分形集在拟对称映射下的变形

项目名称： 分形集在拟对称映射下的变形

项目编号： No.11301092

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 娄曼丽

作者单位： 广东技术师范学院

项目金额： 22万元

中文摘要： 本项目计划研究拟对称映射所导致的欧式空间中分形集的变形，这是一个分形几何和拟对称映射的交叉课题，一方面可以利用拟对称映射的理论刻画分形集的性质，另一方面也可以利用分形几何的手段来研究拟对称映射。 具体研究内容包括紧密联系的三个方面。之一是拟对称映射对维数的改变，包括全面系统地研究直线上的拟对称极小集，以及计算平面上特定分形集的共形维数。之二是在整体上研究特定分形集的拟对称等价类的度量性质，从而得出拟对称映射能保持何种度量性质以及它能改变何种度量性质。之三是利用分形几何研究保持维数不变的拟对称映射的性质。 本项目所涉及的研究是分形几何和拟对称映射相交叉的一个新的方向，是近年来国内外相关方向的一个研究热点。我们的研究更强调融合分形几何与拟对称映射这两门不同学科的思想和方法，它对于理解复杂分形集的结构、进一步丰富和发展拟对称映射的相关理论都具有十分重要的意义。

中文关键词： 分形；拟对称映射；双 Lipschitz嵌入；；

英文摘要： This project is plan to study the distortion of fractals by quasisymmetric mappings on Eulidean spaces. Our study lies on the overlapping region between fractal geometry and the theory of quasisymmetric mapping. On the one hand, we can use the theory of quasisymmetric mapping to describe the property of fractals, on the other hand, we can also use the technique in fractal geometry to study the quasisymmetric mappings. Our study consists of three issues. The first is the distortion of dimension by quasisemmetric mappings, including the study of quasisymmetrically minimal sets on the line and the study of determining the conformal dimension of some specific fractals on the plane. The second is the study of metric properties of fractals in some specific quasisymmetric equivalent classes. We want to conclude from this study that by quasisymmetric mappings what metric properties are preserved and what metric properties are distored. The third is the study of properties of all quasisymmetric mappings which preserve dimension, by making use of theory of fractal geometry. The issues studies in this project have been a subject of interest within fractal geometry and the theory of quasisymmetric mappings in the past a few years. Our study puts emphisis on the fusion of the ideas and the methods of the two subject. It is

英文关键词： fractal；quasi-symmetric mapping；bilipschitz embedding;；；