【作者】 王军；

【导师】 吴敏；

【作者基本信息】 华南理工大学 ， 应用数学， 2011， 博士

【摘要】 本文共四章.第一章为绪论.第二章为预备知识.第三章与第四章为正文.在第三章中,我们在一定条件下,获得了对称Cantor集上的Cantor测度的点态密度的明确公式,并且,我们对公式中的一个关键变量进行了讨论,所得结论推广了已知结果.设Φ0(x)=ax,Φ1(x)=ax+(1-α),这里0<α<1/2.用K(α)表示由φ0与φ1生成的对称Cantor集.令s表示K(α)的Hausdorff维数,μ表示Cantor测度.令(?)*s(μ,x)与(?)*s(μ,x)分别表示点x∈K(a)的s-下密度与s-上密度.用(?)s(μ,x)表示点x∈K(a)的s-密度.对于上述Cantor测度μ,由文献[8,24,25,34]得到下列结论：(1)对每一点(2)存在常数0<d<d≤1,使得对于μ-a.e.x∈K(a)有丰德军,华苏与文志英[12]给出了上述Cantor测度μ的(?)*s(μ,x)与(?)*s(μ,x)的明确公式,这里x∈K(a),a∈(0,1/3],并且证明了公式中的关键变量τ(x)=0对μ-a.e.∈K(a).这是对上述结论(1)与(2)的明确化.本文在的情况下,对每一点x∈K(a)给出了(?)*s(μ,x)与(?)*s(μ,x)的明确公式,并且,对公式中的关键变量τ(x)的值域给出了刻画.在第四章中,我们对一般情形进行了研究.对0<α<1/2及每一点x∈K(a)我们推测出K(α)的s-上密度(?)*s(μ,χ)的明确公式,并且当时.对推测给出了证明.我们所得到的结论推广和丰富了文献[12,22,26]的结果.更多还原

【Abstract】 This dissertation consists of four parts:Chapter One Introduction:Chapter Two Preliminaries; Chapter Three The pointwise densities of Cantor measure (Ⅰ) and Chap-ter Four The pointwise densities of Cantor measure (Ⅱ).In Chapter Three, for a given symmetrical Cantor set. under certain conditions, we obtain the explicit formulae of pointwise densities of the Cantor measure on it. Moreover, we discuss a key quantity in these formulas. Our results generalize the known results.Let K(a) be the symmetrical Cantor set generated byφ0(x)= ax andφ1(x)= ax+(1-a), where 0< a< 1/2. Let s be the Hausdorff dimension of K(a) andμthe Cantor measure.Let (?)*s(μ,x) and (?)*s(μ,x) denote the lower and upper s-densities at x∈K(a) respectively. And let (?)s(μ,x) be s-density at x∈K(a).For the above Cantor measureμ, the following results are found in [8,24,25,34]: (1) For every (2) There exist constants 0< d< d≤1, such thatFeng, Hua and Wen [12] obtained the explicit formulae of (?)*s(μ,x) and (?)*s(μ,x) for every point x∈K(a), where a∈(0,1/3]. Moreover, they proved thatτ(x) = 0 forμ-a.e. x∈K(a), whereτ(x) is a key quantity in these formulae. Their results make the above conclusions (1) and (2) precise.In this thesis, under the hypothesis that we obtain the explicit formulas of the upper and lower s-densities (?)*s(μ,x) and (?)*s(μ,x) for every point x∈K(a). Moreover, we characterize a key quantity (?)(x) in these formulae.In Chapter Four, we investigate a general case for the symmetrical Cantor set K(a). We propose a conjecture about the explicit formulae of s-upper densities (?)*s(μ,x) for every point x∈K(a), where 0< a< 1/2. Moreover, we give the proof under the additional condition thatOur results generalize and enrich the results in [12,22,26].更多还原