共查询到20条相似文献,搜索用时 31 毫秒
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Haudorff测度与等径不等式 总被引:1,自引:0,他引:1
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. 相似文献
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对于Rn 中满足0 < Hs(K) < ∞ 的任意紧致集K, 我们考虑其在共形映射f 作用下的像集的Hausdorff 测度Hs(f(K)). 本文给出了下面结果:
Hs(f(K)) = Hs(K) · ∫K |Dxf|sdμ(x),
其中概率测度μ = (Hs|K/Hs(K)) . 给定满足开集条件的自相似集K, 测度μ 恰好是自相似测度, 因此可以应用上述公式计算f(K) 的Hausdorff 测度, 例如, K 是λ-Sierpinski 地毯, f(z) = z+εz2, 其中0 < λ ≤1/4,复数ε 满足|ε| ≤ 0.1. 而此刻f(K) 恰好是自共形集, 因此我们的算法能计算一类特殊的具有非线性结构的自共形集的Hausdorff 测度. 相似文献
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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. 相似文献
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关于自相似集的Hausdorff测度的一个判据及其应用 总被引:6,自引:1,他引:5
讨论了满足开集条件的自相似集。对于此类分形,用自然覆盖类估计它的Hausdorff测度只能得到一个上限,因而如何判断某一个上限就是它的Hausdorff测度的准确值是一个重要的问题。本文给出了一个判据。作为应用,统一处理了一类自相似集,得到了平面上的一个Cantor集-Cantor尘的Hausdorff测度的准确值,并重新计算了直线上的Cantor集以及一个Sierpinski地毯的Hausdorff测度。 相似文献
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In this paper,we address the problem of exact computation of the Hausdorff measure of a class of Sierpinski carpets E the self-similar sets generating in a unit regular pentagon on the plane.Under some conditions,we show the natural covering is the best one,and the Hausdorff measures of those sets are euqal to | E | s,where s=dim H E. 相似文献
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Pedro Mendes 《Proceedings of the American Mathematical Society》1999,127(11):3305-3308
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).
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Self-similar sets in complete metric spaces 总被引:3,自引:0,他引:3
Andreas Schief 《Proceedings of the American Mathematical Society》1996,124(2):481-490
We develop a theory for Hausdorff dimension and measure of self-similar sets in complete metric spaces. This theory differs significantly from the well-known one for Euclidean spaces. The open set condition no longer implies equality of Hausdorff and similarity dimension of self-similar sets and that has nonzero Hausdorff measure in this dimension. We investigate the relationship between such properties in the general case.
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设Cλ是由迭代函数系统(IFS){f1,f2}生成的对称Cantor集,其中f1(x)=λx, f2(x)=1-λ+λx,0<λ<1/2,x∈[0,1].在压缩比λ满足一定条件时,本文得到了Cλ与其自身的笛卡尔乘积Cλ×Cλ的Hausdorff中心测度的计算公式. 相似文献
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Zuoling Zhou 《中国科学A辑(英文版)》1998,41(7):723-728
The self-similar sets satisfying the open condition have been studied. An estimation of fractal, by the definition can only
give the upper limit of its Hausdorff measure. So to judge if such an upper limit is its exact value or not is important.
A negative criterion has been given. As a consequence, the Marion’s conjecture on the Hausdorff measure of the Koch curve
has been proved invalid.
Project partially supported by the State Scientific Commission and the State Education Commission. 相似文献
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Wolfgang Kreitmeier 《Journal of Mathematical Analysis and Applications》2008,342(1):571-584
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. 相似文献
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本文研究了均匀2n部分康托集的Hausdorff中心测度.利用极大中心密度与Hausdorff 中心测度之间的关系,确定了均匀2n部分康托集Hausdorff中心测度的精确值. 相似文献
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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. 相似文献
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The necessary and sufficient conditions for various self-similar sets and their dimension 总被引:2,自引:0,他引:2
Dihe Hu 《Stochastic Processes and their Applications》2000,90(2):471-262
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. 相似文献
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Balázs Bárány 《Journal of Mathematical Analysis and Applications》2011,383(1):244-258
The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate the properties of the Hausdorff dimension of self-similar sets such that some functions in the corresponding iterated function system share the same fixed point. Then it is not possible to apply directly known techniques. In this paper we are going to calculate the Hausdorff dimension for almost all contracting parameters and calculate the proper dimensional Hausdorff measure of the attractor. 相似文献
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设E是Hausdorff测度正有限的广义齐次自相似集,本文证明了s维Hausdorff测度是E上唯一的非扩张概率测度. 相似文献
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GEOMETRY AND DIMENSION OF SELF—SIMILAR SET 总被引:1,自引:0,他引:1
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. 相似文献