保形图递归集的Hausdorff维数和测度

本文确定了保形图递归集的 Ｈａｕｓｄｏｒｆｆ维数,证明了相应的 Ｈａｕｓｄｏｒｆｆ度是正σ－有限的,并且我们给出了 Ｈａｕｓｄｏｒｆｆ测度为正有限的充分必要条件．     

关于齐次Cantor集的一个注记

设Ｃ是［０,１］上Ｈａｕｓｄｏｒｆｆ测度为正有限的齐次Ｃａｎｔｏｒ集类,本文证明了,这里ｓ是Ｅ的Ｈａｕｓｄｏｒｆｆ维数,Ｈｓ（Ｅ）是Ｅ的ｓ维Ｈａｕｓｄｏｒｆｆ测度,Ｈｓ（Ｅ）的定义见引言,     

一类广义Sierpinski海绵的Hausdorff测度

本文是文［１］的续篇,讨论一类Ｈａｕｓｄｏｒｆｆ测度小于或等于１的广义Ｓｉｅｒｐｉｎｓｋｉ海绵,完全确定了它们的Ｈａｕｓｄｏｒｆｆ测度．     

两条相互独立的非对称Cauchy过程轨道的乘积集的分形性质

设Ｘ１,Ｘ２是Ｒ＾ｄ上两条相互独立的非对称Ｃａｕｃｈｙ过程,我们求出了,两条轨道的乘积集的确切Ｈａｕｓｄｏｒｆｆ测度函数ψ（ｈ）＝ｈ＾２／ｌｏｇ＾ｈ,同时ψ（ｈ）,同时ψ（ｈ）也是图集的乘积集的确切Ｈａｕｓｄｏｒｆｆ测度函数,另外,我们还求出了乘积集的Ｈａｕｓｄｏｒｆｆ维数和Ｐａｃｋｉｎｇ维数均为２,从而证明了乘 集仍然是分形集。     

R~n上分形集的多重维数

本文推广Ｈａｕｓｄｏｒｆｆ测度和维数的概念，引入了被称作为多重维测度和多重维数的概念．文中证明了关于多重维测度的Ｆｒｏｓｔｍａｎ定理，构造了一个例子说明存在一类点集，其Ｈａｕｓｄｏｒｆｆ测度是零或十∞，但其多重维测度是一个正数，并说明了多重维数除第一个分量是正数外，其它分量可以取到任何实数．     

N指标d维广义Brownian Sheet的像集及图集的Hausdorff测度

本文得到了Ｎ指标ｄ维广义Ｂｒｏｗｎｉａｎ Ｓｈｅｅｔ的像集及图集的Ｈａｕｓｄｏｒｆｆ测度的上、下界．     

一类准自相似集的研究

本文引入并研究了准自相似集，利用动力系统技巧讨论了其Ｈａｕｓｄｏｒｆｆ维数的上、下界，得到了一类严格准自相似集的Ｈａｕｓｄｏｒｆｆ维数公式并确定了一类由共形映射族所确定的准自相似集的Ｈａｕｓｄｏｒｆｆ维数．     

Canto型集合C^m（i1,i2,…ik）的Hausdorff维数和Hausdorf…

证明了较一般的Ｃａｎｔｏｒ型集合Ｃ＾ｍ（ｉｄ１，ｉ２，…ｉｋ）的Ｈａｕｓｄｏｒｆｆ维数和它的Ｂｏｕｌｉｑａｎｄ维数，自要上似维数一致，并且给出了估计Ｃ＾ｍ（ｉ１，…ｉｋ）的Ｈａｕｓｄｏｒｆｆ测度上界的方法。     

由表示系统生成的分形的维数

在这篇文章里,由Ｒｎ中点的表示系统所生成的自仿射集中,给出了自仿射集满足Ｍｏｒａｎ开集条件的一个新的判别方法和不满足开集条件的自相似集的Ｈａｕｓｄｏｒｆｆ维数的上界和下界,并根据两个实例估计出其上下界是相等的．     

自相似马氏过程任意紧集逆像集的Hausdorff维数

本文获得乐自相似马氏过程任意紧集过像集的Ｈａｕｓｄｏｒｆｆ维数以及自相似马氏过程水平集的一致 Ｈａｕｓｄｏｒｆｆ维数．     

On the exact Hausdorff measure of a class of self-similar sets satisfying open set condition

In this paper,we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition(OSC).As applications,we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure.     

Haudorff测度与等径不等式

对于:Hausdorff维数为s>0的满足开集条件的自相似集E(?)Rn(n>1),我们引入等径不等式Hs|E(X)≤|X|s,以及使该不等式等号成立而直径大于0的极限集U(?)Rn.这里,Hs|E(·)是限制到集合E上的s维Hausdorff测度,而|X|指集合X在欧氏度量下的直径.当s=n时,n维球是唯一的极限集;当s∈(1,n)时,除去一些反面例子以外,我们对上述等径不等式的极限集的基本性质所知甚少.可以看出,这些不等式与Hs(E)的准确值的计算有密切联系.作为特例,我们将考虑Sierpinski垫片,指出计算这一典型自相似集的In2/In3维Hausdorff测度准确值的困难何在.由此可以大致推想,为什么除去平凡情形以外,至今还没有一个具体的满足开集条件而维数大于1的自相似集的:Hausdorff测度准确值被计算出来.     

Self-Conformal Multifractal Measures

A self-conformal measure is a measure invariant under a set of conformal mappings. In this paper we describe the local structure of self-conformal measures. For such a measure we divide its support into sets of fixed local dimension and give a formula for the Hausdorff and packing dimensions of these sets. Moreover, we compute the generalized dimensions of the self-conformal measure.     

Sum of Cantor sets: Self-similarity and measure

In this note it is shown that the sum of two homogeneous Cantor sets is often a uniformly contracting self-similar set and it is given a sufficient condition for such a set to be of Lebesgue measure zero (in fact, of Hausdorff dimension less than one and positive Hausdorff measure at this dimension).

     

拟相似集的Hausdorff测度的特征

Siegfried GRAF在文献［1］中给出了自相似集上的Hausdorff测度(简称H-测度)的特征.John McLaughlin在文献［2］中引入了拟相似集的概念,K.J.Falconer又在文献［3］中讨论了拟相似集上H-测度和维数的性质.本文研究拟相似集上的H-测度的特征,并得出在一定条件下支撑于其上满足一定条件的测度与H-测度的等价性条件.     

The gonality of smooth curves with plane models

The Hausdorff measure with fractional index is used in order to define a functional on measurable sets of the plane. A fractal set, constructed using the well-known Von Koch set, is involved in the definition. This functional is proved to arise as the limit of a sequence of classical functionals defined on sets of finite perimeter. Thus it is shown that a natural extension of the ordinary functionals of the calculus of variations leads both to fractal sets and to the fractional Hausdorff measure.     

MEASURES AND THEIR DIMENSION SPECTRUMS FOR COOKIE-CUTTER SETS IN R~d

0. IntroductionMany theoretical physicists and mathematicians have studied the Hausdorff dimensions,measures and multifractal decompositions of fractals and obtained a lot of satisfactory results. Particularlyl there hajs given thorough and detailed study for self-similar fractals(of. e.g. [1--7]). mom the dynamical system point of view self-similar sets are regarded asthe attractors of iterated function systems consisting of self-similar cofltraction mappings.However, the researches for measu…     

自相似集的质量分布原理与Hausdorff测度及其应用

本文提出了满足开集条件的自相似集的质量分布原理.作为应用,得到了计算一类满足开集条件的自相似集的Hausdorff测度的准确值的方法,并举例说明了此方法对于计算一类满足开集条件的自相似集的Hausdorff测度的准确值是行之有效的.     

d维平稳高斯过程极集的充分条件及维数

本文研究了d维平稳高斯过程极集的性质,给出了d维平稳高斯过程广义极性的充分条件,并通过一个特殊的Cantor型集的构造将极集的维数与容度巧妙地结合起来,得到了d维平稳高斯过程非极集的Hausdorff维数的下确界.     

The necessary and sufficient conditions for various self-similar sets and their dimension

We have given several necessary and sufficient conditions for statistically self-similar sets and a.s. self-similar sets and have got the Hausdorff dimension and exact Hausdorff measure function of any a.s. self-similar set in this paper. It is useful in the study of probability properties and fractal properties and structure of statistically recursive sets.