分形集上的加倍测度,拟对称映射及测度的维数研究

In this project, we plan to study the classification of Cantor-type sets for doubling measures, quasisymmetric mappings for fractal sets and the dimensions of some Moran measures. The issues studied in this project have been a subject of great importance in fractal geometry recently. Our study consists of three issues. The first is the study of the classification of Cantor-type sets for doubling measures. In our work [31], we gave the characterization of the fatness an thinness of some special Cantor-type sets for doubling measures. We will continue to investigate the classification of more general Cantor-type sets for doubling measures. The second is the distortion of dimension by quasisemmetric mappings, including the study of quasisymmetrically minimal sets on the line. The third is the dimensions of measures in Moran construction in metric space, including the study of local dimensions of measures and the correlation dimension of measures in specific fractal sets. The study of this project is essential to the understanding of the geometric structure of complicated fractal sets and the development of the theory of fractal geometry and related fields.

本项目计划研究Cantor型集基于加倍测度的分类，分形集上的拟对称映射及某些Moran测度的维数问题，这是分形几何与几何测度论中相当重要的研究课题，是近年来国内外相关方向的一个研究热点。具体研究内容包括以下三个方面。第一，研究Cantor型集在加倍测度意义下的胖瘦性分类问题。我们在前期工作[31]中对几类特殊的Cantor型集关于加倍测度的胖瘦性进行了刻画，本项目拟继续研究更一般的Cantor型集在加倍测度下的分类。第二， 研究拟对称映射对维数的改变，包括实直线上的拟对称极小集的判定及其几何性质。第三，研究度量空间中Moran结构集上测度的维数，包括研究测度的局部维数以及计算特定分形集上测度的关联维数。本项目所涉及的研究对于理解复杂分形集的结构、进一步丰富和发展分形几何及相关领域中的理论都具有十分重要的意义。

本项目的研究内容主要涉及Cantor型集基于加倍测度的分类，分形集的拟对称packing极小性及Moran集的Assouad型维数问题，涉及到分形几何，几何测度论，拟共形映射等问题，这是分形几何与几何测度论中相当重要的研究课题，也是国内外同行所关注的研究热点。1）研究了一般Cantor型集在加倍测度意义下的胖瘦性分类问题。2）研究了拟对称packing极小集，并证明了两类Moran集是拟对称packing极小集。3）研究了Moran集的Assouad型维数，包括Assouad维数、下Assouad维数、Assouad维谱等。本项目研究的问题相对独立但密切相关，具有理论及应用双重重要意义。