一类Moran测度的Lq-谱

此文考虑了一类Moran分形，在其生成过程中每一阶的压缩比及压缩比的个数可以是不相同的，证明了支撑在此类Moran分形上的Moran测度的L^q-谱的存在性．     

齐次集及其拟Lipschitz等价

本文定义并研究一类齐次分形,该类分形包含所有的(拟)Ahlfors-David正则集和许多非正则的Moran集,这里如果一个分形的Hausdorff维数与packing维数不相等,则称它是非正则的.对于这类齐次分形,本文得出它们的分形维数,并且给出在适当分离条件下两个齐次分形拟Lipschitz等价的充要条件.随后,本文将这些结果应用到非正则的Moran集上.     

Some properties for the intersection of Moran sets with their translates

Motivated by Mandelbrot’s idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, Nekka and Li studied the properties of these translation sets and showed how they can be used for the classification purpose. In this paper, we pursue this study on a class of Moran sets with their rational translates. We also get the fractal structure of intersection I(x, y) of a class of Moran sets with their rational translates, and the formula of the box-counting dimension. We find that the Hausdorff measures of these sets form a discrete spectrum whose non-zero values come only from shifting vector with the expansion in fraction of (x, y). Concretely, when (x, y) has a finite expansion in fraction, a very brief calculation formula of the measure is given.     

一类齐次Moran集的维数

本文研究了一类特殊的齐次Moran集的维数.将齐次均匀Cantor集通过一系列平移,获得了一类特殊的齐次Moran集并得到了它们维数的精确值,推广了齐次均匀Cantor集维数的计算公式.     

Dimensions of subsets of Moran fractals related to frequencies of their codings

The Moran fractal considered in this paper is an extension of the self-similar sets satisfying the open set condition. We consider those subsets of the Moran fractal that are the union of an uncountable number of sets each of which consists of the points with their location codes having prescribed mixed group frequencies. It is proved that the Hausdorff and packing dimensions of each of these subsets coincide and are equal to the supremum of the Hausdorff (or packing) dimensions of the sets in the union. An approach is given to calculate their Hausdorff and packing dimensions. The main advantage of our approach is that we treat these subsets in a unified manner. Another advantage of this approach is that the values of the Hausdorff and packing dimensions do not need to be guessed a priori.     

拟对称packing极小Moran集

本文研究了一维Moran集的拟对称packing极小性的问题.利用质量分布原理的方法,获得了直线上一类packing维数为1的Moran集为拟对称packing极小集的结果,推广了参考文献中关于拟对称packing极小性的已知结果.     

Some dimensional results for a class of special homogeneous Moran sets

We construct a class of special homogeneous Moran sets, called {m_k}-quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of {m_k}_k?1, we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of ho- mogeneous Moran sets to assume the minimum value, which expands earlier works.     

Hausdorff dimension of Moran sets with increasing spacing

For a family of homogeneous Moran sets, where at each level, subintervals are arranged with in-creasing spaces between neighboring subintervals from left to right, we obtained a formula of the Hausdorff dimensions.     

On the connectedness of limit net sets

In this paper we introduce and study net sets and limit net sets. The construction and geometry of net sets can be described with the help of substitutions with net matrices which we also introduce here. Limit net sets are a special type of Moran fractals. We study connectedness properties of net sets and limit net sets.     

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.