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.     

The packing measure of the Cartesian product of the middle third Cantor set with itself

For the packing measure of the Cartesian product of the middle third Cantor set with itself, the exact value

     

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).

     

Hausdorff dimension of perturbed Cantor sets without some boundedness condition

A perturbed Cantor set (without the uniform boundedness condition away from zero of contraction ratios) whose upper Cantor dimension and lower Cantor dimension coincide has its Hausdorff dimension of the same value of Cantor dimensions. We will show this using an energy theory instead of Frostman's density lemma which was used for the case of the perturbed Cantor set with the uniform boundedness condition. At the end, we will give a nontrivial example of such a perturbed Cantor set. This revised version was published online in August 2006 with corrections to the Cover Date.     

On multifractal of Cantor dust

We consider quasi-self-similar measures with respect to all real numbers on a Cantor dust. We define a local index function on the real numbers for each quasi-self-similar measure at each point in a Cantor dust, The value of the local index function at the real number zero for all the quasi-self-similar measures at each point is the weak local dimension of the point. We also define transformed measures of a quasi-self-similar measure which are closely related to the local index function. We compute the local dimensions of transformed measures of a quasi-self-similar measure to find the multifractal spectrum of the quasi-self-similar measure, Furthermore we give an essential example for the theorem of local dimension of transformed measure. In fact, our result is an ultimate generalization of that of a self- similar measure on a self-similar Cantor set. Furthermore the results also explain the recent results about weak local dimensions on a Cantor dust.     

Some fractals associated with Cantor expansions

In this paper, we consider a class of fractals generated by the Cantor series expansions. By constructing some homogeneous Moran subsets, we prove that these sets have full dimension.     

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.     

关于三分Cantor集的构造的一个基本性质及其应用

本文提出了三分 Cantor集的构造的一个基本性质 .作为应用 ,给出了计算三分 Cantor集的简明的初等的计算方法 ,另外还得到了一列有趣的对数不等式     

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.     

三分Cantor集自乘积的Hausdorff测度的估计

本文证明了三分Cantor集C自乘积集C×C的Hausdorff测度,满足1≤H~((log_3)~4)(C×C)≤1.502879.     

Dimension functions of Cantor sets

We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.

     

齐次Cantor集的Hausdorff测度

本文完全确定了一类齐次Cantor集的Haudsorff测度.     

Projections of random Cantor sets

Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases.     

Hausdorff dimension of Cantor sets and polynomial hulls

We give examples of Cantor sets in of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.

     

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 combinatorial proof of Marstrand’s theorem for products of regular Cantor sets

In a paper from 1954 Marstrand proved that if K⊂R² has a Hausdorff dimension greater than 1, then its one-dimensional projection has a positive Lebesgue measure for almost all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^1+α, α>0, for which the sum of their Hausdorff dimension is greater than 1.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp Lipschitz constant of bi-Lipschitz automorphism on Cantor set

Suppose C _r = (r C _r ) ∪ (r C _r + 1 − r) is a self-similar set with r ∈ (0, 1/2), and Aut(C _r ) is the set of all bi-Lipschitz automorphisms on C _r . This paper proves that there exists f* ∈ Aut(C _r ) such that