_j}_(j≥1)
        3.2.2 Construction of fractal set H
        3.2.3 To estimate the lower bound
    3.3 Applications
Chapter 4 Quantitative recurrence properties in the historic set for sym-bolic systems
    4.1 Preliminaries and main results
    4.2 Some important lemmas
    4.3 Proof of Theorem 4.1.1
        4.3.1 Proof of upper bound
        4.3.2 Proof of lower bound
    4.4 Proof of Theorem 4.1.2
        4.4.1 Proof of upper bound
        4.4.2 Proof of lower bound
Chapter 5 On the topological entropy of the set with a special shadowingtime
    5.1 Preliminaries and main results
    5.2 Proof of Theorem 5.1.2
        5.2.1 Upper bound for h_(top)~B(D_f~(xo))
        5.2.2 Lower bound for h_(top)~B(D_f~(xo))
        5.2.3 Construction of the Fractal F
        5.2.4 Construction of a special sequence of measure μ_k
    5.3 Applications
Chapter 6 Non-dense orbits on topological dynamical systems
    6.1 Preliminaries and main results
    6.2 Proof of Theorem 6.1.1
        6.2.1 Construction of the Fractal F
        6.2.2 Construction of a special sequence of measures μ)k
    6.3 Applications
Bibliography
Acknowledgements
Publications and Preprints
Further researches



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