發(fā)布時(shí)間：2018-06-21 23:39

  本文選題：Moran + 集　； 參考：《華中師范大學(xué)》2017年博士論文

【摘要】：Moran集作為一種典型的分形集,在許多方面都有非常重要的發(fā)展和應(yīng)用,一直備受人們的廣泛關(guān)注.由于Moran集的復(fù)雜性,人們對(duì)Moran集的研究,很重要的一部分是集中在齊次Moran集上.分形幾何的主要問(wèn)題之一就是研究分形集的各種維數(shù),這些維數(shù)用來(lái)度量分形集的不規(guī)則性與裂碎程度,反映了分形集合填充空間的能力,因此是描述集合分形特征的一個(gè)很重要的參數(shù).本論文一共分為七章,主要研究了關(guān)于齊次Moran集維數(shù)的一些問(wèn)題.第一章引言中我們首先簡(jiǎn)要回顧了分形幾何的發(fā)展歷程及現(xiàn)狀,隨后介紹了Moran集與齊次Moran集及其維數(shù)的主要研究結(jié)果和研究現(xiàn)狀,介紹了課題研究的背景,最后陳述了本文所做的主要研究成果.在第二章中,我們簡(jiǎn)單介紹了本文所要涉及到的一些預(yù)備知識(shí).我們首先介紹了分形幾何中常見(jiàn)的幾種維數(shù)——Hausdorff維數(shù),盒維數(shù)和packing維數(shù)的相關(guān)概.念與性質(zhì),以及它們之間的一些聯(lián)系.隨后介紹了迭代函數(shù)系的概念及相關(guān)結(jié)果,最后介紹了符號(hào)空間的概念與性質(zhì).第三章里我們回顧了Moran集的產(chǎn)生、發(fā)展和研究現(xiàn)狀,介紹了一般Moran集與一維齊次Moran集的概念與已有的一些維數(shù)結(jié)果.特別的,在一維的情形下,我們對(duì)一般Moran集的Hausdorff維數(shù)達(dá)到上界的充分條件提供了一個(gè)新的結(jié)論,并僅僅利用質(zhì)量分布原理對(duì)已有的一個(gè)結(jié)論提供了一個(gè)新的證明.與原證明相比,新的證明過(guò)程更為簡(jiǎn)潔且基礎(chǔ)易讀.接下來(lái)三章是本文的主要部分.在第四章中我們考慮了一類特殊齊次Moran集——{mk}-Moran集的構(gòu)造及其Hausdorff維數(shù)估計(jì),進(jìn)一步探討了達(dá)到其Hausdorff維數(shù)上界的{mk}-擬齊次Cantor集的構(gòu)造及性質(zhì).在第五章中我們首先利用第四章中的{mk}-擬齊次Cantor集構(gòu)造性證明了齊次Moran集Hausdorff維數(shù)的介值定理.進(jìn)一步在mk1((?)k≥ 1)的情況下,計(jì)算得出{mk}-擬齊次Cantor集的packing維數(shù).接著在此基礎(chǔ)上,構(gòu)造性證明了齊次Moran集packing維數(shù)的介值定理.最后推導(dǎo)出齊次Moran集維數(shù)達(dá)到最小值的充分條件.在第六章中我們將第五章的結(jié)果推廣到高維情形,證明了d(d≥2)維齊次Moran集Hausdorff維數(shù)的介值定理.在后續(xù)部分探討了平面上一類特殊齊次Moran集,即兩個(gè)一維齊次Moran集的對(duì)應(yīng)階壓縮比ck=ck'4((?)k≥1)時(shí),其卡氏積的packing維數(shù)下界.最后一章里,我們將Moran結(jié)構(gòu)與一些經(jīng)典的分形集結(jié)合起來(lái),研究得到了Moran-Sierpinski 地毯及Moran-Sierpinski 海綿的Hausdorff 維數(shù)、packing 維數(shù)和上盒維數(shù).
[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，

本文編號(hào)：2050541