Moran型剪切集的维数和分类及高斯随机场的分形性质

发布时间：2021-10-13 04:37

　　剪切集是一类非常重要的分形集。在本文第三章,我们考虑一类剪切集称为Moran型剪切集,记为Ea。它是由间隔序列{ai}和正整数序列{nκ}生成的,构造过程见定义（2.2.1）。它是一类非常广泛的分形集,包含一般Moran结构。如果对每个κ=1,2,…,取nκ=2,则坟即为Cantor集Ca。对于一般的正整数序列{nκ},Ea的结构更一般更复杂,获得的主要结果如下：（i）给出了Ea的h-Hausdorff测度和h-packing测度的估计,这个估计是用序列{ai}和{nκ}表示的,推广并包含了已知结果。特别指出的是,我们构造了与Ea是双Lipschitz等价的两个齐次Moran集,由此得到Ea的更好的测度估计,从而得到用{ai}和{nκ}的子序列表示的Ea的Hausdorfl和packing维数公式,推广了Besicovitch和Taylor [5]的结果。（ii）证明存在连续凸函数h是Ea的Hausdorff量纲函数。进一步,在经典的局部点态维数的定义中用h代替rα,用这种点态维数定义的水平集给出了Ea的重分形分解。（iii）对Moran型剪切集Ea,Eb定义三种等价关系,分别用间隔序... 

【文章来源】：华南理工大学广东省 211工程院校 985工程院校 教育部直属院校

【文章页数】：86 页

【学位级别】：博士

【文章目录】：
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


【参考文献】：
期刊论文
[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]. 华苏,饶辉,文志英,吴军.  Science in China,Ser.A. 2000(08)



本文编号：3433963