关于齐次Moran集维数的若干问题研究

发布时间：2018-06-21 23:39

  本文选题：Moran + 集　； 参考：《华中师范大学》2017年博士论文

【摘要】：Moran集作为一种典型的分形集,在许多方面都有非常重要的发展和应用,一直备受人们的广泛关注.由于Moran集的复杂性,人们对Moran集的研究,很重要的一部分是集中在齐次Moran集上.分形几何的主要问题之一就是研究分形集的各种维数,这些维数用来度量分形集的不规则性与裂碎程度,反映了分形集合填充空间的能力,因此是描述集合分形特征的一个很重要的参数.本论文一共分为七章,主要研究了关于齐次Moran集维数的一些问题.第一章引言中我们首先简要回顾了分形几何的发展历程及现状,随后介绍了Moran集与齐次Moran集及其维数的主要研究结果和研究现状,介绍了课题研究的背景,最后陈述了本文所做的主要研究成果.在第二章中,我们简单介绍了本文所要涉及到的一些预备知识.我们首先介绍了分形几何中常见的几种维数——Hausdorff维数,盒维数和packing维数的相关概.念与性质,以及它们之间的一些联系.随后介绍了迭代函数系的概念及相关结果,最后介绍了符号空间的概念与性质.第三章里我们回顾了Moran集的产生、发展和研究现状,介绍了一般Moran集与一维齐次Moran集的概念与已有的一些维数结果.特别的,在一维的情形下,我们对一般Moran集的Hausdorff维数达到上界的充分条件提供了一个新的结论,并仅仅利用质量分布原理对已有的一个结论提供了一个新的证明.与原证明相比,新的证明过程更为简洁且基础易读.接下来三章是本文的主要部分.在第四章中我们考虑了一类特殊齐次Moran集——{mk}-Moran集的构造及其Hausdorff维数估计,进一步探讨了达到其Hausdorff维数上界的{mk}-拟齐次Cantor集的构造及性质.在第五章中我们首先利用第四章中的{mk}-拟齐次Cantor集构造性证明了齐次Moran集Hausdorff维数的介值定理.进一步在mk1((?)k≥ 1)的情况下,计算得出{mk}-拟齐次Cantor集的packing维数.接着在此基础上,构造性证明了齐次Moran集packing维数的介值定理.最后推导出齐次Moran集维数达到最小值的充分条件.在第六章中我们将第五章的结果推广到高维情形,证明了d(d≥2)维齐次Moran集Hausdorff维数的介值定理.在后续部分探讨了平面上一类特殊齐次Moran集,即两个一维齐次Moran集的对应阶压缩比ck=ck'4((?)k≥1)时,其卡氏积的packing维数下界.最后一章里,我们将Moran结构与一些经典的分形集结合起来,研究得到了Moran-Sierpinski 地毯及Moran-Sierpinski 海绵的Hausdorff 维数、packing 维数和上盒维数.
[Abstract]:As a typical fractal set, Moran set has a very important development and application in many aspects. Because of the complexity of Moran sets, a very important part of the study of Moran sets is to concentrate on homogeneous Moran sets. One of the main problems of fractal geometry is to study various dimensions of fractal sets, which are used to measure the irregularity and fragmentation of fractal sets and reflect the ability of fractal sets to fill space. Therefore, it is an important parameter to describe fractal features of sets. This paper is divided into seven chapters. We mainly study some problems about the dimension of homogeneous Moran sets. In the first chapter, we briefly review the development and present situation of fractal geometry, then introduce the main research results and research status of Moran set and homogeneous Moran set and their dimensions, and introduce the background of the research. Finally, the main research results of this paper are presented. In the second chapter, we briefly introduce some of the preparatory knowledge involved in this paper. We first introduce some common dimensions in fractal geometry, such as Hausdorff dimension, box dimension and packing dimension. Read and nature, and some connections between them. Then, the concept of iterative function system and its related results are introduced. Finally, the concept and properties of symbol space are introduced. In chapter 3, we review the generation, development and research status of Moran set, and introduce the concepts of general Moran set and one-dimensional homogeneous Moran set and some existing results of dimension. In particular, in one-dimensional case, we give a new conclusion on the sufficient condition that the Hausdorff dimension of the general Moran set reaches the upper bound, and we only use the mass distribution principle to provide a new proof of the existing conclusion. Compared with the original proof, the new proof process is simpler and easier to read. The following three chapters are the main parts of this paper. In chapter 4, we consider the construction of a special homogeneous Moran set-{mk} -Moran set and its Hausdorff dimension estimation, and further discuss the construction and properties of {mk} -quasi homogeneous Cantor set which reaches the upper bound of its Hausdorff dimension. In chapter 5, we first prove the intermediate value theorem of Hausdorff dimension of homogeneous Moran set by using the constructivity of {mk} -quasi homogeneous Cantor set in Chapter 4. Furthermore, in the case of mk1 (?) k 鈮，

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