高维空间齐次乘积Moran集的维数问题

The project team intends to study the dimensional results of the cartesian products of homogeneous Moran sets. In the doctoral paper, the maximal and minimal values of Hausdorff dimensions for cartesian products of homogeneous Moran sets are obtained, and both the maximal and minimal values can be attained. In this project , we are intend to research the following questions:(1) the continuity of the Hausdorff dimension; (2) the Packing dimension of the cartesian products of homogeneous Moran sets. We will study the the maximal and minimal values of the Packing dimension. Meanwhile, the cartesian products of Cantor set and the cartesian products of partial Cantor set will be constroucted to obtain the maximal and the minimal values. At the same time, the continuity of the Packing dimension will also be studied. The difficult point is that we can not use the method of the one dimension to resolve the question (2), which is the key problem. How to give the minimal value of the Packing dimension ,according to the convergence speed of the contraction ratio, is a difficult problem also. Without any constraint conditions, the high dimensional results of the cartesian products of homogeneous Moran sets are important complements of the results in high dimension. By the study of this project, we can see the fundamental differences between one and the high dimension.

本项目研究高维空间上一类特殊Moran集，齐次乘积Moran集的维数问题。申请人的博士论文中，我们已给出齐次乘积Moran集类Hausdorff维数上、下界，并证明其可达。本项目拟进一步研究：（1）齐次乘积Moran集类Hausdorff维数的连续性；（2）齐次乘积Moran集的Packing维数。包括给出Packing维数的上、下界估计，以及构造合适的(偏)齐次乘积Cantor集达到其上、下界，并对Packing维数的连续性展开研究。其中（2）是本项目拟解决的重点问题，困难在于无法沿用一维的计算方法来计算高维的Packing维数。同时，如何根据压缩比的收敛速度对齐次乘积Moran集做出适当分类，也是得到Packing维数下界的一个关键。无任何限制条件下齐次乘积Moran集维数的研究是对高维空间Moran集现有结果的一个重要补充，也揭示了高维和一维的重大差异。

Moran集突破自相似集的正则性，是自相似集的一类最重要的推广。对一维齐次Moran集的Hausdorff维数，Packing维数的研究已经有了一系列的结果；但高维空间无限制条件的Moran集维数的结果却很少。本项目对高维空间一类齐次乘积Moran集的维数问题展开研究，主要包括Hausdorff维数的连续性的证明以及Packing维数的上、下界的估计。通过对平面上的一类齐次乘积Moran集的压缩比的分情况讨论，给出了Packing维数的上、下界的精确估计。同时，给出了齐次乘积Cantor集的Packing维数的计算公式，并且通过构造具体例子，指出偏齐次乘积Cantor集的Packing维数与其结构密切相关。平面上无任何限制条件的乘积Moran集维数的研究是对现有结果的一个重要补充，一方面揭示了一维和高维中偏齐次乘积Cantor集的Packing维数的不同，另一方面也揭示了一维和高维中Hausdorff维数和Packing维数的不同。