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.     

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).

     

m分非均匀Cantor集的Hausdorff测度

证明了m分非均匀Cantor集的E的H ausdorff测度HS(E)=1.     

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．     

Dimensions of weighted Besicovitch sets

We consider weighted Besicovitch sets which are defined in terms of the weighted frequency of digit 1 in the dyadic expansion of real numbers. Explicit formulas for their Hausdorff dimensions are given.     

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.     

一类Sierpinski垫的Hausdorff测度

设S_λ为压缩比为λ(λ≤1/3)的一类Sierpinski垫,s=-log_λ3为S_λ的Hausdorff维数,N为产生S_λ的所有基本三角形的集合.本文使用网测度方法,获得了S_λ的s-维Hausdorff测度的精确值H~s(S_λ)=1,同时证明了H~s(S_λ)可由S_λ关于网N的s-维Hausdorff测度H_N~s(S_λ)确定,获得了S_λ的非平凡的最佳覆盖.     

Estimating the Hausdorff dimensions of univoque sets for self-similar sets

An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.     

Bounds of Hausdorff measure of the Sierpinski gasket

By a new method, we obtain the lower and upper bounds of the Hausdorff measure of the Sierpinski gasket, which can approach the Hausdorff measure of the Sierpinski gasket infinitely.     

GEOMETRY AND DIMENSION OF SELF—SIMILAR SET

The authors show that the self-similar set for a finite family of contractive similitudes (similarities, i.e., |fi(x) - fi(y)| = αi|x - y|, x,y ∈ RN, where 0 < αi < 1) is uniformly perfect except the case that it is a singleton. As a corollary, it is proved that this self-similar set has positive Hausdorff dimension provided that it is not a singleton. And a lower bound of the upper box dimension of the uniformly perfect sets is given. Meanwhile the uniformly perfect set with Hausdorff measure zero in its Hausdorff dimension is given.     

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。     

一类递归集的Hausdorff维数及Bouligand维数

本文给出递归集的Ｈａｕｓｄｏｒｆｆ维数的下界估计，并由此确定了一类递归集的维数，所获结果包含并推广了Ｂｅｄｆｏｒｄ，Ｄｅｋｋｉｎｇ及文志英、钟红柳等人的有关结果。     

平面上一类自相似集的Hausdorff测度与上凸密度

考虑平面单位正方形内生成的一类自相似集的Hausdorff测度的计算问题.在满足强分离条件及维数小于1的条件下,当相似比满足某些条件时,证明了自然覆盖为其实现上凸密度1计算的最好形状,因而自然覆盖即是最好覆盖.而作为它的直接推论得到该类自相似集的Hausdorff测度的精确值为(2s)~/(1/2),其中s为其Hausdorff维数.     

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.     

Hausdorff measures for a class of homogeneous cantor sets

We consider the homogeneous Cantor sets which are generalization of symmetric perfect sets, and give a formula of the exact Hausdorff measures for a class of such sets.     

一类广义齐次Moran集的Hausdorff维数

本文构造了一类具有类似高维Moran结构的集合,给出一些充分条件来计算其Hausdorff维数.     

扩散 Cantor 集的最小Hausdorff 测度

In this paper, we construct a scattered Cantor set having the value 1/2 of log2/log3- dimensional Hausdorff measure. Combining a theorem of Lee and Baek, we can see the value 21 is the minimal Hausdorff measure of the scattered Cantor sets, and our result solves a conjecture of Lee and Baek.     

A dimensional result for random self-similar sets

A very important property of a deterministic self-similar set is that its Hausdorff dimension and upper box-counting dimension coincide. This paper considers the random case. We show that for a random self-similar set, its Hausdorff dimension and upper box-counting dimension are equal

     

由Cantor展式定义的Besicovitch-Eggleston子集

本文研究了由Cantor展式所确定的一类Besicovitch-Eggleston子集.应用Billingsley定理,得到了这类集合的维数.并且表明无穷符号空间和有限符号空间上的Besicovitch-Eggleston子集的性质是有区别的.     

关于不变集的一个注记

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＲｎｂｅｎｄｉｍｅｎｓｉｏｎａｌＥｕｃｌｉｄｅａｎｓｐａｃｅ,ａｎｄ (ｆ1,… ,ｆｍ,Ｒｎ)ａｃｏｎｔｒａｃｔｉｏｎｉｔｅｒａｔｅｄｆｕｎｃｔｉｏｎｓｙｓｔｅｍ .Ｉｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔｔｈｅｒｅｅｘｉｓｔｓａｕｎｉｑｕｅｎｏｎ ｅｍｐｔｙｃｏｍｐａｃｔｓｅｔＥｓｕｃｈｔｈａｔＥ =∪ｍｉ=1 ｆｉ(Ｅ) .ＷｅｃａｌｌｔｈｅｓｅｔＥｉｎｖａｒｉｎｔｓｅｔｆｏｒ (ｆ1,… ,ｆｍ,Ｒｎ) .　Ｌｅｔ (ｆ1,… ,ｆｍ,Ｒｎ)ｂｅａｃｏｎｔｒａｃｔｉｏｎｉｔｅｒａｔｅｄｆｕｎｃｔ…