统计自相似测度的局部性质

本文讨论了由随机归结构所决定的统计自相似测度的局部性质，证明了统计自相似测度在多重ｆｒａｃｔａｌ分解的意义为平凡的，作为基推论，得到了随机相似集的ｐａｃｋｉｎｇ维数。     

随机收缩与随机集值算子方程解集的稳定性

本文应用随机收缩,建立具有随机定义域的随机集值算子方程随机解集的稳定性定理,其推论给出了随机集值算子随机不动点集的稳定性。     

统计自相似集和测芳的概率特征

证明了统计自相似集和测度是由一族独立同分布的随机压缩算子所构成的随机递归集和它的分布，此处ｆ０是概率空间（Ω，Ｆ，Ｐ）到ｃｏｍ（Ｅ）的随机元，而ｃｏｎ（Ｅ）是完备可分距离空间Ｅ到Ｅ的压缩算子全体。     

统计自相似集的重分形分解

设K是由Graf定义的统计自相似集.本文构造了一支撑在K上的随机测 度p,并研究了结合P的K的重分形分解,得到了分形谱函数f(α)的表达式.     

随机自仿射集的Hausdorff维数估计

定义了一类广泛的随机自仿射集,得到了此类集合的Ｈａｕｓｄｏｒｆｆ维数估计．此前的随机自相似（包括Ｇｒａｆ,Ｍａｕｌｄｉｎ与Ｆａｌｃｏｎｅｒ等定义的随机自相似情形）和Ｆａｌｃｏｎｅｒ定义的（严格）自仿射以及作者定义的μ 统计自仿射情形均成为该文结果的特例．     

统计自相似集和测度的概率特征

证明了统计自相似集和测度是由一族独立同分布的随机压缩算子｛fσ,σ∈D｝所构成的随机递归集和它的分布．此处fσ是概率空间（Ω,F,P）到con（E）的随机元,con（E）是完备可分距离空间E到E的压缩算子全体．     

随机1—集压缩算子的随机不动点指数和随机不动定理

在［１］中我们建立了随拓扑度并得到了系列新的随机不动点定理。本文建立了随机１－集压缩算子的随机不动点指数理论，得到一些新的随机不动点定理，为研究各类随机方程提供一些存在性原理，给出了在随机Ｈａｍｍｅｒｓｔｅｉｎ积分方程的应用。     

多分支自相似集的自仿嵌入

本文研究了一类多分支自相似集的自仿嵌入.利用压缩映射的不动点,在一定条件下,证明了若一个自相似集能自仿嵌入到另一个自相似集中,则它们对应的迭代函数系的压缩比满足某种代数性质.     

1—集区缩型随机算子方程的若干定理

本文提出了几个新概念，研究了１－集压缩型随机算子方程的随机解。同时还推广了若干重要定理。     

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.     

广义统计自相似集的Hausdorff维数估计

作者进一步研究了在文章［１］中构造的广义统计自相似集的分形性质，得到了这类集合的Hausdorff维数和确切Hausdorff测度函数。文中的结果是［4］中结果的延拓。     

广义Cantor集的自相似性

本文考虑一类广义Cantor集Γ_(β,の)={∞∑n=1dnβn:dn∈Dn,n≥1}的自相似性,其中0β1且对任意的n≥1,D_n为整数集Z的非空有限子集;并且给出Γ_(β,の)为齐次生成自相似集的充分必要条件.作为应用,本文考虑一类广义Cantor集交的自相似性,部分推广了Li,Yao和Zhang(2011)关于自相似性的结果.     

I.I.D. STATISTICAL CONTRACTION OPERATORS AND STATISTICALLY SELF－SIMILAR SETS

I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.     

On different forms of self similarity

Fractal geometry is mainly based on the idea of self-similar forms. To be self-similar, a shape must able to be divided into parts that are smaller copies, which are more or less similar to the whole. There are different forms of self similarity in nature and mathematics. In this paper, some of the topological properties of super self similar sets are discussed. It is proved that in a complete metric space with two or more elements, the set of all non super self similar sets are dense in the set of all non-empty compact sub sets. It is also proved that the product of self similar sets are super self similar in product metric spaces and that the super self similarity is preserved under isometry. A characterization of super self similar sets using contracting sub self similarity is also presented. Some relevant counterexamples are provided. The concepts of exact super and sub self similarity are introduced and a necessary and sufficient condition for a set to be exact super self similar in terms of condensation iterated function systems (Condensation IFS’s) is obtained. A method to generate exact sub self similar sets using condensation IFS’s and the denseness of exact super self similar sets are also discussed.     

Dimensions for random self–conformal sets

A set is called regular if its Hausdorff dimension and upper box–counting dimension coincide. In this paper, we prove that the random self–conformal set is regular almost surely. Also we determine the dimensions for a class of random self–conformal sets.     

On the Lipschitz equivalence of self‐affine sets

Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.     

Cantor尘的Hausdorff测度的初等证明

本文从自相似集的几何性质出发 ,用初等的方法得到了 Cantor尘的 Hausdorff测度 .     

Gauges for the self‐similar sets

For a self‐similar set E with the open set condition we completely determine the class of its Hausdorff gauges and the class of its prepacking gauges. Moreover, its Hausdorff measures and its packing premeasures with respect to the corresponding gauges are estimated. Without the open set condition we prove that a doubling gauge function is a packing gauge of E if and only if it is a prepacking gauge of E. Also, we give some extensions and applications of these results. Here a gauge function is called a Hausdorff, a prepacking, and a packing gauge of a set, if with respect to the function the set has positive and finite Hausdorff measure, packing premeasure, and packing measure, respectively. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Growth of Self‐Similar Graphs

Locally finite self‐similar graphs with bounded geometry and without bounded geometry as well as non‐locally finite self‐similar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The length scaling factor ν and the volume scaling factor μ can be defined similarly to the corresponding parameters of continuous self‐similar sets. There are different notions of growth dimensions of graphs. For a rather general class of self‐similar graphs, it is proved that all these dimensions coincide and that they can be calculated in the same way as the Hausdorff dimension of continuous self‐similar fractals: . © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 224–239, 2004     

SPLIT INCLUSION AND METRICALLY NUCLEAR MAP

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