随机分形

本文概括了随机分形的主要结果，综述了随机分形的最新进展和目前的动态，提出了一些末解决的问题，全文共分为三部分：（１）由随机过程和随机场（如Ｌｅｖｙ过程，Ｇａｕｓｓ场，自相似过程等）产生的各种随机分形集（如象集、水平集、Ｋ重点集等）的Ｈａｕｓｄｏｒｆｆ维数、测度和ｐａｃｋｉｎｇ维数、测试；（２）随机Ｃａｎｔｏｒ型集和统计自相似集的维数和测试；（３）分形集（如Ｓｐｉｅｒｐｉｎｓｋｉ ｇａｓｋｅｔ，ｎｅ     

关于满足弱开集条件的自相似测度的局部维数的注记

对于自相似集合,已知开集条件与强开集条件是等价的,我们讨论了强开集条件的基些性质,并给出了自似测度局部维数研究的一个应用。     

Exact packing dimension in random recursive constructions

We explore the exact packing dimension of certain random recursive constructions. In case of polynomial decay at 0 of the distribution function of random variable X, associated with the construction, we prove that it does not exist, and in case of exponential decay it is t|log|logt||, where is the fractal dimension of the limit set and 1/ is the rate of exponential decay.Research supported by the Department of Mathematics and Statistics (Mathematics) at University of Jyväskylä.Mathematics Subject Classification (2000):Primary 28A78, 28A80; Secondary 60D05, 60J80     

广义谢尔宾斯基海绵的Packing测度

讨论了三维欧氏空间上的一类自仿射集──广义谢尔宾斯基(Sierpinski)海绵的填充测度(Packing Measure),对φ(t)=t^θ,φ(t)=t^θ/|logt|及更一般的情况,证明了填充测度P_ψ[K(T,D)]为无穷或有限的条件,     

满Parry测度的有限型子集的Hausdorff维数与测度

All the full Parry measure subsets of a given subshift of finite type determined by an irreducible 0-1 matrix have the same Hausdorrf dimension and Hausdorff measure which coincide with those of the set of finite type.     

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.     

N-d维广义Brownian单的像集一致Packing测度和一致Packing维数

我们获得了Ｎ－ｄ维广义Ｂｒｏｗｎｉａｎ单的像集的一致Ｐａｃｋｉｎｇ测度和一致Ｐａｃｋｉｎｇ维数。     

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.     

关于满足强分离开集条件的自相似集的Hausdorff测度

设E是Rn中由相似压缩S1,S2,…,Sm所确定的满足开集条件的自相似集,其Hausdorff维数为s,其s-维Hausdorff测度记为Hs(E).利用部分估计原理得到了本文的主要结果:若E满足强分离开集条件,则在E中存在一个压缩拷贝串序列{Ui}和紧集U(|U|>0),使得Hs(U)等于|U|s,并且{Ui}按Hausdorff度量收敛到U,进而证明了由U可以构造一个数列,使得该数列正好收敛到Hs(E);另外,引入了自相似集的相似压缩不动点,得到了等式Hs(E∩U)=|U|s 成立的一个必要条件.     

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．     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

This paper dealswith non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s ∈ (0,1). We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set. Then we establish the existence and uniqueness of tempered pullback random attractors. Finally, the finiteness of fractal dimension of the random attractors is proved.     

概率空间（Ω，F，μ）中关于μ的packing维数与经典的实直线上的packing维数之间的关系

本文将概率空刚（Ω，μ）中ｐａｃｋｉｎｇ维数的定义与经典的实直线上的ｐａｃｋｉｎｇ维数的定义相联系，证明了在Ｌｅｂｅｓｇｕｅ情形，对所有的Ａ∈，关于μ的ｐａｃｋｉｎｇ维数Ｄｉｍ＿μ（Ａ）与被Ｔａｙｌｏｒ和Ｔｒｉｃｏｔ所定义的ｐａｃｋｉｎｇ维数Ｄｉｍ（Ａ）是一致的。Ｂｉｌｌｉｎｇｓｌｅｙ的结果与我们的结果相结合，表明在Ｌｅｂｅｓｇｕｅ情形，关于μ的分形与被Ｔａｙｌｏｒ所定义的分形是一致的。     

Dimension Results for Space-anisotropic Gaussian Random Fields

Let X = {X(t) ∈ R~d, t ∈ R~N} be a centered space-anisotropic Gaussian random field whose components satisfy some mild conditions. By introducing a new anisotropic metric in R~d, we obtain the Hausdorff and packing dimension in the new metric for the image of X. Moreover, the Hausdorff dimension in the new metric for the image of X has a uniform version.     

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．     

自相似测度的联合发散点和填充维数

近年来,发散点引起了数学界的广泛关注,对单测度的发散点,前人已经作了很完备的研究,对有限多个自相似测度的联合发散点的集合,仅其豪斯多夫维数被L．Olsen研究并给出了,然而,我们对其填充维数仍一无所知,因此,本文主要研究联合发散点集合的填充维数。     

自相似测度的联合发散点和填充维数(英文)

近年来,发散点引起了数学界的广泛关注.对单测度的发散点,前人已经作了很完备的研究.对有限多个自相似测度的联合发散点的集合,仅其豪斯多夫维数被L.Olsen研究并给出了,然而,我们对其填充维数仍一无所知.因此,本文主要研究联合发散点集合的填充维数.     

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.     

一般随机缺项三角级数表示断片的Bouligand维数

该文对［１］中的Ｂｏｕｌｉｇａｎｄ维数计算公式进行了改进，用对称原理和简化原理，得到了一般随机缺项三角级数所表示断片的Ｂｏｕｌｉｇａｎｄ维数的一些计算公式．     

Pointwise dimensions of general Moran measures with open set condition

Recently Lou and Wu obtained the formulas of pointwise dimensions of some Moran measures on Moran sets in Rd under the strong separation condition.In this paper,we prove that the result is still true under the open set condition.Due to the lack of the strong separation condition,our approach is essentially different from that used by Lou and Wu.We also obtain the formulas of the Hausdorff and packing dimensions of the Moran measures and discuss some interesting examples.     

The OSC does not imply the SOSC for infinite iterated function systems

It is shown that every class of contracting similitudes on satisfying the OSC and such that , where denotes the corresponding fractal, can be extended to an infinite family of contracting similitudes which still satisfies the OSC but the SOSC does not hold.