剪切集是一類非常重要的分形集。在本文第三章,我們考慮一類剪切集稱為Moran型剪切集,記為Ea。它是由間隔序列{ai}和正整數(shù)序列{nκ}生成的,構(gòu)造過程見定義（2.2.1）。它是一類非常廣泛的分形集,包含一般Moran結(jié)構(gòu)。如果對(duì)每個(gè)κ=1,2,…,取nκ=2,則墳即為Cantor集Ca。對(duì)于一般的正整數(shù)序列{nκ},Ea的結(jié)構(gòu)更一般更復(fù)雜,獲得的主要結(jié)果如下：（i）給出了Ea的h-Hausdorff測(cè)度和h-packing測(cè)度的估計(jì),這個(gè)估計(jì)是用序列{ai}和{nκ}表示的,推廣并包含了已知結(jié)果。特別指出的是,我們構(gòu)造了與Ea是雙Lipschitz等價(jià)的兩個(gè)齊次Moran集,由此得到Ea的更好的測(cè)度估計(jì),從而得到用{ai}和{nκ}的子序列表示的Ea的Hausdorfl和packing維數(shù)公式,推廣了Besicovitch和Taylor [5]的結(jié)果。（ii）證明存在連續(xù)凸函數(shù)h是Ea的Hausdorff量綱函數(shù)。進(jìn)一步,在經(jīng)典的局部點(diǎn)態(tài)維數(shù)的定義中用h代替rα,用這種點(diǎn)態(tài)維數(shù)定義的水平集給出了Ea的重分形分解。（iii）對(duì)Moran型剪切集Ea,Eb定義三種等價(jià)關(guān)系,分別用間隔序... 

【文章來源】：華南理工大學(xué)廣東省 211工程院校 985工程院校 教育部直屬院校

【文章頁數(shù)】：86 頁

【學(xué)位級(jí)別】：博士

【文章目錄】：
摘要
Abstract
Chapter 1 Introduction
Chapter 2 Preliminaries
    2.1 Measures and dimensions
        2.1.1 Hausdorff measure and Hausdorff dimension
        2.1.2 Box dimension
        2.1.3 Packing measure and packing dimension
    2.2 Moran-type sets
        2.2.1 Definition and notations
        2.2.2 Equivalent relationships
    2.3 Packing dimension profiles
        2.3.1 Falconer and Howroyd-type packing dimension profiles
        2.3.2 Howroyd-type box dimension profiles on(R~N,ρ)
        2.3.3 Howroyd-type packing measure and dimension on(R~N,ρ)
        2.3.4 The relationships among the three packing dimension profiles
Chapter 3 Dimensions and equivalences ofMoran-type cut-out set
    3.1 Background
    3.2 M ain results
        3.2.1 Upper and lower bounds for h-Hausdorff and h-packing measures of E_α
        3.2.2 Level set of Moran-type cut-out set
        3.2.3 The equivalences of Moran-type cut-out set
Chapter 4 Packing Dimensions of theImages of Gaussian RandomFields
    4.1 Background and main results
    4.2 Packing dimension of X(E)
        4.2.1 Packing dimension of X(E)in terms of Dim_d~ρE
        4.2.2 Packing dimension of X(E)in terms of P-dim_d~ρE
Chapter 5 Packinggraphs of
    5.1 Background and main results
    5.2 Properties of packing dimension profiles on(R~N+d,τ)
    5.3 Proof of Theorem 5.1.1
        5.3.1 Preliminary lemmas
        5.3.2 Proof of Theorem 5.1.1
    5.4 Packing dimension profiles of the Cartesian product
Bibliography
Publications
Acknowledgements
中文概要
附件


【參考文獻(xiàn)】：
期刊論文
[1]Pointwise dimensions of general Moran measures with open set condition[J]. LI JinJun 1,2 & WU Min 1*,1 Department of Mathematics,South China University of Technology,Guangzhou 510640,China;2 Department of Mathematics,Zhangzhou Normal University,Zhangzhou 363000,China.  Science China（Mathematics）. 2011(04)
[2]Uniform dimension results for Gaussian random fields[J]. WU DongSheng1 & XIAO YiMin2,31 Department of Mathematical Sciences,University of Alabama in Huntsville,Huntsville,AL 35899,USA 2 Department of Statistics and Probability,Michigan State University,East Lansing,MI 48824,USA 3 College of Mathematics and Computer Science,Anhui Normal University,Wuhu 241000,China.  Science in China（Series A:Mathematics）. 2009(07)
[3]The multifractal spectrum of some Moran measures[J]. WU Min School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China.  Science in China,Ser.A. 2005(08)
[4]Moran sets and Moran classes[J]. WEN ZhiyingDepartment of Mathematics, Tsinghua University, Beijing 100084, China.  Chinese Science Bulletin. 2001(22)
[5]On the structures and dimensions of Moran sets[J]. 華蘇,饒輝,文志英,吳軍.  Science in China,Ser.A. 2000(08)



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